# Video Content by Date

### 2022

**Nov, 28**: Theta-finite pro-Hermitian vector bundles from loop groups elements

Speaker:

*Mathieu Dutour*

In the finite-dimensional situation, Lie's third theorem provides a correspondence between Lie groups and Lie algebras. Going from the latter to the former is the more complicated construction, requiring a suitable representation, and taking exponentials of the endomorphisms induced by elements of the group.

As shown by Garland, this construction can be adapted for some Kac-Moody algebras, obtained as (central extensions of) loop algebras. The resulting group is called a loop group. One also obtains a relevant infinite-rank Chevalley lattice, endowed with a metric. Recent work by Bost and Charles provide a natural setting, that of pro-Hermitian vector bundles and theta invariants, in which to study these objects related to loop groups. More precisely, we will see in this talk how to define theta-finite pro-Hermitian vector bundles from elements in a loop group. Similar constructions are expected, in the future, to be useful to study loop Eisenstein series for number fields.

This is joint work with Manish M. Patnaik.

**Nov, 21**: On the Quality of the ABC-Solutions

Speaker:

*Solaleh Bolvardizadeh*

The quality of the triplet $(a,b,c)$, where $\gcd(a,b,c) = 1$, satisfying $a + b = c$ is defined as

$$

q(a,b,c) = \frac{\max\{\log |a|, \log |b|, \log |c|\}}{\log \mathrm{rad}(|abc|)},

$$

where $\mathrm{rad}(|abc|)$ is the product of distinct prime factors of $|abc|$. We call such a triplet an $ABC$-solution. The $ABC$-conjecture states that given $\epsilon > 0$ the number of the $ABC$-solutions $(a,b,c)$ with $q(a,b,c) \geq 1 + \epsilon$ is finite.

In the first part of this talk, under the $ABC$-conjecture, we explore the quality of certain families of the $ABC$-solutions formed by terms in Lucas and associated Lucas sequences. We also introduce, unconditionally, a new family of $ABC$-solutions that has quality $> 1$.

In the remaining of the talk, we prove a conjecture of Erd\"os on the solutions of the Brocard-Ramanujan equation

$$

n! + 1 = m^2

$$

by assuming an explicit version of the $ABC$-conjecture proposed by Baker.

**Nov, 17**: Torsion points and concurrent lines on Del Pezzo surfaces of degree one

Speaker:

*Julie Desjardins*

The blow up of the anticanonical base point on X, a del Pezzo surface of degree 1, gives rise to a rational elliptic surface E with only irreducible fibers. The sections of minimal height of E are in correspondence with the 240 exceptional curves on X.

A natural question arises when studying the configuration of those curves: If a point of X is contained in “many” exceptional curves, is it torsion on its fiber on E?

In 2005, Kuwata proved for del Pezzo surfaces of degree 2 (where there is 56 exceptional curves) that if “many” equals 4 or more, then yes. In a joint paper with Rosa Winter, we prove that for del Pezzo surfaces of degree 1, if “many” equals 9 or more, then yes. Moreover, we find counterexamples where a torsion point lies at the intersection of 7 exceptional curves.

**Nov, 16**: The question of q, a look at the interplay of number theory and ergodic theory in continued fractions

Speaker:

*Joseph Vandehey*

In the theory of continued fractions, the denominator of the truncated fraction (often denoted q) contains a great deal of information important in applications. However, q is a surprisingly complicated object from the point of view of ergodic theory. We will look at a few problems related to q and see how different techniques have overcome these difficulties, including modular properties (Moeckel, Fisher-Schmidt), renewal-type theorems (Sinai-Ulcigrai, Ustinov), and "nonstandard" arrangements of points (Avdeeva-Bykovskii).

**Nov, 3**: Quadratic Twists of Modular L-functions

Speaker:

*Xiannan Li*

The behavior of quadratic twists of modular L-functions at the critical point is related both to coefficients of half integer weight modular forms and data on elliptic curves. Here we describe a proof of an asymptotic for the second moment of this family of L-functions, previously available conditionally on the Generalized Riemann Hypothesis by the work of Soundararajan and Young. Our proof depends on deriving

an optimal large sieve type bound.

This event is part of the PIMS CRG Group on L-Functions in Analytic Number Theory. More details can be found on the webpage here: https://sites.google.com/view/crgl-functions/crg-weekly-seminar

**Oct, 27**: Learning Tasks in the Wasserstein Space

Speaker:

*Caroline Moosmueller*

Detecting differences and building classifiers between distributions, given only finite samples, are important tasks in a number of scientific fields. Optimal transport (OT) has evolved as the most natural concept to measure the distance between distributions and has gained significant importance in machine learning in recent years. There are some drawbacks to OT: computing OT can be slow, and it often fails to exploit reduced complexity in case the family of distributions is generated by simple group actions.

If we make no assumptions on the family of distributions, these drawbacks are difficult to overcome. However, in the case that the measures are generated by push-forwards by elementary transformations, forming a low-dimensional submanifold of the Wasserstein manifold, we can deal with both of these issues on a theoretical and on a computational level. In this talk, we’ll show how to embed the space of distributions into a Hilbert space via linearized optimal transport (LOT), and how linear techniques can be used to classify different families of distributions generated by elementary transformations and perturbations. The proposed framework significantly reduces both the computational effort and the required training data in supervised learning settings. We demonstrate the algorithms in pattern recognition tasks in imaging and provide some medical applications.

This is joint work with Alex Cloninger, Keaton Hamm, Harish Kannan, Varun Khurana, and Jinjie Zhang.

**Oct, 26**: Influence of the endothelial surface layer on the motion of red blood cells

Speaker:

*Ying Zhang*

The endothelial lining of blood vessels presents a large surface area for exchanging materials between blood and tissues. The endothelial surface layer (ESL) plays a critical role in regulating vascular permeability, hindering leukocyte adhesion as well as inhibiting coagulation during inflammation. Changes in the ESL structure are believed to cause vascular hyperpermeability and induce thrombus formation during sepsis. In addition, ESL topography is relevant for the interactions between red blood cells (RBCs) and the vessel wall, including the wall-induced migration of RBCs and formation of a cell-free layer. To investigate the influence of the ESL on the motion of RBCs, we construct two models to represent the ESL using the immersed boundary method in two dimensions. In particular, we use simulations to study how lift force and drag force change over time when a RBC is placed close to the ESL as thethickness, spatial variation, and permeability of the ESL vary. We find that spatial variation has a significant effect on the wall-induced migration of the RBC when the ESL is highly permeable and that the wall-induced migration can be significantly inhibited by the presence of a thick ESL.

**Oct, 25**: Pointwise ergodic theorem along a subsequence of integers

Speaker:

*Sovanlal Mondal*

After Birkhoff’s Pointwise Ergodic Theorem was proved in 1931, there have been many attempts to generalize the theorem along a subsequence of the integers instead of taking the entire se- quence (n). In this talk, we will present the following result of Roger Jones and Máté Wierdl:

If a sequence (an) satisfies an+1/an ≥ +1 + 1/(log n)12−ε , for some ε > 0, then in any aperiodic dynamical system (X, Σ, μ, T), we can always find a function f ∈ L2 such that the Cesàro averages along the se- quence (an) which is defined by An∈[N] f (Tan x) := N1 ∑ f (Tan x) (0.1) n∈[N] fail to converge in a set of positive measure.

**Oct, 24**: On vertex-transitive graphs with a unique hamiltonian circle

Speaker:

*Dave Morris*

We will discuss graphs that have a unique hamiltonian cycle and are vertex-transitive, which means there is an automorphism that takes any vertex to any other vertex. Cycles are the only examples with finitely many vertices, but the situation is more interesting for infinite graphs. (Infinite graphs do not have "hamiltonian cycles," but there are natural analogues.) The case where the graph has only finitely many ends is not difficult, but we do not know whether there are examples with infinitely many ends. This is joint work in progress with Bobby Miraftab.

**Oct, 20**: AI for Science; and the Implication for Mathematics

Speaker:

*Weinan E*

Modern machine learning has had remarkable success in all kinds of AI applications, and is also poised to change fundamentally the way we do research in traditional areas of science and engineering. In this talk, I will give an overview of some of the recent progress made in this exciting new direction and the theoretical and practical issues that I consider most important.

**Oct, 20**: Moments and Periods for GL(3)

Speaker:

*Chung-Hang (Kevin) Kwan*

In the past century, the studies of moments of L-functions have been important in number theory and are well-motivated by a variety of arithmetic applications. This talk will begin with two problems in elementary number theory, followed by a survey of techniques in the past and the present. We will slowly move towards the perspectives of period integrals which will be used to illustrate the interesting structures behind moments. In particular, we shall focus on the “Motohashi phenomena”.

**Oct, 19**: Agent-based models: from bacterial aggregation to wealth hot-spots

Speaker:

*Theodore Kolokolnikov*

Agent-based models are widely used in numerous applications. They have an advantage of being easy to formulate and to implement on a computer. On the other hand, to get any mathematical insight (motivated by, but going beyond computer simulations) often requires looking at the continuum limit where the number of agents becomes large. In this talk I give several examples of agent- based modelling, including bacterial aggregation, spatio-temporal SIR model, and wealth hotspots in society; starting from their derivation to taking their continuum limit, to analysis of the resulting continuum equations.

**Oct, 17**: Sums of Fibonacci numbers close to a power of 2

Speaker:

*Elchin Hasanalizade*

The Fibonacci sequence \(F(n) : (n\geq 0) is the binary recurrence sequence defined by

$$

F(0) = F(1) = 1 \qquad \mbox{and} \\

F(n+2) = F(n+1) + F(n) \qquad \forall n \geq 0.

$$

There is a broad literature on the Diophantine equations involving the Fibonacci numbers. In this talk, we will study the Diophantine inequality

$$

\left\lvert F(n) + F(m) − 2a\right\rvert < 2a/2

$$

in positive integers n,m and a with $n \geq m$. The main tools used are lower bounds for linear forms in logarithms due to Matveev and Dujella-Petho version of the Baker-Davenport reduction method in Diophantine approximation.

**Oct, 14**: Resource-mediated competition between two plant species with different rates of water intake

Speaker:

*Chunyi Gai*

We propose an extension of the well-known Klausmeier model of vegetation to two plant species that consume water at different rates. Rather than competing directly, the plants compete through their intake of water, which is a shared resource between them. In semi-arid regions, the Klausmeier model produces vegetation spot patterns. We are interested in how the competition for water affects the co-existence and stability of patches of different plant species. We consider two plant types: a “thirsty” species and a “frugal” species, that only differ by the amount of water they consume per unit growth, while being identical in other aspects. We find that there is a finite range of precipitation rate for which two species can co-exist. Outside of that range (when the rate is either sufficiently low or high), the frugal species outcompetes the thirsty species. As the precipitation rate is decreased, there is a sequence of stability thresholds such that thirsty plant patches are the first to die off, while the frugal spots remain resilient for longer. The pattern consisting of only frugal spots is the most resilient. The next-most-resilient pattern consists of all-thirsty patches, with the mixed pattern being less resilient than either of the homogeneous patterns. We also examine numerically what happens for very large precipitation rates. We find that for a sufficiently high rate, the frugal plant takes over the entire range, outcompeting the thirsty plant.

**Oct, 13**: Extreme Values of the Riemann Zeta Function and Dirichlet L-functions at the Critical Points of the Zeta Function

Speaker:

*Shashank Chorge*

We compute extreme values of the Riemann Zeta function at the critical points of the zeta function in the critical strip. i.e. the points where $\zeta'(s) = 0$ and $\mathfrak{R}s< 1.$. We show that the values taken by the zeta function at these points are very similar to the extreme values taken without any restrictions. We will show geometric significance of such points.

We also compute extreme values of Dirichlet L-functions at the critical points of the zeta function, to the right of $\mathfrak{R}s=1$. It shows statistical independence of L-functions and zet function in a certain way as these values are very similar to the values taken by L-functions without any restriction.

**Oct, 11**: Siegel-Veech transform

Speaker:

*Sayantan Khan*

In this talk, I will talk about the Siegel-Veech transform, and how it can be used to count the number of cylinders (of bounded length) on a translation surface. This counting result relies amongst other tools, on the ergodicity of the SL(2, R) action on the moduli space of translation surfaces. This talk will not assume prior knowledge of translation surfaces, and most of the techniques used will be techniques coming from homogeneous dynamics.

**Oct, 6**: Multiplicative functions in short intervals

Speaker:

*Paranedu Darbar*

In this talk, we are interested in a general class of multiplicative functions. For a function that belongs to this class, we will relate its “short average” to its “long average”. More precisely, we will compute the variance of such a function over short intervals by using Fourier analysis and by counting rational points on certain binary forms. The discussion is applicable to some interesting multiplicative functions such as

$$

\mu_k(n), \frac{\phi (n)}{n}, \frac{n}{\phi (n)}, \mu^2(n)\frac{\phi(n)}{n},

\sigma_\alpha (n), (-1)^{\#\left\{p: p^k | n \right\}}

$$

and many others and it provides various new results and improvements to the previous result

in the literature. This is a joint work with Mithun Kumar Das.

This event is part of the PIMS CRG Group on *L*-Functions in Analytic Number Theory. More details can be found on the webpage here: https://sites.google.com/view/crgl-functions/crg-weekly-seminar

**Oct, 5**: Actomyosin cables by mechanical self-organization

Speaker:

*Mingfeng Qiu*

Supracellular actomyosin cables often drive morphogenesis in development. The origin of these cables is poorly understood. We show theoretically and computationally that under external loading, cell-cell junctions capable of mechanical feedback could undergo spontaneous symmetry breaking and establish a dominant path through which tension propagates, giving rise to a contractile cable. This type of cables transmit force perturbation over a long range, and can be modulated by the tissue properties and the external loading magnitude. Our theory is general and highlights the potential role of mechanical signals in guiding development.

**Oct, 3**: Gaps in the sequence square root n mod 1

Speaker:

*Keivan Mallahi-Karai*

In this talk I will present some of aspects of the proof of a theorem of Elkies and McMullen (Duke Math Journal, 2004) on the asymptotic distribution of the gap sizes for the finite sequence ( √n (mod 1) : 1 ≤ n ≤ N ) as N goes to infinity. The proof relies, among other things, on tools from homogenous dynamics and relates the problem to one in the geometry of numbers.

**Sep, 29**: Effect of Dependence on the Convergence of Empirical Wasserstein Distance

Speaker:

*Nabarun Deb*

The Wasserstein distance is a powerful tool in modern machine learning to metrize the space of probability distributions in a way that takes into account the geometry of the domain. Therefore, a lot of attention has been devoted in the literature to understanding rates of convergence for Wasserstein distances based on i.i.d data. However, often in machine learning applications, especially in reinforcement learning, object tracking, performative prediction, and other online learning problems, observations are received sequentially, rendering some inherent temporal dependence. Motivated by this observation, we attempt to understand the problem of estimating Wasserstein distances using the natural plug-in estimator based on stationary beta-mixing sequences, a widely used assumption in the study of dependent processes. Our rates of convergence results are applicable under both short and long-range dependence. As expected, under short-range dependence, the rates match those observed in the i.i.d. case. Interestingly, however, even under long-range dependence, we can show that the rates can match those in the i.i.d. case provided the (intrinsic) dimension is large enough. Our analysis establishes a non-trivial trade-off between the degree of dependence and the complexity of certain function classes on the domain. The key technique in our proofs is a blend of the big-block-small-block method coupled with Berbee’s lemma and chaining arguments for suprema of empirical processes.

**Sep, 28**: Primes, postdocs and pretentiousness

Speaker:

*Andrew Granville*

Reflections on the research developments that have contributed to this award, mostly to do with the distribution of primes and multiplicative functions, discussing my research team's contributions, and the possible future for several of these questions.

**Sep, 28**: Rotary Molecular Motors Driven By Transmembrane Ionic Currents

Speaker:

*Charles S Peskin*

There are two rotary motors in biology, ATP synthase and the bacterial flagellar motor. Both are driven by transmembrane ionic currents. We consider an idealized model of such a motor, essentially an electrostatic turbine. The model has a rotor and a stator, which are closely fitting cylinders. Attached to the rotor is a fixed density of negative charge, with helical symmetry. Positive ions move longitudinally by drift and diffusion on the stator. A key assumption is local electroneutrality of the combined charge distribution. With this setup we derive explicit formulae for the transmembrane current and the angular velocity of the rotor in terms of the transmembrane electrochemical potential difference of the positive ions and the mechanical torque on the motor. This relationship between "forces" and "fluxes" turns out to be linear, and given by a symmetric positive definite matrix, as anticipated by non-equilibrium thermodynamics, although we do not make any use of that formalism in deriving the result. The equal off-diagonal terms of this 2x2 matrix describe the electromechanical coupling of the motor. Although macroscopic, the model can be used as a foundation for stochastic simulation via the Einstein relation.

**Sep, 26**: Height gaps for coefficients of D-finite power series

Speaker:

*Khoa D. Nguyen*

A power series $f(x_1,\ldots,x_m)\in \mathbb{C}[[x_1,\ldots,x_m]]$ is said to be D-finite if all the partial derivatives of $f$ span a finite dimensional vector space over the field $\mathbb{C}(x_1,\ldots,x_m)$. For the univariate series $f(x)=\sum a_nx^n$, this is equivalent to the condition that the sequence $(a_n)$ is P-recursive meaning a non-trivial linear recurrence relation of the form:

$$P_d(n)a_{n+d}+\cdots+P_0(n)a_n=0$$ where the $P_i$'s are polynomials. In this talk, we consider D-finite power series with algebraic coefficients and discuss the growth of the Weil height of these coefficients. This is from a joint work with Jason Bell and Umberto Zannier in 2019 and a more recent work in June 2022.

**Sep, 23**: Random plane geometry -- a gentle introduction

Speaker:

*Bálint Virág*

Consider Z^2, and assign a random length of 1 or 2 to each edge based on independent fair coin tosses. The resulting random geometry, first passage percloation, is conjectured to have a scaling limit. Most random plane geometric models (including hidden geometries) should have the same scaling limit. I will explain the basics of the limiting geometry, the "directed landscape", the central object in the class of models named after Kardar, Parisi and Zhang.

**Sep, 22**: Joint value distribution of L-functions

Speaker:

*Junxian Li*

It is believed that distinct primitive L-functions are “statistically independent”. The independence can be interpreted in many different ways. We are interested in the joint value distributions and their applications in moments and extreme values for distinct L-functions. We discuss some large deviation estimates in Selberg and Bombieri-Hejhal’s central limit theorem for values of several L-functions. On the critical line, values of distinct primitive L-functions behave independently in a strong sense. However, away from the critical line, values of distinct Dirichlet L-functions begin to exhibit some correlations.

This is based on joint works with Shota Inoue.

This event is part of the PIMS CRG Group on L-Functions in Analytic Number Theory. More details can be found on the webpage here: https://sites.google.com/view/crgl-functions/crg-weekly-seminar

**Sep, 21**: Agent-based modelling and topological data analysis of zebrafish patterns

Speaker:

*Bjorn Sandstede*

Patterns are widespread in nature and often form during early development due to the self-organization of cells or other independent agents. One example are zebrafish (Danio rerio): wild-type zebrafish have regular black and gold stripes, while mutants and other fish feature spotty and patchy patterns. Qualitatively, these patterns display impressive consistency and redundancy, yet variability inevitably exists on both microscopic and macroscopic scales. I will first discuss an agent-based model that suggests that both consistency and richness of patterning on zebrafish stems from the presence of redundancy in iridophore interactions. In the second part of my talk, I will focus on how we can quantify features and variability of patterns to facilitate predictive analyses. I will discuss an approach based on topological data analysis for quantifying both agent-level features and global pattern attributes on a large scale. The proposed methodology is able to quantify the differential impact of stochasticity in cell interactions on wild-type and mutant patterns and predicts stripe and spot statistics as a function of varying cellular communication. This is joint work with Alexandria Volkening and Melissa McGuirl.

**Sep, 13**: Fubini foiled: pathological foliations from symbolic codings

Speaker:

*Aaron Calderon*

In this talk, I'll present a counterintuitive construction of A. Katok (exposited by Milnor) which at first glance seems to contradict Fubini's theorem. In particular, one can build a full-measure set E in the unit square and a foliation of the square by smooth curves such that any leaf of the foliation meets E in exactly one point. If time permits, I'll also mention work of Ruelle-Wilkinson and Shub-Wilkinson that shows these sorts of pathological examples are common in non-uniformly hyperbolic dynamics.

**Sep, 7**: Combined modeling and experimental study of the interplay between tissue growth and shape regulation during Drosophila wing disc development

Speaker:

*Mark Alber*

The regulation and maintenance of an organ’s shape is a major outstanding question in developmental biology. The Drosophila wing imaginal disc serves as a powerful system for elucidating design principles of the shape formation in epithelial morphogenesis. Yet, even simple epithelial systems such as the wing disc are extremely complex. A tissue’s shape emerges from the integration of many biochemical and biophysical interactions between proteins, subcellular components, and cell-cell and cell-ECM interactions. How cellular mechanical properties affect tissue size and patterning of cell identities on the apical surface of the wing disc pouch has been intensively investigated. However, less effort has focused on studying the mechanisms governing the shape of the wing disc in the cross-section. Both the significance and difficulty of such studies are due in part to the need to consider the composite nature of the material consisting of multiple cell layers and cell-ECM interactions as well as the elongated shape of columnar cells. Results obtained using iterative approach combining multiscale computational modelling and quantitative experimental approach will be used in this talk to discuss direct and indirect roles of subcellular mechanical forces, nuclear positioning, and extracellular matrix in shaping the major axis of the wing pouch during the larval stage in fruit flies, which serves as a prototypical system for investigating epithelial morphogenesis. The research findings demonstrate that subcellular mechanical forces can effectively generate the curved tissue profile, while extracellular matrix is necessary for preserving the bent shape even in the absence of subcellular mechanical forces once the shape is generated. The developed integrated multiscale modelling environment can be readily extended to generate and test hypothesized novel mechanisms of developmental dynamics of other systems, including organoids that consist of several cellular and extracellular matrix layers.

**Sep, 6**: No IET is Mixing

Speaker:

*Gianluca Faraco*

In 1980, Katok proved that no interval exchange transformation (IET) is mixing for any Borel invariant measure. The same holds for any special flow constructed by mean of any IET and a roof function of bounded variation. In this talk, I aim to explain to you the proof of these results.

**Aug, 25**: Statistical Estimation with Differential Privacy

Speaker:

*Gautam Kamath*

Naively implemented, statistical procedures are prone to leaking information about their training data, which can be problematic if the data is sensitive. Differential privacy, a rigorous notion of data privacy, offers a principled framework to dealing with these issues. I will survey recent results in differential private statistical estimation, presenting a few vignettes which highlight novel challenges for even the most fundamental problems, and suggesting solutions to address them. Along the way, I’ll mention connections to tools and techniques in a number of fields, including information theory and robust statistics.

**Aug, 16**: Furstenberg's topological x2 x3 result

Speaker:

*Barak Weiss*

In 1967 Furstenberg proved that any closed subset of the one dimensional torus R/Z, invariant under the two maps x -> 2x mod 1, x -> 3x mod 1, is either finite or the entire torus. I will explain a proof of this result due to Boshernitzan (1994). Furstenberg's proof is slightly longer but perhaps more conceptual. I will explain the main steps in Furstenberg's approach and their connection to joinings.

**Aug, 2**: Multiple mixing for SL(2,R)

Speaker:

*Jon Chaika*

We present a special case of an argument of Mozes that mixing implies mixing of all orders for certain Lie groups.

**Jul, 29**: Moments of the Hurwitz zeta function

Speaker:

*Anurag Sahay*

The Hurwitz zeta function is a shifted integer analogue of the Riemann zeta function, for shift parameters $0<\alpha\leqslant 1$. We consider the integral moments of the Hurwitz zeta function on the critical line $\Re(s)=\frac12$. We focus on rational $\alpha$. In this case, the Hurwitz zeta function decomposes as a linear combination of Dirichlet $L$-functions, which leads us into investigating moments of products of $L$-functions. Using heuristics from random matrix theory, we conjecture an asymptotic of the same form as the moments of the Riemann zeta function. If time permits, we will discuss the case of irrational shift parameters $\alpha$, which will include some joint work with Winston Heap and Trevor Wooley and some ongoing work with Heap.

**Jul, 29**: An extension of Venkatesh's converse theorem to the Selberg class

Speaker:

*Min Lee*

In his thesis, Venkatesh gave a new proof of the classical converse theorem for modular forms of level~$1$ in the context of Langlands' ``Beyond Endoscopy". We extend his approach to arbitrary levels and characters. The method of proof, via the Petersson trace formula, allows us to treat arbitrary degree~$2$ gamma factors of Selberg class type.

This is joint work with Andrew R. Booker and Michael Farmer.

**Jul, 29**: Averages of long Dirichlet polynomials with modular coefficients

Speaker:

*Alessandro Fazzari*

We study the moments of $L$-functions associated with primitive cusp forms, in the weight aspect. In particular, we present recent joint work with Brian Conrey, where we obtain an asymptotic formula for the twisted $r$-th moment of a long Dirichlet polynomial approximation of such $L$-functions. This result, which is conditional on the Generalized Lindel\"of Hypothesis, agrees with the prediction of the recipe by Conrey, Farmer, Keating, Rubinstein and Snaith.

**Jul, 29**: Local statistics for zeros of Artin--Schreier $L$-functions

Speaker:

*Alexei Entin*

We discuss the local statistics of zeros of $L$-functions attached to Artin--Scheier curves over finite fields, that is, curves defined by equations of the form $y^p-y=f(x)$, where $f$ is a rational function with coefficients in $F_q$ ($q$ a power of~$p$).

We consider three families of Artin--Schreier $L$-functions: the ordinary, polynomial (the $p$-rank $0$ stratum) and odd-polynomial families.

We present recent results on the $1$-level zero-density of the first and third families and the $2$-level density of the second family, for test functions with Fourier transform supported in suitable intervals. In each case we obtain agreement with a unitary or symplectic random matrix model.

**Jul, 29**: Moments of $L$-functions in the world of number field counting

Speaker:

*Brandon Alberts*

We discuss some appearances of $L$-function moments in number field counting problems, with a particular focus on counting abelian extensions of number fields with restricted ramification.

**Jul, 29**: The eighth moment of the Riemann zeta function

Speaker:

*Quanli Shen*

I will talk about recent work joint with Nathan Ng and Peng-Jie Wong. We established an asymptotic formula for the eighth moment of the Riemann zeta function, assuming the Riemann hypothesis and a quaternary additive divisor conjecture.

**Jul, 28**: Geodesic restrictions of Maass forms and moments of Hecke $L$-functions

Speaker:

*Peter Humphries*

How large are the $L^2$-restrictions of automorphic forms to closed geodesics? I will discuss how this problem can be shown to be equivalent to proving bounds for certain weighted moments of Hecke $L$-functions, and how the lattice structure of the ring of integers of real quadratic numbers fields can be exploited to obtain essentially optimal upper bounds for these weighted moments.

**Jul, 28**: Twisted first moment of $GL(3)\times GL(2)$ $L$-function

Speaker:

*Jakob Streipel*

We compute a first moment of $GL(3)\times GL(2)$ $L$-functions twisted by a $GL(2)$ Hecke eigenvalue at a prime. We talk about the ideas behind the proof, ways in which it can be generalised or extended, and obstacles for doing so in other directions. We also talk a bit about why such moments are interesting, briefly discussing some applications.

**Jul, 28**: Moments and periods for $GL(3)$

Speaker:

*Chung-Hang Kwan*

The celebrated Motohashi phenomenon concerns the duality between the fourth moment of the Riemann zeta function and the cubic moment of automorphic $L$-functions of $GL(2)$. Apart from its structural elegance, such a duality plays a very important role in various moment problems. In this talk, we will discuss the generalized Motohashi phenomena for the group $GL(3)$ through the lenses of period integrals and the method of unfolding. As a consequence, the Kuznetsov and the Voronoi formulae are not needed in our argument.

**Jul, 28**: Double square moments and bounds for resonance sums for cusp forms

Speaker:

*Praneel Samanta*

Let $f$ and $g$ be holomorphic cusp forms for the modular group $SL_2(\mathbb Z)$ of weight $k_1$ and $k_2$ with

Fourier coefficients $\lambda_f(n)$ and $\lambda_g(n)$, respectively. For real $\alpha\neq0$ and $0<\beta\leq1$, consider a smooth resonance sum $S_X(f,g;\alpha,\beta)$ of $\lambda_f(n)\lambda_g(n)$ against $e(\alpha n^\beta)$ over $X\leq n\leq2X$. Double square moments of $S_X(f,g;\alpha,\beta)$ over both $f$ and $g$ are nontrivially bounded when their weights $k_1$ and $k_2$ tend to infinity together. By allowing both $f$ and $g$ to move, these double moments are indeed square moments associated with automorphic forms for $GL(4)$. These bounds reveal insights into the size and oscillation of the resonance sums and their potential resonance for $GL(4)$ forms when $k_1$ and $k_2$ are large.

**Jul, 28**: Math to Power Industry 2022 Gala

Speaker:

*Kristine Bauer, Dhavide Aruliah*

Math^Industry is an annual workshop organized by the Pacific Institute for the Mathematical Sciences to bring together graduate students, academics and industrial partners to work on real world problems. Practical problems from industry are framed by industry partners and project teams are formed to tackle them. The workshop culminates in a gala event where the results of the work on each project is presented.

The gala event took place on zoom and the project reports were presented in three breakout rooms. In the video below, the recordings from these rooms have been placed one after the other and the start time of each is give in square brackets below.

- Introduction
- Graduation Address by Dhavide Aruliah [11:15]
- Project Presentations
- Room 1 [43:32]
- Aerium Analytics (project description )
- Awesense (project description )
- Perfit (project description )

- Room 2 [1:43:40]
- Novion (project description )
- IOTO International (project description )
- Big River Analytics (project description )

- Room 3 [2:38:23]
- Cedar Academy (project description )
- Natural Resources Canada (project description )
- Environmental Instruments Canada (project description )

- Remarks by M2PI Alumnus Erik Chan [3:42:18]
- Closing remarks [4:00:53]

**Jul, 28**: $L^p$-norm bounds for automorphic forms

Speaker:

*Rizwanur Khan*

A fundamental problem in analysis is understanding the distribution of mass of Laplacian eigenfunctions via bounds for their $L^p$ norms in terms of the size of their Laplacian eigenvalue. Number theorists are interested in the Laplacian eigenfunctions on the modular surface that are additionally joint eigenfunctions of every Hecke operator---namely the Hecke--Maass cusp forms. In this talk, I will describe joint work with Peter Humphries in which we prove new bounds for $L^p$ norms in this situation. This is achieved by using $L$-functions and their reciprocity formulae: certain special identities between two different moments of central values of $L$-functions.

**Jul, 27**: Moments of large families of Dirichlet $L$-functions

Speaker:

*Vorrapan Chandee*

Sixth and higher moments of $L$-functions are important and challenging problems in analytic number theory. In this talk, I will discuss my recent joint works with Xiannan Li, Kaisa Matom\"aki, and Maksym Radziwi\l\l~on an asymptotic formula of the sixth and the eighth moment of Dirichlet $L$-functions averaged over primitive characters mod~$q$ over all moduli $q \leq Q$ (and with a short average over critical line for the eighth moment). Unlike the previous works, we do not need to include an average on the critical line for the sixth moment, and we can obtain the eighth moment result without the Generalized Riemann Hypothesis.

**Jul, 27**: Discrete Moments

Speaker:

*Fatma Cicek*

This talk aims to provide an overview of discrete moment computations, specifically, moments of objects related to the Riemann zeta-function when they are sampled at the nontrivial zeros of the zeta-function. We will discuss methods that have been used to do such calculations and will mention their applications.

**Jul, 27**: Limitations to equidistribution in arithmetic progressions

Speaker:

*Aditi Savalia*

It is well known that the prime numbers are equidistributed in arithmetic progressions. Such a phenomenon is also observed more generally for a class of arithmetic functions. A key result in this context is the Bombieri--Vinogradov theorem which establishes that the primes are equidistributed in arithmetic progressions ``on average" for moduli $q$ in the range $q\leq x^{1/2-\epsilon}$ for any $\epsilon > 0 $. Building on an idea of Maier, Friedlander--Granville showed that such equidistribution results fail if the range of the moduli $q$ is extended to $q\leq x/(\log x)^B$ for any $B>1$. We discuss variants of this result and give some applications. This is joint work with my supervisor Akshaa Vatwani

**Jul, 27**: Quantum variance for automorphic forms

Speaker:

*Bingrong Huang,*

In this talk, I will discuss the quantum variances for families of automorphic forms on modular surfaces. The resulting quadratic forms are compared with the classical variance. The proofs depend on moments of central $L$-values and estimates of the shifted convolution sums/non-split sums. (Based on joint work with Stephen Lester.)

**Jul, 26**: Logging of the zeta-function, but only for a few moments!

Speaker:

*Tim Trudgian*

When we're between friends, we often throw in an $\epsilon$ here or there, and why not? Whether something grows like $(\log T)^{100}$ or just $T^{\epsilon}$ doesn?t often make much difference. I shall outline some current work, with Aleks Simoni\v{c}, on the error term in the fourth-moment of the Riemann zeta-function. We know that the $T^{\epsilon}$ in this problem can be replaced by a power of $\log T$ ? but which power? Tune in to find out.

**Jul, 26**: The third moment of quadratic $L$-Functions

Speaker:

*Ian Whitehead*

I will present a smoothed asymptotic formula for the third moment of Dirichlet $L$-functions associated to real characters. Beyond the main term, which was known, the formula has an unexpected secondary term of size $x^{3/4}$ and an error of size $x^{2/3}$. I will give background on the multiple Dirichlet series techniques that motivated this result. And I will describe the new ideas about local and global multiple Dirichlet series that made the final, sieving step in the proof possible. This is joint work with Adrian Diaconu.

**Jul, 26**: Asymptotic mean square of product of higher derivatives of the zeta-function and Dirichlet polynomials

Speaker:

*Mithun Das*

We discuss the asymptotic behavior of the mean square of higher derivatives of the Riemann zeta function or Hardy's $Z$-function product with a Dirichlet polynomial in a short interval. As an application, we obtain a refinement of some results by Levinson--Montgomery as well as Ki--Lee on zero density estimates of higher derivatives of the Riemann zeta function near the critical line. Also, we obtain a zero distribution result for Matsumoto--Tanigawa's $\eta_k$-function. This is joint work with S. Pujahari.

**Jul, 26**: Lambert series of logarithm and a mean value theorem for $\zeta(\frac{1}{2}-it)\zeta'(\frac{1}{2}+it)$

Speaker:

*Atul Dixit*

**Jul, 25**: Negative moments of the Riemann zeta function

Speaker:

*Alexandra Florea*

**Jul, 25**: The recipe for moments of $L$-functions

Speaker:

*Siegfried Baluyot*

In 2005, Conrey, Farmer, Keating, Rubinstein, and Snaith formulated a `recipe' that leads to detailed conjectures for the asymptotic behavior of moments of various families of $L$-functions. In this talk, we will survey recent progress towards their conjectures and explore connections with different subjects.

**Jul, 25**: The generalised Shanks's conjecture

Speaker:

*Andrew Pearce-Crump*

**Jul, 25**: One-level density of zeros of Dirichlet L-functions over function fields

Speaker:

*Hua Lin*

We compute the one-level density of zeros of order-$\ell$ Dirichlet $L$-functions over function fields $\mathbb{F}_q[t]$ for $\ell=3,4$ in the Kummer setting ($q\equiv1\pmod{\ell}$) and for $\ell=3,4,6$ in the non-Kummer setting ($q\not\equiv1\pmod{\ell}$). In each case, we obtain a main term predicted by Random Matrix Theory (RMT) and a lower order term not predicted by RMT. We also confirm the symmetry type of the family is unitary, supporting the Katz and Sarnak philosophy.

**Jul, 25**: Distributions of sums of the divisor function over function fields

Speaker:

*Matilde Lalín*

**Jul, 25**: Selberg's central limit theorem for quadratic Dirichlet $L$-functions over function fields

Speaker:

*Allysa Lumley*

**Jul, 22**: Floer Homology Applications 3

Speaker:

*Jeff Hicks*

A lecture titled "Floer Homology Applications" by Jeff Hicks, University of Edinburgh. This is the 3rd in a series of 3.

General Description:

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

The goal of the summer school is to provide participants the tools in symplectic geometry and stable homotopy theory required to work on Floer homotopy theory. Students will come away with a basic understanding of some of the key techniques, questions, and challenges in both of these fields. The summer school may be particularly valuable for participants with a solid understanding of one of the two fields who want to learn more about the other and the connections between them.

**Jul, 22**: Spectra and Smash Products 4

Speaker:

*Cary Malkiewich*

A lecture titled "Spectra and Smash Products" by Cary Malkiewich, Binghamton University. This is the 4th in a series of 4.

General Description:

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

The goal of the summer school is to provide participants the tools in symplectic geometry and stable homotopy theory required to work on Floer homotopy theory. Students will come away with a basic understanding of some of the key techniques, questions, and challenges in both of these fields. The summer school may be particularly valuable for participants with a solid understanding of one of the two fields who want to learn more about the other and the connections between them.

**Jul, 21**: The equivalence of the Ekeland-Hofer and equivariant symplectic homology capacities

Speaker:

*Vinicius G. B. Ramos*

The Ekeland-Hofer capacities are some of the earliest symplectic capacities. They were defined without Floer theory and their calculation for ellipsoids and polydisks laid the foundation for the understanding of symplectic embeddings for a long time. More recently, Gutt and Hutchings defined a sequence of capacities using positive S^1 equivariant symplectic homology, which are harder to define, but much easier to compute. In this talk, I will explain how there is an isomorphism from the Hamiltonian Floer homology of a class of Hamiltonians to its H^{1/2}-Morse homology and how this implies that those two sequences of capacities coincide. This is joint work with J. Gutt.

**Jul, 21**: Floer Homology Applications 2

Speaker:

*Jeff Hicks*

A lecture titled "Floer Homology Applications" by Jeff Hicks, University of Edinburgh. This is the 2nd in a series of 3.

General Description:

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

The goal of the summer school is to provide participants the tools in symplectic geometry and stable homotopy theory required to work on Floer homotopy theory. Students will come away with a basic understanding of some of the key techniques, questions, and challenges in both of these fields. The summer school may be particularly valuable for participants with a solid understanding of one of the two fields who want to learn more about the other and the connections between them.

**Jul, 21**: Spectra and Smash Products 3

Speaker:

*Cary Malkiewich*

A lecture titled "Spectra and Smash Products" by Cary Malkiewich, Binghamton University. This is the 3rd in a series of 4.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 20**: Floer Homotopy 4

Speaker:

*Mohammed Abouzaid*

A lecture titled "Floer Homotopy" by Mohammed Abouzaid, Columbia University. This is the 4th in a series of 4.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 20**: Floer Homology Applications 1

Speaker:

*Jeff Hicks*

A lecture titled "Floer Homology Applications" by Jeff Hicks, University of Edinburgh. This is the 1st in a series of 3.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 20**: Spectra and Smash Products 2

Speaker:

*Cary Malkiewich*

A lecture titled "Spectra and Smash Products" by Cary Malkiewich, Binghamton University. This is the 2nd in a series of 4.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 19**: Floer Homotopy 3

Speaker:

*Mohammed Abouzaid*

A lecture titled "Floer Homotopy" by Mohammed Abouzaid, Columbia University. This is the 3rd in a series of 4.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 19**: Floer Homotopy 2

Speaker:

*Mohammed Abouzaid*

A lecture titled "Floer Homotopy" by Mohammed Abouzaid, Columbia University. This is the 2nd in a series of 4.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 18**: A knot Floer stable homotopy type

Speaker:

*Ciprian Manolescu*

Given a grid diagram for a knot or link K in the three-sphere, we construct a spectrum whose homology is the knot Floer homology of K. We conjecture that the homotopy type of the spectrum is an invariant of K. Our construction does not use holomorphic geometry, but rather builds on the combinatorial definition of grid homology. We inductively define models for the moduli spaces of pseudo-holomorphic strips and disk bubbles, and patch them together into a framed flow category. The inductive step relies on the vanishing of an obstruction class that takes values in a complex of positive domains with partitions. (This is joint work with Sucharit Sarkar.)

**Jul, 18**: Floer Homotopy 1

Speaker:

*Mohammed Abouzaid*

A lecture titled "Floer Homotopy" by Mohammed Abouzaid, Columbia University. This is the 1st in a series of 4.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 18**: Spectra and Smash Products 1

Speaker:

*Cary Malkiewich*

A lecture titled "Spectra and Smash Products" by Cary Malkiewich, Binghamton University. This is the 1st in a series of 4.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 15**: Floer Homology Fundamentals 9

Speaker:

*Catherine Cannizzo*

A lecture titled "Floer Homology Fundamentals" by Catherine Cannizzo, SCGP. This is the 9th in a series of 9.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 15**: Floer Homology Fundamentals 8

Speaker:

*Nate Bottman*

A lecture titled "Floer Homology Fundamentals" by Nate Bottman, Max Planck. This is the 8th in a series of 9.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 14**: Symplectomorphisms mirror to birational transformations of P^2

Speaker:

*Abigail Ward*

We construct a non-finite type four-dimensional Weinstein domain M_{univ} and describe a HMS-type correspondence between certain birational transformations of P^2 preserving a standard holomorphic volume form and symplectomorphisms of M_{univ}. The space M_{univ} is universal in the sense it admits every Liouville four-manifold mirror to a log Calabi-Yau surface as a Weinstein subdomain; our construction recovers a mirror correspondence between the automorphism group of any open log Calabi-Yau surface and the group of symplectomorphisms of its mirror by restriction to these subdomains. This is joint work in progress with Ailsa Keating.

**Jul, 14**: Floer Homology Fundamentals 7

Speaker:

*Nate Bottman*

A lecture titled "Floer Homology Fundamentals" by Nate Bottman, Max Planck. This is the 7th in a series of 9.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 14**: String Topology 3

Speaker:

*Katherine Poirier*

A lecture titled "String Topology" by Katherine Poirier, New York City College of Technology. This is the 3rd in a series of 3.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 14**: Floer Homology Fundamentals 6

Speaker:

*Nate Bottman*

A lecture titled "Floer Homology Fundamentals" by Nate Bottman, Max Planck. This is the 6th in a series of 9.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 13**: String Topology 2

Speaker:

*Katherine Poirier*

A lecture titled "String Topology" by Katherine Poirier, New York City College of Technology. This is the 2nd in a series of 3.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 13**: Floer Homology Fundamentals 5

Speaker:

*Catherine Cannizzo*

A lecture titled "Floer Homology Fundamentals" by Catherine Cannizzo, SCGP. This is the 5th in a series of 9.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 12**: Floer Homology Fundamentals 4

Speaker:

*Nate Bottman*

A lecture titled "Floer Homology Fundamentals" by Nate Bottman, Max Planck. This is the 4th in a series of 9.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 12**: String Topology 1

Speaker:

*Katherine Poirier*

A lecture titled "String Topology" by Katherine Poirier, New York City College of Technology. This is the 1st in a series of 3.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 12**: Floer Homology Fundamentals 3

Speaker:

*Catherine Cannizzo*

A lecture titled "Floer Homology Fundamentals" by Catherine Cannizzo, SCGP. This is the 3rd in a series of 9.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 11**: Floer Homology Fundamentals 2

Speaker:

*Nate Bottman*

A lecture titled "Floer Homology Fundamentals" by Nate Bottman, Max Planck. This is the 2nd in a series of 9.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 11**: Floer Homology Fundamentals 1

Speaker:

*Catherine Cannizzo*

A lecture titled "Floer Homology Fundamentals"by Catherine Cannizzo, SCGP. This is the 1st in a series of 9.

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

**Jul, 1**: Recent advances in dynamical optimal transport: Lecture 3

Speaker:

*Jan Maas*

In this lecture series we present an overview of dynamical optimal transport and some of its applications to discrete probability and non-commutative analysis. Particular focus is on gradient structures and functional inequalities for dissipative quantum systems, and on homogenisation results for dynamical optimal transport.

**Jun, 30**: Recent advances in dynamical optimal transport: Lecture 2

Speaker:

*Jan Maas*

In this lecture series we present an overview of dynamical optimal transport and some of its applications to discrete probability and non-commutative analysis. Particular focus is on gradient structures and functional inequalities for dissipative quantum systems, and on homogenisation results for dynamical optimal transport.

**Jun, 30**: Optimal Transport for Machine Learning: Lecture 3

Speaker:

*Gabriel Peyré*

Optimal transport (OT) has recently gained lot of interest in machine learning. It is a natural tool to compare in a geometrically faithful way probability distributions. It finds applications in both supervised learning (using geometric loss functions) and unsupervised learning (to perform generative model fitting). OT is however plagued by the curse of dimensionality, since it might require a number of samples which grows exponentially with the dimension. In this course, I will explain how to leverage entropic regularization methods to define computationally efficient loss functions, approximating OT with a better sample complexity. More information and references can be found on the website of our book “Computational Optimal Transport”.

**Jun, 29**: Gross substitutes, optimal transport and matching models: Lecture 3

Speaker:

*Alfred Galichon*

Gross substitutes is a fundamental property in mathematics, economics and computation, almost as important as convexity. It is at the heart of optimal transport theory – although this is often underrecognized – and understanding the connection key to understanding the extension of optimal transport to other models of matching.

**Jun, 29**: Recent advances in dynamical optimal transport: Lecture 1

Speaker:

*Jan Maas*

In this lecture series we present an overview of dynamical optimal transport and some of its applications to discrete probability and non-commutative analysis. Particular focus is on gradient structures and functional inequalities for dissipative quantum systems, and on homogenisation results for dynamical optimal transport.

**Jun, 28**: Gross substitutes, optimal transport and matching models: Lecture 2

Speaker:

*Alfred Galichon*

Gross substitutes is a fundamental property in mathematics, economics and computation, almost as important as convexity. It is at the heart of optimal transport theory – although this is often underrecognized – and understanding the connection key to understanding the extension of optimal transport to other models of matching.

**Jun, 28**: Optimal Transport for Machine Learning: Lecture 2

Speaker:

*Gabriel Peyré*

Optimal transport (OT) has recently gained lot of interest in machine learning. It is a natural tool to compare in a geometrically faithful way probability distributions. It finds applications in both supervised learning (using geometric loss functions) and unsupervised learning (to perform generative model fitting). OT is however plagued by the curse of dimensionality, since it might require a number of samples which grows exponentially with the dimension. In this course, I will explain how to leverage entropic regularization methods to define computationally efficient loss functions, approximating OT with a better sample complexity. More information and references can be found on the website of our book “Computational Optimal Transport”.

**Jun, 27**: Optimal Transport for Machine Learning: Lecture 1

Speaker:

*Gabriel Peyré*

Optimal transport (OT) has recently gained lot of interest in machine learning. It is a natural tool to compare in a geometrically faithful way probability distributions. It finds applications in both supervised learning (using geometric loss functions) and unsupervised learning (to perform generative model fitting). OT is however plagued by the curse of dimensionality, since it might require a number of samples which grows exponentially with the dimension. In this course, I will explain how to leverage entropic regularization methods to define computationally efficient loss functions, approximating OT with a better sample complexity. More information and references can be found on the website of our book “Computational Optimal Transport”.

**Jun, 27**: Gross substitutes, optimal transport and matching models: Lecture 1

Speaker:

*Alfred Galichon*

Gross substitutes is a fundamental property in mathematics, economics and computation, almost as important as convexity. It is at the heart of optimal transport theory – although this is often underrecognized – and understanding the connection key to understanding the extension of optimal transport to other models of matching.

**Jun, 24**: Recent results in interface motions in the framework of optimal transport: Lecture 3

Speaker:

*Inwon Kim*

In the lectures we will discuss recent results obtained on interface motions in the framework of optimal transport. We intend to (time allowing) discuss the following problems:

The Hele-Shaw type flows in the context of tumor growth. Here the flow describe the growth of tumor cells with contact inhibition. The tumor cells then form a congested zone, which evolves by the pressure generated by the constraint on maximal density. We start with a simple mechanical model, and discuss the effects of nutrients and surface tension in the context of minimizing movements. While the well-posedness would be established by minimizing movements, we will also explore qualitative properties of solutions such as regularity of the interface.

The Stefan problem, in the framework of optimal stopping time. Our focus will be on the well-posedness of the supercooled Stefan problem, which describes freezing of supercooled fluid. The interface between the fluid and ice, as it freezes, exhibits a high degree of irregularity. Our goal is to introduce a notion of solutions that are physically meaningful and stable. We will start with a quick introduction of the necessary background on the optimal stopping time between probability measures. We will establish the well-posedness, and discuss qualitative behavior of solutions.

**Jun, 24**: A variational approach to the regularity theory for optimal transportation: Lecture 3

Speaker:

*Felix Otto*

In this mini-course, we shall explain the variational approach to regularity

theory for optimal transportation introduced in [8]. This approach does

completely bypass the celebrated regularity theory of Caffarelli [2], which is

based on the regularity theory for the Monge-Amp ere equation as a fully

nonlinear elliptic equation with a comparison principle. Nonetheless, one

recovers the same partial regularity theory [5, 4].

The advantage of the variational approach resides in its robustness regarding

the regularity of the measures, which can be arbitrary measures [7][Theorem

1.4], and in terms of the problem formulation, e.g. by its extension to almost

minimizers [10]. The former for instance is crucial in order to tackle the

widely popular matching problem [3, 1] e.g. the optimal transportation between

(random) point clouds, as carried out in [7, 6, 9]. The latter is convenient

when treating more general than square Euclidean cost functions.

The variational approach follows de Giorgi’s philosophy for minimal surfaces.

At its core is the approximation of the displacement by the gradient of a

harmonic function. This approximation is based on the Eulerian formulation of

optimal transportation, which reveals its strict convexity and the proximity to

the $H^{-1}$-norm. In this mini-course, we shall give a pretty self-contained

derivation of this harmonic approximation result, and establish applications to

the matching problem.

#### References

- [1] L. Ambrosio, F. Stra, D. Trevisan: A PDE approach to a 2-dimensional

matching problem. Probab. Theory Relat. Fields 173, 433–477 (2019). - [2] L.A. Caffarelli: The regularity of mappings with a convex potential.

Journal of the American Mathematical Society 5 (1992), no. 1, 99–104. - [3] S. Caracciolo, C. Lucibello, G. Parisi, G. Sicuro: Scaling hypothesis for

the Euclidean bipartite matching problem. Physical Review E, 90(1), 2014. - [4] G. De Philippis, A. Figalli: Partial regularity for optimal transport

maps. Publications Mathématiques. Institut de Hautes Études Scientifiques

121 (2015), 81–112. - [5] A. Figalli, Y.-H. Kim: Partial regularity of Brenier solutions of the

Monge-Amépre equation. Discrete and Continuous Dynamical Systems (Series A)

28 (2010), 559–565. - [6] M. Goldman, M. Huesmann: A fluctuation result for the displacement in the

optimal matching problem. arXiv e-prints, May 2021. arXiv:2105.02915. - [7] M. Goldman, M. Huesmann, F. Otto: Quantitative linearization results for

the Monge-Amp`ere equation. Communications on Pure and Applied Mathematics

(2021). - [8] M. Goldman, F. Otto: A variational proof of partial regularity for optimal

transportation maps. Annales Scientifiques de l’Ećole Normale Supérieure.

Quatriéme Série 53 (2020), no. 5, 1209–1233. - [9] M. Huesmann, F. Mattesini, F. Otto: There is no stationary cyclically

monotone Poisson matching in 2d. arXiv e-prints, September 2021.

arXiv:2109.13590. - [10] F. Otto, M. Prod’homme, T. Ried: Variational approach to regularity of

optimal transport maps: general cost functions. (English summary) Ann. PDE 7

(2021), no. 2, Paper No. 17, 74 pp.

**Jun, 23**: A variational approach to the regularity theory for optimal transportation: Lecture 2

Speaker:

*Felix Otto*

In this mini-course, we shall explain the variational approach to regularity

theory for optimal transportation introduced in [8]. This approach does

completely bypass the celebrated regularity theory of Caffarelli [2], which is

based on the regularity theory for the Monge-Amp ere equation as a fully

nonlinear elliptic equation with a comparison principle. Nonetheless, one

recovers the same partial regularity theory [5, 4].

The advantage of the variational approach resides in its robustness regarding

the regularity of the measures, which can be arbitrary measures [7][Theorem

1.4], and in terms of the problem formulation, e.g. by its extension to almost

minimizers [10]. The former for instance is crucial in order to tackle the

widely popular matching problem [3, 1] e.g. the optimal transportation between

(random) point clouds, as carried out in [7, 6, 9]. The latter is convenient

when treating more general than square Euclidean cost functions.

The variational approach follows de Giorgi’s philosophy for minimal surfaces.

At its core is the approximation of the displacement by the gradient of a

harmonic function. This approximation is based on the Eulerian formulation of

optimal transportation, which reveals its strict convexity and the proximity to

the $H^{-1}$-norm. In this mini-course, we shall give a pretty self-contained

derivation of this harmonic approximation result, and establish applications to

the matching problem.

#### References

- [1] L. Ambrosio, F. Stra, D. Trevisan: A PDE approach to a 2-dimensional

matching problem. Probab. Theory Relat. Fields 173, 433–477 (2019). - [2] L.A. Caffarelli: The regularity of mappings with a convex potential.

Journal of the American Mathematical Society 5 (1992), no. 1, 99–104. - [3] S. Caracciolo, C. Lucibello, G. Parisi, G. Sicuro: Scaling hypothesis for

the Euclidean bipartite matching problem. Physical Review E, 90(1), 2014. - [4] G. De Philippis, A. Figalli: Partial regularity for optimal transport

maps. Publications Mathématiques. Institut de Hautes Études Scientifiques

121 (2015), 81–112. - [5] A. Figalli, Y.-H. Kim: Partial regularity of Brenier solutions of the

Monge-Amépre equation. Discrete and Continuous Dynamical Systems (Series A)

28 (2010), 559–565. - [6] M. Goldman, M. Huesmann: A fluctuation result for the displacement in the

optimal matching problem. arXiv e-prints, May 2021. arXiv:2105.02915. - [7] M. Goldman, M. Huesmann, F. Otto: Quantitative linearization results for

the Monge-Amp`ere equation. Communications on Pure and Applied Mathematics

(2021). - [8] M. Goldman, F. Otto: A variational proof of partial regularity for optimal

transportation maps. Annales Scientifiques de l’Ećole Normale Supérieure.

Quatriéme Série 53 (2020), no. 5, 1209–1233. - [9] M. Huesmann, F. Mattesini, F. Otto: There is no stationary cyclically

monotone Poisson matching in 2d. arXiv e-prints, September 2021.

arXiv:2109.13590. - [10] F. Otto, M. Prod’homme, T. Ried: Variational approach to regularity of

optimal transport maps: general cost functions. (English summary) Ann. PDE 7

(2021), no. 2, Paper No. 17, 74 pp.

**Jun, 22**: A variational approach to the regularity theory for optimal transportation: Lecture 1

Speaker:

*Felix Otto*

In this mini-course, we shall explain the variational approach to regularity

theory for optimal transportation introduced in [8]. This approach does

completely bypass the celebrated regularity theory of Caffarelli [2], which is

based on the regularity theory for the Monge-Amp ere equation as a fully

nonlinear elliptic equation with a comparison principle. Nonetheless, one

recovers the same partial regularity theory [5, 4].

The advantage of the variational approach resides in its robustness regarding

the regularity of the measures, which can be arbitrary measures [7][Theorem

1.4], and in terms of the problem formulation, e.g. by its extension to almost

minimizers [10]. The former for instance is crucial in order to tackle the

widely popular matching problem [3, 1] e.g. the optimal transportation between

(random) point clouds, as carried out in [7, 6, 9]. The latter is convenient

when treating more general than square Euclidean cost functions.

The variational approach follows de Giorgi’s philosophy for minimal surfaces.

At its core is the approximation of the displacement by the gradient of a

harmonic function. This approximation is based on the Eulerian formulation of

optimal transportation, which reveals its strict convexity and the proximity to

the $H^{-1}$-norm. In this mini-course, we shall give a pretty self-contained

derivation of this harmonic approximation result, and establish applications to

the matching problem.

#### References

- [1] L. Ambrosio, F. Stra, D. Trevisan: A PDE approach to a 2-dimensional

matching problem. Probab. Theory Relat. Fields 173, 433–477 (2019). - [2] L.A. Caffarelli: The regularity of mappings with a convex potential.

Journal of the American Mathematical Society 5 (1992), no. 1, 99–104. - [3] S. Caracciolo, C. Lucibello, G. Parisi, G. Sicuro: Scaling hypothesis for

the Euclidean bipartite matching problem. Physical Review E, 90(1), 2014. - [4] G. De Philippis, A. Figalli: Partial regularity for optimal transport

maps. Publications Mathématiques. Institut de Hautes Études Scientifiques

121 (2015), 81–112. - [5] A. Figalli, Y.-H. Kim: Partial regularity of Brenier solutions of the

Monge-Amépre equation. Discrete and Continuous Dynamical Systems (Series A)

28 (2010), 559–565. - [6] M. Goldman, M. Huesmann: A fluctuation result for the displacement in the

optimal matching problem. arXiv e-prints, May 2021. arXiv:2105.02915. - [7] M. Goldman, M. Huesmann, F. Otto: Quantitative linearization results for

the Monge-Amp`ere equation. Communications on Pure and Applied Mathematics

(2021). - [8] M. Goldman, F. Otto: A variational proof of partial regularity for optimal

transportation maps. Annales Scientifiques de l’Ećole Normale Supérieure.

Quatriéme Série 53 (2020), no. 5, 1209–1233. - [9] M. Huesmann, F. Mattesini, F. Otto: There is no stationary cyclically

monotone Poisson matching in 2d. arXiv e-prints, September 2021.

arXiv:2109.13590. - [10] F. Otto, M. Prod’homme, T. Ried: Variational approach to regularity of

optimal transport maps: general cost functions. (English summary) Ann. PDE 7

(2021), no. 2, Paper No. 17, 74 pp.

**Jun, 22**: Recent results in interface motions in the framework of optimal transport: Lecture 2

Speaker:

*Inwon Kim*

In the lectures we will discuss recent results obtained on interface motions in the framework of optimal transport. We intend to (time allowing) discuss the following problems:

The Hele-Shaw type flows in the context of tumor growth. Here the flow describe the growth of tumor cells with contact inhibition. The tumor cells then form a congested zone, which evolves by the pressure generated by the constraint on maximal density. We start with a simple mechanical model, and discuss the effects of nutrients and surface tension in the context of minimizing movements. While the well-posedness would be established by minimizing movements, we will also explore qualitative properties of solutions such as regularity of the interface.

The Stefan problem, in the framework of optimal stopping time. Our focus will be on the well-posedness of the supercooled Stefan problem, which describes freezing of supercooled fluid. The interface between the fluid and ice, as it freezes, exhibits a high degree of irregularity. Our goal is to introduce a notion of solutions that are physically meaningful and stable. We will start with a quick introduction of the necessary background on the optimal stopping time between probability measures. We will establish the well-posedness, and discuss qualitative behavior of solutions.

**Jun, 21**: Recent results in interface motions in the framework of optimal transport: Lecture 1

Speaker:

*Inwon Kim*

In the lectures we will discuss recent results obtained on interface motions in the framework of optimal transport. We intend to (time allowing) discuss the following problems:

The Hele-Shaw type flows in the context of tumor growth. Here the flow describe the growth of tumor cells with contact inhibition. The tumor cells then form a congested zone, which evolves by the pressure generated by the constraint on maximal density. We start with a simple mechanical model, and discuss the effects of nutrients and surface tension in the context of minimizing movements. While the well-posedness would be established by minimizing movements, we will also explore qualitative properties of solutions such as regularity of the interface.

The Stefan problem, in the framework of optimal stopping time. Our focus will be on the well-posedness of the supercooled Stefan problem, which describes freezing of supercooled fluid. The interface between the fluid and ice, as it freezes, exhibits a high degree of irregularity. Our goal is to introduce a notion of solutions that are physically meaningful and stable. We will start with a quick introduction of the necessary background on the optimal stopping time between probability measures. We will establish the well-posedness, and discuss qualitative behavior of solutions.

**Jun, 20**: Towards a Mathematical Theory of Developmental Biology: Lecture 1

Speaker:

*Geoff Schiebinger*

New measurement technologies like single-cell RNA sequencing are bringing ‘big data’ to biology. One of the most exciting prospects associated with this new trove of data is the possibility of studying temporal processes, such as differentiation and development. In this talk, we introduce the basic elements of a mathematical theory to answer questions like How does a stem cell transform into a muscle cell, a skin cell, or a neuron? How can we reprogram a skin cell into a neuron? We model a developing population of cells with a curve in the space of probability distributions on a high-dimensional gene expression space. We design algorithms to recover these curves from samples at various time-points and we collaborate closely with experimentalists to test these ideas on real data.

**May, 25**: Subgraphs in Semi-random Graphs

Speaker:

*Natalie Behague*

The semi-random graph process can be thought of as a one player game. Starting with an empty graph on n vertices, in each round a random vertex u is presented to the player, who chooses a vertex v and adds the edge uv to the graph (hence 'semi-random'). The goal of the player is to construct a small fixed graph G as a subgraph of the semi-random graph in as few steps as possible. I will discuss this process, and in particular the asympotically tight bounds we have found on how many steps the player needs to win. This is joint work with Trent Marbach, Pawel Pralat and Andrzej Rucinski.

**May, 20**: 2022 PIMS Education Prize: Sean Graves

Speaker:

*Sean Graves*

PIMS is glad to announce that Sean Graves is the winner of the 2022 Education Prize. Graves is a faculty lecturer in the Department of Mathematical and Statistical Sciences and the Coordinator for the Decima Robinson Support Centre at the University of Alberta. The selection committee was extremely impressed by his energy and enthusiasm towards teaching, and the impact of his work developing mathematical talent through outreach. This prize, awarded annually by PIMS, recognizes individuals and groups in the PIMS network, Western Canada and Washington State who have played a major role in encouraging activities which have enhanced public awareness and appreciation of mathematics.

“(Graves’) hands-on training focuses on communication, diversity, professionalism, and pedagogically strong teaching techniques. Any person who spends time with Sean talking about mathematics perceives that there is an intrinsic beauty within this discipline: a magic of sorts,” noted Arturo Pianzola, Department Chair at UAlberta.

Sean Graves has been a faculty lecturer since 2011 and has received numerous awards from the University of Alberta for his teaching and service. In 2017 he was awarded the William Hardy Alexander Award for Excellence in Undergraduate Teaching. He has been passionate about training future educators in his teaching of math and developed a new course focused on mathematical reasoning for elementary teachers. Sean has also been the lead organizer for UAlberta SNAP Math Fairs each year since 2007, and a co-organizer of the Canadian Mathematics Society’s Alberta Math Summer Camp, for students aged 12-15 years. His continuous dedication to mathematics students of all ages, as well as teachers is inspiring to many.

**May, 20**: Changing the Culture 2022 Plenary: Logic in K − 12

Speaker:

*John Baldwin*

We will give examples from grade 1 to through high school where the logical insights of the last century impact classroom teaching. We include both "do's and don'ts". These examples range through such topics as "equals" vs "evaluate" vs "solve", "why multiplication is not JUST repeated addition", "lies my teacher told me", "identities, equalities and quantifiers", and "Is it true that the sum of the angles of a triangle is 90o". We will briefly discuss the place of formal logic in the secondary school.

**May, 18**: Exact results in quantum field theory from differential systems

Speaker:

*Raphaël Belliard*

Despite being the most efficient set of computational techniques available to the theoretical physicist, quantum field theory (QFT) does not describe all the observed features of the quantum interactions of our universe. At the same time, its mathematical formulation beyond the approximation scheme of perturbation theory is yet to be understood as a whole. I am following a path that tries to solve these two parallel problems at once and I will tell the story of how that way is paved by the study of equivariant differential systems and homology with local coefficients. More precisely, I will introduce these main characters in two space-time dimensions and describe how their symplectic geometry contains the data of correlation functions in conformally invariant QFT. If time allows, I will discuss how the Lax formulation of integrable systems in terms of Higgs bundles gives us hints as per how to extend the method to cases with four space-time dimensions.

**May, 12**: 2022 Celebration of Women in Mathematics - Panel Discussion

Speaker:

*Manuela Golban, Avleen Kaur, Deniz Sezer, Rekha R. Thomas*

This panel discussion took part as part of the 2022 Celebration of Women in Mathematics event.

**May, 12**: A moment with L-functions

Speaker:

*Matilde Lalín*

The Riemann zeta function plays a central role in our understanding of the prime numbers. In this talk we will review some of its amazing properties as well as properties of other similar functions, the Dirichlet L-functions. We will then see how the method of moments can help us in the study of L-functions and some surprising properties of their values. This talk will be accessible to advanced undergraduate students and is part of the May12, Celebration of Women in Mathematics.

**May, 4**: Mesoscale Imaging Reveals the Markovian Dynamics of the Brain

Speaker:

*Jordan Culp*

With mesoscale imaging, we can optogenetically record the calcium signals from the entire cortical surface of the mouse brain. However, a mathematical analysis to assess the stability or changes to the brains dynamics remains elusive due to the size and complexity of the underlying data. Here, we apply a novel Continuous-Time Markov Chain approach to assess changes to the dynamics of the mouse brain under the application of different drugs, visual stimulation, and seizure induction. In all cases we can create a kind of dynamical bar-code of the brain dynamics of the mouse by computing Markov transition probability matrices and occupancy distributions. This dynamical bar-code is unique and reproducible for each mouse, yet changes in consistent ways as a result of our experimental manipulations. Thus, we argue that a Markovian description of the mesoscale brain is sufficient for detecting dynamical changes. In this talk, I will describe the experimental background and significance of our results, along with the derivation and detailed presentation of our mathematical model. This is joint work with the McGirr, Teskey, and Nicola labs at the University of Calgary.

**Apr, 28**: OT techniques in data driven methodology: theory and practice from mathematical finance and statistics

Speaker:

*Jan Obloj*

Wasserstein distances, or Optimal Transport methods more generally, offer a powerful non-parametric toolbox to conceptualise and quantify model uncertainty in diverse applications. Importantly, they work across the spectrum: from small uncertainty around a selected model (e.g., the empirical measure) to large uncertainty of considering all models consistent with the data. I will showcase this using examples from mathematical finance (pricing and hedging of options, optimal investment) and statistics (non-parametric estimators, regularised regression methods). I will illustrate the large uncertainty regime using Martingale OT problems. For the small uncertainty regime I will consider a generic stochastic optimization problem and its distributionally robust version using Wasserstein balls. I will derive explicit formulae for the first order correction to both the value function and the optimizer. Throughout, I will present both theoretical result, as well as comments on the available numerical methods.

The talk will be borrow from many joint works, including with Daniel Bartl, Samuel Drapeau, Stephan Eckstein, Gaoyue Guo, Tongseok Lim and Johannes Wiesel.

**Apr, 27**: Shift operators and their adjoints in several contexts

Speaker:

*Meredith Sargent*

I will give a very broad overview discussing various uses and generalizations of the shift operator (and its adjoint). In the classical case we consider the Hardy space of analytic functions on the complex disk with square summable Taylor coefficients. The shift operator is simply multiplication by z and this "shifts" the coefficients of the function. The backward shift does the opposite, and in the case of the Hardy space, it's actually the adjoint of the shift. (This doesn't happen in every function space!) There are many classical results about subspaces that are invariant under the shift or its adjoint and connecting these to functions and operators. I'll discuss some of the generalizations of the shift operators and some of my recent and current projects and how they connect to the classical theory.

**Apr, 20**: Non-realizability of polytopes via linear programming

Speaker:

*Amy Wiebe*

A classical question in polytope theory is whether an abstract polytope can be realized as a concrete convex object. Beyond dimension 3, there seems to be no concise answer to this question in general. In specific instances, answering the question in the negative is often done via “final polynomials” introduced by Bokowski and Sturmfels. This method involves finding a polynomial which, based on the structure of a polytope if realizable, must be simultaneously zero and positive, a clear contradiction. The search space for these polynomials is ideal of Grassmann-Plücker relations, which quickly becomes too large to efficiently search, and in most instances where this technique is used, additional assumptions on the structure of the desired polynomial are necessary.

In this talk, I will describe how by changing the search space, we are able to use linear programming to exhaustively search for similar polynomial certificates of non-realizability without any assumed structure. We will see that, perhaps surprisingly, this elementary strategy yields results that are competitive with more elaborate alternatives and allows us to prove non-realizability of several interesting polytopes.

**Apr, 7**: Projections and circles

Speaker:

*Malabika Pramanik*

Large sets in Euclidean space should have large projections in most directions. Projection theorems in geometric measure theory make this intuition precise, by quantifying the words “large” and “most”.

How large can a planar set be if it contains a circle of every radius? This is the quintessential example of a curvilinear Kakeya problem, central to many areas of harmonic analysis and incidence geometry.

What do projections have to do with circles?

The talk will survey a few landmark results in these areas and point to a newly discovered connection between the two.

**Apr, 6**: Positivity preservers forbidden to operate on diagonal blocks

Speaker:

*Prateek Vishwakarma*

The question of which functions acting entrywise preserve positive semidefiniteness has a long history, beginning with the Schur product theorem [Crelle 1911], which implies that absolutely monotonic functions (i.e., power series with nonnegative coefficients) preserve positivity on matrices of all dimensions. A famous result of Schoenberg and of Rudin [Duke Math. J. 1942, 1959] shows the converse: there are no other such functions. Motivated by modern applications, Guillot and Rajaratnam [Trans. Amer. Math. Soc. 2015] classified the entrywise positivity preservers in all dimensions, which act only on the off-diagonal entries. These two results are at "opposite ends", and in both cases the preservers have to be absolutely monotonic. We complete the classification of positivity preservers that act entrywise except on specified "diagonal/principal blocks", in every case other than the two above. (In fact we achieve this in a more general framework.) The ensuing analysis yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic.

**Mar, 24**: Transcendental values of power series and dynamical degrees

Speaker:

*Holly Kreiger*

Abstract: In the study of a discrete dynamical system defined by polynomials, we hope as a starting point to understand the growth of the degrees of the iterates of the map. This growth is measured by the dynamical degree, an invariant which controls the topological, arithmetic, and algebraic complexity of the system. I will discuss the history of this question and the recent surprising construction, joint with Bell, Diller, and Jonsson, of a transcendental dynamical degree for an invertible map of this type, and how our work fits into the general phenomenon of power series taking transcendental values at algebraic inputs.

### Speaker Biography

Holly Krieger is a leader in the area of arithmetic dynamics. She received a Ph.D. from the University of Illinois at Chicago, and was a postdoc at MIT before starting her present position in Cambridge. She was the Australian Mathematical Society's Mahler Lecturer in 2019, and received a Whitehead Prize from the London Mathematical Society in 2020.

**Mar, 23**: Thunderstorms in the present, past and future

Speaker:

*Courtney Schumacher*

- What do thunderstorms look like on the inside?
- Were they any different 30 to 50 thousand years ago?
- How might they change in the next 100 years as global temperatures continue to rise?

The presentation will start with how a thunderstorm looks in 3-D using radar technology and lightning mapping arrays. We will then travel tens of thousands of years into the past using chemistry analysis of cave stalactites in Texas to see how storms behaved as the climate underwent large shifts in temperature driven by glacial variability. I will end the talk with predictions of how lightning frequency may change over North America by the end of the century using numerical models run on supercomputers, and the potential impacts to humans and ecosystems.

**Mar, 23**: Directional sensing and signal integration by immune cells

Speaker:

*Sean Collins*

Human neutrophils and other immune cells sense chemical gradients to navigate to sites of injury, infection, and inflammation in the body. Impressively, these cells can detect gradients that differ by as little as about 1% in concentration across the length of the cell. Abstract models suggest that they may do this by integrating opposing local positive and long-range negative signals generated by receptors. However, the molecular basis for signal processing remains unclear. To investigate models of sensing, we developed experimental tools to control receptors with light while measuring downstream signaling responses with spatial resolution in single cells. We are directly measuring responses to both local and cell-wide receptor activation to determine the wiring of signal processing. While we do not see evidence for long-range negative signals, we do see a subcellular context-dependence of signal transmission. We propose that signal transmission from receptors happens locally, but cell-wide polarity biases sensing to maintain persistent migration and achieve temporal averaging to promote directional accuracy.

**Mar, 23**: From liquid fuel injection to blood flow in human body

Speaker:

*Anirudh Asuri Mukundan*

With the advancement in the high performance computing (HPC), it has become feasible to simulate various physical processes and phenomena. Such processes have applications ranging from energy & transportation sector to biological research. The process of liquid fuel injection and atomization forming fuel drops in aircraft engines is central to the formation of pollutants, therefore, it is crucial to study and control this process. The atomization is a physical process in which bulk liquid breaks up into small drops, further breaking up into even smaller drops finally leading to their evaporation. Quite often these drops are studied in an Eulerian fashion. Another approach to investigate the drops or deformable capsules is in a Lagrangian fashion. In this approach, each drop/capsule is tracked separately and is assumed to be either a rigid sphere or a deformable thin membrane. The latter has the direct application to the investigations of red blood cells (RBC) in biological systems. In fact, a RBC has a visco-hyperelastic thin membrane rendering it to be transported through capillary blood vessels of two times smaller its own size. By studying the dynamics of deformation of this membrane, it is possible to extract vital mechanical properties and develop a generalizable numerical model. This model has the potential to be employed to predict blocks in blood vessel the knowledge of which is helpful in improving the measurement of blood pressure. In this talk, I will be presenting two accurate, efficient, and robust numerical methods for simulating liquid fuel atomization process along with showcasing their engineering applications for subsonic & supersonic aircrafts. Furthermore, I will be giving a brief introduction to my current research work on the development of a numerical membrane model (NMM) for studying RBC deformation dynamics.

**Mar, 17**: Adventures with Partial Identification in Studies of Marked Individuals

Speaker:

*Simon Bonner*

Monitoring marked individuals is a common strategy in studies of wild animals (referred to as mark-recapture or capture-recapture experiments) and hard to track human populations (referred to as multi-list methods or multiple-systems estimation). A standard assumption of these techniques is that individuals can be identified uniquely and without error, but this can be violated in many ways. In some cases, it may not be possible to identify individuals uniquely because of the study design or the choice of marks. Other times, errors may occur so that individuals are incorrectly identified. I will discuss work with my collaborators over the past 10 years developing methods to account for problems that arise when are only individuals are only partially identified. I will present theoretical aspects of this research, including an introduction to the latent multinomial model and algebraic statistics, and also describe applications to studies of species ranging from the golden mantella (an endangered frog endemic to Madagascar measuring only 20 mm) to the whale shark (the largest know species of fish, measuring up to 19m).

**Mar, 17**: Predicting rain and lightning using statistical and machine learning techniques

Speaker:

*Courtney Schumacher*

Convective storms are highly intermittent and intense, making their occurrence and strength difficult to predict. This is especially true for climate models, which have grid resolutions much coarser (e.g., 100 km) than the scale of a storm’s microphysical and dynamical processes (< 1 km). Physically-based parameterizations struggle to account for this scale mismatch, causing large model errors in rain and lightning. This talk will explore some avenues of using statistical techniques (such as generalized linear and log-Gaussian Cox process models) and machine learning methods (such as random forests and neural networks) that are trained by satellite observations of thunderstorms to see how well they can improve upon existing physical parameterizations in producing accurate rain and lightning characteristics given a set of large-scale environmental conditions.

**Mar, 17**: Wasserstein gradient flows for machine learning

Speaker:

*Anna Korba*

An important problem in machine learning and computational statistics is to sample from an intractable target distribution, e.g. to sample or compute functionals (expectations, normalizing constants) of the target distribution. This sampling problem can be cast as the optimization of a dissimilarity functional, seen as a loss, over the space of probability measures. In particular, one can leverage the geometry of Optimal transport and consider Wasserstein gradient flows for the loss functional, that find continuous path of probability distributions decreasing this loss. Different algorithms to approximate the target distribution result from the choice of the loss, a time and space discretization; and results in practice to the simulation of interacting particle systems. Motivated in particular by two machine learning applications, namely bayesian inference and optimization of shallow neural networks, we will present recent convergence results obtained for algorithms derived from Wasserstein gradient flows.

**Mar, 16**: Feelling Fundamental Principles of Bacterial Cell Physiology using Long-Term Time-Lapse Atomic Force Microscopy

Speaker:

*Haig Alexander Eskandarian*

Exposure of bacteria to cidal stresses typically select for the emergence of stress-tolerant cells refractory to killing. Stress tolerance has historically been attributed to the regulation of discrete molecular mechanisms, including though not limited to regulating pro-drug activation or pumps abrogating antibiotic accumulation. However, fractions of mycobacterial mutants lacking these molecular mechanisms still maintain the capacity to broadly tolerate stresses. We have sought to understand the nature of stress tolerance through a largely overlooked axis of mycobacterial-environmental interactions, namely microbial biomechanics. We developed Long-Term Time-Lapse Atomic Force Microscopy (LTTL-AFM) to dynamically characterize nanoscale surface mechanical properties that are otherwise unobservable using other established advanced imaging modalities. LTTL-AFM has allowed us to revisit and redefine fundamental biophysical principles underlying critical bacterial cell processes targeted by a variety of cidal stresses and for which no molecular mechanisms have previously been described. I aim to highlighting the disruptive power of LTTL-AFM to revisit dogmas of fundamental cell processes like cell growth, division, and death. Our studies aim to uncover new molecular paradigms for how mycobacteria physically adapt to stress and provide expanded avenues for the development of novel treatments of microbial infections.

**Mar, 9**: Modular forms and their role in counting combinatorial and topological objects

Speaker:

*Josh Males*

I will begin by introducing some of the most basic combinatorial objects - partitions. It turns out that their generating function is a prototypical example of a modular form. These are objects with infinite symmetry, in turn giving them extraordinary properties. I'll then talk about the asymptotic behaviour of various modular-type objects arising from combinatorics and topology using the Circle Method of Hardy-Ramanujan and Wright, as well as one can even obtain exact formulae. In particular, I'll highlight the asymptotic (non)-equidistribution properties of Betti numbers of various Hilbert schemes as well as t-hooks in partitions. This talk will include various works with configurations of my collaborators Kathrin Bringmann, Giulia Cesana, William Craig, Daniel Johnston, Ken Ono, and Aleksander Simonič.

### Speaker Biography:

Joshua Males received his MMath (masters + bachelors) degree from Durham University, UK under the supervision of Jens Funke, before taking a year sabbatical in Durham. In late 2017 he joined Kathrin Bringmann's number theory group at the University of Cologne, Germany, where he earned his PhD in May 2021. Since August 2021, Joshua has been a PIMS postdoctoral fellow at the University of Manitoba, working under his mentor Siddarth Sankaran. His research focuses on modular forms and their use in number theory and beyond, with connections to combinatorics, topology, and arithmetic geometry. At the time of writing, Joshua has 8 published articles (4 solo author) and 6 preprints (1 solo author) as well as 3 more articles in the latter stages of preparation.

**Mar, 3**: Meta-Analytic Inference for the COVID-19 Infection Fatality Rate

Speaker:

*Paul Gustafson*

Estimating the COVID-19 infection fatality rate (IFR) has proven to be challenging, since data on deaths and data on the number of infections are subject to various biases. I will describe some joint work with Harlan Campbell and others on both methodological and applied aspects of meeting this challenge, in a meta-analytic framework of combining data from different populations. I will start with the easier case when the infection data are obtained via random sampling. Then I will discuss drawing in additional infection data obtained in decidedly non-random manner.

**Feb, 24**: Mathematician Helping Art Historians and Art Conservators

Speaker:

*Ingrid Daubechies*

Mathematics can help Art Historians and Art Conservators in studying and understanding art works, their manufacture process and their state of conservation. The presentation will review several instances of such collaborations, explaining the role of mathematics in each instance, and illustrating the approach with extensive documentation of the art works.

### Speaker Biography

Ingrid Daubechies is a Belgian Physicist and Mathematician, one of the leaders in the area of wavelets, a part of applied harmonic analysis. Wavelets are widely used in data compression and image encoding. Indeed, a wavelet pioneered by Daubechies is the basis of the standard for digital cinema. Ingrid Daubechies has held positions at the Free University in Brussels, Princeton University, and is currently James B. Duke Professor at Duke University. She is a Member of the National Academy of Sciences and of the National Academy of Engineering and a Fellow of the American Association for the Advancement of Science. Ingrid Daubechies has received many awards including the Leroy P. Steele Prize for Seminal Contribution to Research of the American Mathematical Society.

**Feb, 24**: Optimal transport theory in incomplete econometric models

Speaker:

*Marc Henry*

This talk focuses on the central role played by optimal transport theory in the study of incomplete econometric models. Incomplete econometric models are designed to analyze microeconomic data within the constraints of microeconomic theoretic principles, such as maximization, equilibrium and stability. These models are called incomplete because they do not predict a single distribution for the variables observed in the data. Incompleteness arises because of multiple equilibria in game theoretic solutions, unobserved heterogeneity in choice sets, interval predictions in auctions, and unknown sample selection mechanisms. The problem of confronting the model parameters (possibly infinite dimensional) and the data can be formulated as an optimal transport problem, where the transport cost is some measure of departure from the microeconomic theoretic principles. We will discuss a selection of inference methodologies on the model parameter based on different choices of transport cost, and applications to industrial organization, consumer demand theory and network formation.

**Feb, 23**: Small prime k-th power residues modulo p

Speaker:

*Kübra Benli*

Let \(p\) be a prime number. For each positive integer \(k\geq 2\), it is widely believed that the smallest prime that is a k-th power residue modulo p should be \(O(p^{\epsilon})\), for any \(\epsilon>0\). Elliott proved that such a prime is at most \(p^{\frac{k-1}{4}+\epsilon}\), for each \(\epsilon > 0\). In this talk, we discuss the number of prime k-th power residues modulo p in the interval \([1,p^{\frac{k-1}{4}+\epsilon}]\) for \(\epsilon > 0\).

**Feb, 16**: Humans Make Things Messy

Speaker:

*Shelby M. Scott*

Models become notably more complex when stochasticity is introduced. One of the best ways to add frustrating amounts of randomness to your model: incorporate humans. In this talk, I discuss three different ways in which humans have made things messy in my mathematical models, statistical models, and data science work. Despite the fact that humans do, indeed, make things messy, they also make our models so much more realistic, interesting, and intriguing. So while humans make things messy, it is so worth it to bring them into your work.

**Feb, 16**: The Geometric Structure of Possible Singularities for the Navier-Stokes and Euler Equations

Speaker:

*Evan Miller*

I will discuss several geometric constraints of the finite-time blowup of smooth solutions of the Navier-Stokes equation in the regularity criteria related to the eigenvalue structure of the strain matrix and to the vorticity direction. These regularity criteria suggest that strain self-amplification via axial compression/planar stretching drives any possible blowup. I will also discuss model equations where this form of blowup does indeed occur.

### Speaker Biography:

Evan Miller received his PhD in mathematics from the University of Toronto under the supervision of Prof. Robert McCann in 2019. He was then a postdoc at McMaster University, working with Prof. Eric Sawyer. He was also a visiting postdoc at the Fields Institute in Toronto and the Mathematical Sciences Research Institute in Berkeley for thematic programs in mathematical fluid mechanics. At MSRI, he worked with Prof. Jean-Yves Chemin. Evan is now a PIMS postdoctoral fellow at the University of British Columbia working with Prof. Tai-Peng Tsai and Prof. Stephen Gustafson.

**Feb, 9**: Asymptotic analysis of the concentration difference due to diffusive fluxes across narrow windows

Speaker:

*Frédéric Paquin-Lefebvre*

How far inside a domain does a flux of Brownian particles perturb a background concentration when particles can escape through a neighboring window? What motivates this question is the dynamics of ions entering and exiting nanoregions of excitable cells through ionic membrane channels. Here this is explored using a simple diffusion model consisting of the Laplace's equation in a domain whose boundary is everywhere reflective except for a collection of narrow circular windows, where either flux or absorbing boundary conditions are prescribed. We derive asymptotic formulas revealing the role of the influx amplitude, the diffusion properties, and the geometry, on the concentration difference. Lastly, a length scale to estimate how deep inside a domain a local diffusion current can spread is introduced. This is joint work with David Holcman at ENS.

**Feb, 9**: Knot Floer homology of satellite knots

Speaker:

*Wenzhao Chen*

Knot Floer homology is a package of widely-used knot invariants constructed via pseudo-holomorphic curves. In this talk, we will restrict our attention to the knot Floer homology of a class of knots called satellite knots; understanding these invariants figure prominently in studying 4-dimensional questions in knot theory, such as analyzing surfaces bounded by knots in 4-manifolds. However, previous methods of computing these invariants are rather involved. In this talk, I will present a new and more effective way to compute the knot Floer homology of satellite knots; our approach is built on the immersed-curve technique introduced by Hanselman-Rasmussen-Watson in bordered Heegaard Floer homology. This talk is based on joint work in progress with Jonathan Hanselman.

### Speaker Biography:

Wenzhao Chen obtained his Ph.D. at Michigan State University in 2019, where he studied Heegaard Floer homology and low dimensional topology under the supervision of Dr. Matt Hedden. He was a postdoc in the Max Planck Institue for Mathematics in Bonn from 2019 to 2021. Currently, He is a PIMS Postdoctoral Fellow at the University of British Columbia. He is working with Dr. Liam Watson in low-dimensional topology.

**Jan, 27**: A survey on weak optimal transport

Speaker:

*Nathael Gozlan*

This talk will present the framework of weak optimal transport which allows to incorporate more general penalizations on elementary mass transports. After recalling general duality results and different optimality criteria, we will focus on recent applications of weak optimal transport. We will see in particular how a weak variant of the squared Wasserstein distance can be used to characterize the Gaussian concentration of measure phenomenon for convex functions or to study the contraction properties of the Brenier map. If time permits we will also discuss a new variant of the weak transport problem which has applications in economy. Based on joint works with P. Choné, M. Fathi, N. Juillet, F. Kramarz, M. Prodhomme, C. Roberto, P-M Samson, Y. Shu and P. Tetali.

**Jan, 26**: EKR-Module Property

Speaker:

*Venkata Pantangi*

Let \(G\) be a finite group acting transitively on \(X\). We say \(g,h \in G\) are intersecting if \(gh^{-1}\) fixes a point in \(X\). A subset \(S\) of \(G\) is said to be an intersecting set if every pair of elements in \(S\) intersect. Cosets of point stabilizers are canonical examples of intersecting sets. The group action version of the classical Erdos-Ko-Rado problem asks about the size and characterization of intersecting sets of maximum possible size. A group action is said to satisfy the EKR property if the size of every intersecting set is bounded above by the size of a point stabilizer. A group action is said to satisfy the strict-EKR property if every maximum intersecting set is a coset of a point stabilizer. It is an active line of research to find group actions satisfying these properties. It was shown that all \(2\)-transitive satisfy the EKR property. While some \(2\)-transitive groups satisfy the strict-EKR property, not all of them do. However a recent result shows that all \(2\)-transitive groups satisfy the slightly weaker "EKR-module property"(EKRM), that is, the characteristic vector of a maximum intersecting set is a linear span of characteristic vectors of cosets of point stabilizers. We will discuss about a few more infinite classes of group actions that satisfy the EKRM property. I will also provide a few non-examples and a characterization of the EKRM property using characters of \(G\) .

**Jan, 20**: Monge-Kantorovich distance and PDEs

Speaker:

*Benoît Perthame*

The Monge transfer problem goes back to the 18th century. It consists in minimizing the transport cost of a material from a place to another (and changing the shape). Monge could not solve the problem and the next significant step was achieved 150 years later by Kantorovich who introduced the transport distance between two probability measures as well as the dual problem.

The Monge-Kantorovich distance is not easy to use for Partial Differential Equations and the method of doubling the variables is one of them. It is very intuitive in terms of stochastic processes and this provides us with a method for conservative PDEs as parabolic equations (possibly fractional), homogeneous Boltzman equation, scattering equation or porous medium equation...

Structured equations, as they appear in mathematical biology, is a particular class where the method can be used.

### Speaker Biography

Benoît Perthame studied at the École Normale Supérieure, and has been a Professor at the University of Orléans, the École Normale Supérieure and Paris VI. He is a leader in the area of non-linear partial differential equations, and has made important contributions both to the theory of differential equations. He has also played a pioneering role in applying differential equations to problems of modeling in biology and other sciences. He has written several research monographs, as well as close to 300 papers.

Benoît Perthame was an Invited Speaker at the ICM in 1994, and gave a plenary lecture at the ICM in 2014. He has received the Peccot Prize from the Collège de France, and is a member of the French Academiy of Sciences.

**Jan, 12**: Knotted Objects Confined to Tubes in the Simple Cubic Lattice

Speaker:

*Puttipong Pongtanapaisan*

Motivated by biological questions related to DNA packing and the movement of molecules through channels, it is of interest to determine whether a specific knot or link type can be realized in a confined volume. In this talk, we will discuss the size of the smallest lattice tube that can contain certain families of knotted objects. We will take advantage of a theorem of Arsuaga et al., which allows us to study entanglements in lattice tubes by analyzing how level spheres coming from the standard height function intersect the knotted object. We conclude by discussing the exponential growth rate of links in the smallest lattice tube which admits nontrivial knotting and linking. This talk is based on joint work with Jeremy Eng, Robert Scharein, and Chris Soteros.