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Euler’s famous formula tells us that (with appropriate caveats), a map on the sphere with f countries (faces), e borders (edges), and v border-ends (vertices) will satisfy v-e+f=2. And more generally, for a map on a surface with g holes, v-e+f=2-2g. Thus we can figure out the genus of a surface by cutting it into pieces (faces, edges, vertices), and just counting the pieces appropriately. This is an example of the topological maxim “think globally, act locally”. A starting point for modern algebraic geometry can be understood as the realization that when geometric objects are actually algebraic, then cutting and pasting tells you far more than it does in “usual” geometry. I will describe some easy-to-understand statements (with hard-to-understand proofs), as well as easy-to-understand conjectures (some with very clever counterexamples, by M. Larsen, V. Lunts, L. Borisov, and others). I may also discuss some joint work with Melanie Matchett Wood.

Speaker biography:

Ravi Vakil is a Professor of Mathematics and the Robert K. Packard University Fellow at Stanford University, and was the David Huntington Faculty Scholar. He received the Dean's Award for Distinguished Teaching, an American Mathematical Society Centennial Fellowship, a Frederick E. Terman fellowship, an Alfred P. Sloan Research Fellowship, a National Science Foundation CAREER grant, the presidential award PECASE, and the Brown Faculty Fellowship. Vakil also received the Coxeter-James Prize from the Canadian Mathematical Society, and the André-Aisenstadt Prize from the CRM in Montréal. He was the 2009 Earle Raymond Hedrick Lecturer at Mathfest, and a Mathematical Association of America's Pólya Lecturer 2012-2014. The article based on this lecture has won the Lester R. Ford Award in 2012 and the Chauvenet Prize in 2014. In 2013, he was a Simons Fellow in Mathematics.

Gabriel Verret (University of Auckland, New Zealand)

Many results in algebraic graph theory can be viewed as upper bounds on the size of the automorphism group of graphs satisfying various hypotheses. These kinds of results have many applications. For example, Tutte's classical theorem on 3-valent arc-transitive graphs led to many other important results about these graphs, including enumeration, both of small order and in the asymptotical sense. This naturally leads to trying to understand barriers to this type of results, namely graphs with large automorphism groups. We will discuss this, especially in the context of vertex-transitive graphs of fixed valency. We will highlight the apparent dichotomy between graphs with automorphism group of polynomial (with respect to the order of the graph) size, versus ones with exponential size.

In this talk, we introduce some recent regularity results of free boundary in
optimal transportation. Particularly for higher order regularity, when
densities are Hölder continuous and domains are $C^2$, uniformly convex, we obtain the free boundary is $C^{2,\alpha}$ smooth. We also consider another mode
case that the target consists of two disjoint convex sets, in which
singularities of optimal transport mapping arise. Under similar assumptions,
we show that the singular set of the optimal mapping is an $(n-1)$-dimensional
$C^{2,\alpha}$ regular sub-manifold of $\mathbb{R}^n$. These are based on a
series of joint work with Shibing Chen and Xu-Jia Wang.

High resolution climate simulations (horizontal resolutions of 25km or smaller) are a desired product for policy makers, the public, and renewable energy applications. Since running climate models at high resolution is not feasible, dynamical and statistical downscaling methods are applied. The latest version of the Canadian regional climate model (CanRCM5) is specifically designed to dynamically downscale future climate projections of its parent Canadian Earth System Model (CanESM5). The close relationship between these two Canadian models allows for improved RCM driving relative to independent RCM modelling centres, as all required prognostic variables are available on CanRCM5's lateral boundaries from its parent global model. Coupling different scale size models is challenging as regional climate models can easily develop their own climate. To keep the features of CanRCM5 consistent with CanESM5, the large-scale dynamics of CanRCM5 are typically nudged towards its parent model. We present a framework that identifies appreciable differences between the regional and global model and apply it to near-surface wind and precipitation fields showing that particularly the influence of better resolved topography yields substantial differences in the climate projections of CanRCM and CanESM. Finally, we discuss latest developments on research on bias correcting CanESM and CanRCM which allows for more accurate representations of the climate state. (Joint work with John Scinocca and Slava Kharin)

Growing evidence is demonstrating the startling reach of wind farm wakes, which can often exceed 100 kilometers under frequently occurring stable atmospheric conditions. These "long wakes" can significantly affect the annual energy yield of neighboring wind farms. Unfortunately, this impact is not accounted for in industry-standard wind resource assessment methods, leading to consistent overestimations of energy production and the suboptimal placement of new wind farms. As the global wind farm fleet continues to rapidly grow, the development of tools to assess these impacts is crucial prior to making any major investments in new project sites.

In this presentation, we will elucidate how numerical weather prediction (NWP) modeling is emerging as the preferred tool for simulating long-wake propagation. We will initially discuss the physical mechanisms underlying long-wake propagation, and provide observational evidence substantiating their occurrence. Subsequently, we will delve into the suitability of the Weather Research and Forecasting (WRF) NWP model for long-wake modeling, specifically its ability to accurately simulate diurnal and seasonal atmospheric stability fluctuations and its capacity to model wake propagation through its wind farm parameterization option. In conclusion, we will present encouraging validation results from five onshore wind farms in the U.S., reinforcing the potential and accuracy of the WRF model in tackling this growing issue.

This panel was organized as part of the PIMS Workshop on Forecasting and Mathematical Modeling for Renewable Energy and Public Panel Discussion on Climate Change. Ahead of the panel, the panelists were invited to give a short talk on their areas of expertise.

• Réne Aïd: More electricity demand response for less carbon emissions
• Gaël Giraud: Macroeconomics and Climate
• Seth Klein: Mobilizing Canada for the Climate Emergency
• Judith Sayers: First Nation Leadership in Clean Energy and Climate Action
• Andrew Weaver: Privilege, agency, and the climate scientist’s role in the global warming debate

Judith Sayers is the President of the Nuu-chah-nulth Tribal Council, a lawyer, renewable energy leader, and chancellor of Vancouver Island University. This presentation was given ahead of a panel discussion on Tackling Climate Change and the Just Transition to Renewable Energy.

Gaël Giraud is the Founding director of the Georgetown University Environmental Justice Program. He is also a professor at Georgetown and the former chief economist of the French Development agency CNRS. This presentation was given ahead of a panel discussion on Tackling Climate Change and the Just Transition to Renewable Energy in which Dr. Giraud also participated.

We formulate an equilibrium model of intraday trading in electricity markets. Agents face balancing constraints between their customers consumption plus intraday sales and their production plus intraday purchases. They have continuously updated forecast of their customers consumption at maturity. Forecasts are prone to idiosyncratic noise as well as common noise (weather). Agents production capacities are subject to independent random outages, which are each modeled by a Markov chain. The equilibrium price is defined as the price that minimizes trading cost plus imbalance cost of each agent and satisfies the usual market clearing condition. Existence and uniqueness of the equilibrium are proved, and we show that the equilibrium price and the optimal trading strategies are martingales. The main economic insights are the following. (i) When there is no uncertainty on generation, it is shown that the market price is a convex combination of forecasted marginal cost of each agent, with deterministic weights. Furthermore, the equilibrium market price is consistent with Almgren and Chriss's model, and we identify the fundamental part as well as the permanent market impact. It turns out that heterogeneity across agents is a necessary condition for Samuelson's effect to hold. We show that when heterogeneity lies only on costs, Samuelson's effect holds true. A similar result stands when heterogeneity lies only on market access quality. (ii) When there is production uncertainty only, we provide an approximation of the equilibrium for large number of players. The resulting price exhibits increasing volatility with time. (Joint work with Andrea Cosso and René Aïd)

Electricity markets balance an increasingly intermittent supply with price-inelastic demand, while climate change and electrification of mobility are contributing to transforming diurnal and seasonal demand patterns. Electricity systems face an increasing level of stochasticity, and market participants need to inform their dispatch decisions based on 24-hour price forecasts for participation in Day-Ahead Markets, which in turn depend on supply and demand forecasts. The arrival of grid-scale electricity storage is also creating new scope for prices forecasts, while smaller scale storage systems act as price takers. In the long term, large-scale deployment of grid-scale electricity storage has the potential of significantly reducing price variation through arbitrage, which could shift the “value added” of forecasting from short-term (intra-day) to long-term predictions, and from supporting operational (dispatch) decisions to supporting capacity (investment) decisions.

The rising U.S. offshore wind sector holds great promise, both environmentally and economically, to unlock vast supplies of clean, domestic, and renewable energy. To harness this valuable resource, Gigawatt (GW)-scale offshore wind projects are already under way at several locations off of the U.S. coastline. This promising future, however, is still clouded with uncertainties on how to optimally manage those ultra-scale offshore wind assets, which would be operating under harsh environmental and operational conditions, in relatively under-explored territories, and at unprecedented scales. I will present some of our progress in formulating tailored forecasting and optimization models aimed at minimizing some of those uncertainties. Our models and analyses are largely tailored and tested using data from the U.S. Mid-Atlantic—where several GW-scale wind projects are currently under development.

How much wind power will a turbine generate over its lifetime? To answer such questions, we can consider climate model output to generate very long-term wind power forecasts on the scale of years to decades. One major limitation of the data projected by climate models is their coarse temporal resolution that is usually not finer than three hours and can be as coarse as one month. However, wind speed distributions of low temporal resolution might not be able to account for high frequency variability which can lead to distributional shifts in the projected wind speeds. Even if these changes are small this can have a huge impact due to the highly non-linear relationship between wind and wind power and the long forecast horizons we consider. In my talk, I will discuss how the resolution of wind speed data from climate projections affects wind power forecasts.

The ability to forecast solar irradiance plays an indispensable role in solar power forecasting, which constitutes an essential step in planning and operating power systems under high penetration of solar power generation. Since solar radiation is an atmospheric process, solar irradiance forecasting, and thus solar power forecasting, can benefit from the participation of atmospheric scientists. In this talk, the two fields, namely, atmospheric science and power system engineering are jointly discussed with respect to how solar forecasting plays a part. Firstly, the state of affairs in solar forecasting is elaborated; some common misconceptions are clarified; and salient features of solar irradiance are explained. Next, five technical aspects of solar forecasting: (1) base forecasting methods, (2) post-processing, (3) irradiance-to-power conversion, (4) verification, and (5) grid-side implications, are reviewed. Following that, ten research topics moving into the future are enumerated; they are related to (1) data and tools, (2) numerical weather prediction, (3) forecast downscaling, (4) large eddy simulation, (5) dimming and brightening, (6) aerosols, (7) spatial forecast verification, (8) multivariate probabilistic forecast verification, (9) predictability, and (10) extreme weather events. Last but not least, a pathway towards ultra-high PV penetration is laid out, based on a recently proposed concept of firm generation and forecasting.

The classical notion of symmetry can be formalized by actions of groups. Quantum symmetry is a generalization of the notion of symmetry to the quantum setting, where symmetries can no longer be completely described by the actions of groups. In this setting, quantum symmetries are given by Hopf actions of quantum groups on algebras. I will start with background on quantum groups and Hopf actions and then give examples of quantum symmetries of quiver path algebras. Path algebras can be described in terms of directed graphs and play an important role in the representation theory of finite-dimensional algebras. While quantum symmetries are not straightforward to visualize, path algebras give us a nice tool for doing so. Then, I will discuss a tensor categorical perspective for understanding quantum symmetry and how this perspective can be applied to quantum symmetries of path algebras and finite-dimensional algebras.

We talk about how some simple sounding problems about straight line paths on surfaces require many different kinds of mathematical thinking to solve, focusing on the example of understanding straight line paths on Platonic solids. We'll use this to start a discussion of how we can emphasize teaching different ways of thinking, and why geometry is an important resource for students. There will be lots and lots of fun pictures and hopefully interesting and provocative ideas!

The following resources are referenced during this talk:

• http://userhome.brooklyn.cuny.edu/aulicino/dodecahedron/

The under representation of women, especially women of color, has been persistently well documented (see for example the data dashboard on www.womendomath.org). One reason that this is a problem is that it can be difficult for women to identify role models - this can in turn make it harder for women to envision their own success. During my career, I found it very helpful to learn the stories of women in STEM and to draw on aspects of their success to try to invent my own path. In this talk, I will retell twelve stories of women in STEM that influenced me. I can’t promise that the stories will be historically accurate, but I will try to say what I learned from the stories as I heard them and what lessons I hope others might take from them as well.

Let the triple $(a, b, c)$ of integers be such that $\gcd{\left(a, b, c\right)} = 1$ and $a + b = c$. We call such a triple an ABC-solution. The quality of an ABC-solution $(a, b, c)$ is defined as
$$
q(a, b, c) = \frac{\max\left\{\log |a|, \log |b|, \log |c|\right\}}{\log \operatorname{rad}\left({|abc|}\right)}
$$
where $\operatorname{rad} (|abc|)$ is the product of distinct prime factors of $|abc|$. 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 remainder of the talk, we provide a result on a conjecture of Erdõs on the solutions of the Brocard-Ramanujan equation
$$
n! + 1 = m^2
$$
by assuming an explicit version of the ABC-conjecture proposed by Baker.

Borsuk’s number b(n) is the smallest integer such that any set of diameter 1 in the n-dimensional space can be covered by b(n) sets of a smaller diameter. Exponential upper bounds on b(n) were first obtained by Shramm (1988) and later by Bourgain and Lindenstrauss (1989).

To obtain an upper bound on b(n), Bourgain and Lindenstrauss provided exponential bounds (both upper and lower) in Grünbaum's problem – the problem of determining the minimal number of open balls of diameter 1 needed to cover a set of diameter 1. On the other hand, Schramm provided an exponential upper bound on the illumination number of n-dimensional bodies of constant width. In 2015 Kalai asked if there exist n-dimensional convex bodies of constant width with illumination number exponentially large in n.

In this talk I will answer Kalai’s question in the affirmative and provide a new lower bound in the Grünbaum’s problem. This talk is based on a joint work with Andriy Bondarenko and Andriy Prymak.

It is well-known that quadratic forms can be diagonalized over fields and that they are in a one-to-one correspondence with bilinear forms; the algebraic theory of quadratic forms is build on these two properties. But there is a catch -- they require division by two. Over a field of characteristic 2, neither of them is true, and the whole quadratic form theory needs to be rebuilt from scratch.
In the talk, we will give a brief introduction to the theory of quadratic forms in characteristic 2. Then we will focus on isotropy -- that is, whether we can find elements of the field on which the quadratic form in question gains the value zero. One of the classical problems is to describe, for any given quadratic form, "how much" it is isotropic over any field extension. We will see that there is basically only one type of field extensions that are relevant for this problem.

Work of Thurston and Perelman implies that every compact 3-manifold decomposes into pieces each of which supports one of eight possible geometric structures. Among these eight geometries, the hyperbolic geometry leads to the richest and least well understood class of manifolds. Moreover, Mostow-Prasad rigidity implies that any such hyperbolic structure is unique in stark contrast to the situation in dimension 2. This rigidity also gives rise to number-theoretic invariants of hyperbolic 3-manifolds, and my talk will focus on these. In particular, associated to any finite volume hyperbolic 3-manifold is a number field called the trace field and a quaternion algebra over that trace field. For knot complements, this quaternion algebra is trivial in the sense that it is always a matrix algebra. However, for closed orbifolds such as those obtained by hyperbolic Dehn surgery on a hyperbolic knot complement, the algebra is often nontrivial. A conjecture of Chinburg, Reid, and Stover relates the algebras one can obtain by surgery to the Alexander polynomial of the knot. This problem involves the character variety of the knot and a generalization of quaternion algebras called Azumaya algebras. I will discuss the interplay of these objects as well as some work on the conjecture.

Given an smooth function h, this talk will focus on solving the equation \psi(Rz)-\psi(z) = h(z) for circle rotations. We will see how the Diophantine condition on the rotation implies smooth solutions.

Vertex operator algebras are the symmetry algebras of two dimensional conformal field theory. In a famous series of papers, Kazhdan and Lusztig proved an equivalence between particular semisimple categories of modules over affine Lie algebras and quantum groups, the former of which can also be realized as modules over a corresponding vertex operator algebra. Such equivalences between representation categories of vertex operator algebras and quantum groups are now broadly referred to as the Kazhdan-Lusztig correspondence. There has been substantial research interest over the last two decades in understanding the Kazhdan-Lusztig correspondence for vertex operator algebras with non-semisimple representation theory. In this talk I will present an overview of this research area and discuss recent results and future directions.

We prove an explicit formula for the first moment of Maass form symmetric square L-functions defined over Gaussian integers. As a consequence, we derive a new upper bound for the second moment. This is joint work with Dmitry Frolenkov.

People interact with each other in social and communication networks, which affect the processes that occur on them. In this talk, I will give an introduction to dynamical proceses on networks. I will focus my discussion on opinion dynamics, and I will also discuss coupled opinion and disease dynamics on networks. Time-permitting, I may also briefly discuss a model of COVID-19 that centers on disabled people and their caregivers.

The classical obstacle problem consists of finding the equilibrium position of an elastic membrane whose boundary is held fixed and which is constrained to lie above a given obstacle. By classical results of Caffarelli, the free boundary is smooth outside a set of singular points. However, explicit examples show that the singular set could be, in general, as large as the regular set. This talk aims to introduce this beautiful problem and describe some classical and recent results on the regularity of the free boundary.

Speaker Biography: Alessio Figalli is a leading figure in the areas of Optimal Transport, partial differential equations and the calculus of variations. He received his Ph.D. from the Scuola Normale Superiore di Pisa and the Ecole Normale Superieur de Lyon and has held positions in Paris and Austin, Texas. He is currently a Professor at ETH Zurich. His work has been recognized with many awards including the Prize of the European Mathematical Society in 2012 and the Fields Medal in 2018.

Joshua Males (University of Manitoba, Canada)

In his famous '86 paper, Andrews made several conjectures on the function σ(q) of Ramanujan, including that it has coefficients (which count certain partition-theoretic objects) whose sup grows in absolute value, and that it has infinitely many Fourier coefficients that vanish. These conjectures were famously proved by Andrews-Dyson-Hickerson in their '88 Invent. paper, and the function σ has been related to the arithmetic of Z[6–√]by Cohen (and extensions by Zwegers), and is an important first example of quantum modular forms introduced by Zagier.

A closer inspection of Andrews' '86 paper reveals several more functions that have been a little left in the shadow of their sibling σ , but which also exhibit extraordinary behaviour. In an ongoing project with Folsom, Rolen, and Storzer, we study the function v1(q) which is given by a Nahm-type sum and whose coefficients count certain differences of partition-theoretic objects. We give explanations of four conjectures made by Andrews on v1, which require a blend of novel and well-known techniques, and reveal that v1 should be intimately linked to the arithmetic of the imaginary quadratic field Q[−3−−−√]
.

Constructing and training the neural network depends on various types of Stochastic Gradient Descent (SGD) methods, with adaptations that help with convergence by boosting the speed of the gradient search. Convergence for existing algorithms requires a large number of observations to achieve high accuracy with certain classes of functions. We work with a different, non-curve-tracking technique with the potential of achieving better speeds of convergence. In this talk, the new idea of 'decoupling' hidden layers by bootstrapping and using linear stochastic approximation is introduced. By utilizing resampled observations, the convergence of this process is quick and requires a lower number of data points. This proposed bootstrap learning algorithm can deliver quick and accurate estimates. This boost in speed allows the approximation of classes of functions within a fraction of the observations required with traditional neural network training methods.

In a 1992 article where she surveyed her recent breakthrough on unipotent flows on homogeneous spaces, Ratner presented an argument for the equidistribution of horospherical orbits in the context of horocycle flow on SL(2,R)/Lattice. This idea is separate from the ideas in her celebrated work on unipotent flows and I will present her argument for horospherical equidistribution in the simplest situation I can think of: proving the ergodicity of a particular directional flow on the flat two torus. Ratner's argument has similarities to Masur's criterion for unique ergodicity of translation flows, proven around the same time. Time permitting I will comment on Masur's criterion as well.

Euler's divergent series $\sum_{n>0} n!z^n$ which converges only for $z = 0$ becomes an interesting object when evaluated with respect to a p-adic norm (which will be introduced in the talk). Very little is known about the values of the series. For example, it is an open question whether the value at one is irrational (or even non-zero). As individual values are difficult to reach, it makes sense to try to say something about collections of values over sufficiently large sets of primes. This leads to looking at primes in arithmetic progressions, which is in turn raises a need for an explicit bound for the number of primes in an arithmetic progression under the generalized Riemann hypothesis.
During the talk, I will speak about both sides of the story: why we needed good explicit bounds for the number of primes in arithmetic progressions while working with questions about irrationality, and how we then proved such a bound.

The talk is joint work with Tapani Matala-aho, Neea Palojärvi and Louna Seppälä. (Questions about irrationality with T. M. and L. S. and primes in arithmetic progressions with N. P.)

The strong Allee effect plays an important role on the evolution of population in ecological systems. One important concept is the Allee threshold that determines the persistence or extinction of the population in a long time. In general, a small initial population size is harmful to the survival of a species since when the initial data is below the Allee threshold, the population tends to extinction, rather than persistence. Another interesting feature of population evolution is that a species whose movement strategy follows a conditional dispersal strategy is more likely to persist. To study the interaction between Allee effect and the biased movement strategy, we mainly consider the pattern formation and local dynamics for a class of single species population models that is subject to the strong Allee effect. We first rigorously show the existence of multiple localized solutions when the directed movement is strong enough. Next, the spectrum analysis of the associated linear eigenvalue problem is established and used to investigate the stability properties of these interior spikes. This analysis proves that there exist not only unstable but also linear stable steady states. Finally, we extend results of the single equation to coupled systems for two interacting species, each with different advective terms, and competing for the same resources. We also construct several non-constant steady states and analyze their stability.

This is a work in progress talk by a local graduate student.

Elliptic curves are one of the major objects of study in number theory. Over finite fields, their zeta functions were proven to be rational by F. K. Schmidt in 1931. In 1985, R. Schoof devised an algorithm to compute zeta functions of elliptic curves over finite fields by directly computing the numerators of these rational functions modulo sufficiently many primes. Over function fields of positive characteristic p, we know from the work of A. Grothendieck, M. Artin, J.L. Verdier (1964/1965) and others, that their L-functions are rational. They are even polynomials with integer coefficients if we assume that their j-invariants are nonconstant rational functions, as shown by P. Deligne in 1980 using a result of J.-I. Igusa (1959).

Therefore, we can meaningfully study the reduction of the L-function of an elliptic curve E with nonconstant j-invariant modulo an integer N. In 2003, C. Hall gave a formula for that reduction modulo N, provided the elliptic curve had rational N-torsion.

In this talk, we first obtain, under the assumptions of C. Hall, a formula for the L-function of any of the infinitely many quadratic twists of E. Secondly, without any condition on the rational 2-torsion subgroup of E, we give a formula for the quotient modulo 2 of L-functions of any two quadratic twists of E. Thirdly, we illustrate that sometimes the reduced L-function is enough to determine important properties of the L-function itself. More precisely, we use the previous results to compute the global root numbers of an infinite family of quadratic twists of some elliptic curve and, under extra assumptions, find in most cases the exact analytic rank of each of these quadratic twists. Finally, we use our formulas to compute directly some degree 2 L-functions, in analogy with the algorithm of Schoof.

Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications are essential. This has motivated researchers to investigate the problem of adversarial training —or how to make models robust to adversarial attacks— but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore two questions: 1) Can we use analytical tools to find lower bounds for adversarial robustness problems?, and 2) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? In this talk I will showcase how ideas from optimal transport theory can provide answers to these questions.

This talk is based on joint works with Camilo Andrés García Trillos, Matt Jacobs, and Jakwang Kim.

The past twenty-five years have heralded an unparalleled increase in understanding of cancer. At the same time, mathematical modelling has emerged as a natural tool for unravelling the complex processes that contribute to the initiation and progression of tumours, for testing hypotheses about experimental and clinical observations, and assisting with the development of new approaches for improving its treatment. In this talk I will reflect on how increased access to experimental data is stimulating the application of new theoretical approaches for studying tumour growth. I will focus on two case studies which illustrate how mathematical approaches can be used to characterise and quantify tumour vascular networks, and to understand how microstructural features of these networks affect tumour blood flow.

In an introduction to proofs course, students learn to write proofs informally in the language of set theory and classical logic. In this talk, I'll explore the alternate possibility of teaching students to write proofs informally in the language of dependent type theory. I'll argue that the intuitions suggested by this formal system are closer to the intuitions mathematicians have about their praxis. Furthermore, dependent type theory is the formal system used by many computer proof assistants both "under the hood" to verify the correctness of proofs and in the vernacular language with which they interact with the user. Thus, students could practice writing proofs in this formal system by interacting with computer proof assistants such as Coq and Lean.

We improve the best known to date result of Dress-Iwaniec-Tenenbaum, getting ($\log
x)^2$ instead of $\left(log x\right)^(5/2)$. We use a weighted form of Vaughan's identity, allowing a smooth truncation inside the procedure, and an estimate due to Barban-Vehov and Graham related to Selberg's sieve. We give effective and non-effective versions of the result.

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?authuser=0

Cristhian Garay (CIMAT Guanajuato, Mexico)

Classical valuation theory has proved to be a valuable tool in number theory, algebraic geometry and singularity theory. For example, one can enrich spectra of rings with new points coming from valuations defined on them and taking values in totally ordered abelian groups.

Totally ordered groups are examples of idempotent semirings, and generalized valuations appear when we replace totally ordered abelian groups with more general idempotent semirings. An important example of idempotent semiring is the tropical semifield.

As an application of this set of ideas, we show how to associate an idempotent version of the structure sheaf of a scheme, which behaves particularly well with respect to idempotization of closed subschemes.

This is a joint work with Félix Baril Boudreau.

In recent years, the problem of optimal transport has received significant attention in statistics and machine learning due to its powerful geometric properties. In this talk, we introduce the optimal transport problem and present concrete applications of this theory in statistics. In particular, we will propose a general framework for distribution-free nonparametric testing in multi-dimensions, based on a notion of "multivariate ranks" defined using the theory of optimal transport. We demonstrate the applicability of this approach by constructing exactly distribution-free tests for testing the equality of two multivariate distributions. We investigate the consistency and asymptotic distributions of these tests, both under the null and local contiguous alternatives. We further study their local power and asymptotic (Pitman) efficiency, and show that a subclass of these tests achieve attractive efficiency lower bounds that mimic the classical efficiency results of Hodges and Lehmann (1956) and Chernoff and Savage (1958).

In 1896, the prime number theorem was established, showing that π(x) ∼ li(x). Perhaps the most widely used estimates in explicit analytic number theory are bounds on |π(x)-li(x)| or the related error term |θ(x)-x|. In this talk we discuss methods one can use to obtain good bounds on these error terms when x is large. Moreover, we will explore the many ways in which these bounds could be improved in the future.

Antonella Perucca (University of Luxembourg, Luxembourg)

Kummer theory is a classical theory about radical extensions of fields in the case where suitable roots of unity are present in the base field. Motivated by problems close to Artin's primitive root conjecture, we have investigated the degree of families of general Kummer extensions of number fields, providing parametric closed formulas. We present a series of papers that are in part joint work with Christophe Debry, Fritz Hörmann, Pietro Sgobba, and Sebastiano Tronto.

Since Alan Turing's pioneering publication on morphogenetic pattern formation obtained with reaction-diffusion (RD) systems, it has been the prevailing belief that two-component reaction diffusion systems have to include a fast diffusing inhibiting component (inhibitor) and a much slower diffusing activating component (activator) in order to break symmetry from a uniform steady-state. This time-scale separation is often unbiological for cell signal transduction pathways.
We modify the traditional RD paradigm by considering nonlinear reaction kinetics only inside compartments with reactive boundary conditions to the extra-compartmental space that provides a two-species diffusive coupling. The construction of a nonlinear algebraic system for all existing steady-states enables us to derive a globally coupled matrix eigenvalue problem for the growth rates of eigenperturbations from the symmetric steady-state, on finite domains in 1-D and 2-D and a periodically extended version in 1-D.

We show that the membrane reaction rate ratio of inhibitor rate to activator rate is a key bifurcation parameter leading to robust symmetry-breaking of the compartments. Illustrated with Gierer-Meinhardt, FitzHugh-Nagumo and Rauch-Millonas intra-compartmental reaction kinetics, our compartmental-reaction diffusion system does not require diffusion of inhibitor and activator on vastly different time scales.
Our results elucidate a possible mechanism of the ubiquitous biological cell specialization observed in nature.