# Video Content by Date

### 2022

**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, 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.