## North Tower, Grand Ballroom Level

All plenary and public lectures will take place in the Grand Ballroom

### 5 December, 2022

##### Abstract

Over the last forty years, most progress in four-dimensional topology came from gauge theory and related invariants. Khovanov homology is an invariant of knots in R^3 of a different kind: its construction is combinatorial, and connected to ideas from representation theory. There is hope that it can tell us more about smooth 4-manifolds; for example, Freedman, Gompf, Morrison and Walker suggested a strategy to disprove the 4D Poincare conjecture using Rasmussen’s invariant from Khovanov homology. It is yet unclear whether their strategy can work. I will explain a new attempt to pursue it (joint work with Lisa Piccirillo) and some of the challenges we encountered. I will also review other topological applications of Khovanov homology, with regard to smoothly embedded surfaces in 4-manifolds.

##### Abstract

The COVID-19 pandemic has made some ideas in infectious disease modelling household names: $R_0$, $R_t$ and herd immunity were very much in the public eye for many months. But the pandemic also raised new challenges for infectious disease modelling. Much of ““classical”” infectious disease modelling focused on setting up a model, determining its (usually local asymptotic) behaviour, and discovering how its basic reproduction number depended on its parameters. But in the era of the pandemic, we need models that can help us interpret data in real time, that can cope with heterogeneity, that are suitable for modelling feasible actions at a range of scales, and we need models that can incorporate data describing viral diversity. Some of the challenges call for juxtapositions of mechanistic models and statistical models, describing the distributions of variables we can observe. In this talk, I will describe new modelling and estimation approaches that we have developed in this context. First, I will introduce ““eventR””, which is like a basic reproduction number for a specific event, and outline how it can be used to help compare strategies for preventing transmission. Next, I will describe a mechanistic-statistical model with which we can estimate a key parameter for any infectious disease simulation: the per unit time, per contact transmission rate, as long as we know enough about the transmission setting. I will link these two through an analysis of the cluster size distributions in schools in four Canadian provinces in 2021, and will describe and interpret our findings about the transmission rates. Finally, I will discuss the broader challenges for infectious disease modelling in this pandemic and beyond.

### 6 December, 2022

##### Abstract

Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and they give rise to a variety of questions pertaining to analysis, Partial Differential Equations and probability. We will first review these motivations, then present the ‘‘mean-field’’ derivation of effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order behavior, giving information on the configurations at the microscopic level and connecting with crystallization questions, and finish with the description of the effect of temperature.

##### Abstract

Deep generative models have achieved enormous success in learning the underlying high-dimensional data distribution from samples. In this talk, we will introduce two methods to learn deep generative models. First, we will introduce variational gradient flow (VGrow) which can be used to minimize the f-divergence between the evolving distribution and the target distribution. In particular, we showed that the commonly used logD-trick indeed belongs to f-divergence. Second, we will introduce a Schrödinger Bridge approach to learning deep generative models. Our theoretical results guarantee that the distribution learned by our approach converges to the target distribution. Experimental results on multimodal synthetic data and benchmark data support our theoretical findings and indicate that the generative model via Schrödinger Bridge is comparable with state-of-the-art GANs, suggesting a new formulation of generative learning. We demonstrate its usefulness in image interpolation and image inpainting.

##### Abstract

Some of the most remarkable achievements of science are the mind-boggling measurements it has contributed throughout history to our collective understanding of the universe. For example, a simple google search will reveal that the radius of our galaxy is 52850 light years, and that the Andromeda galaxy is 2.537 million light years away. You can also find out that the mass of an electron is 0.00000000000000000000000000091093837 grams. Sadly, I don’t think I will have time to discuss these great numbers in this lecture. However, I will mention a few simpler, but still great measurements in history and how they involved calculating with precision. I will also reflect on what our knowledge of such matters tells us about the nature of mathematics as human heritage.

### 7 December, 2022

##### Abstract

I will present some recent results on rigidity of group actions and flows that come from looking at induced actions on “boundaries at infinity”. This idea has a long history, going back to Selberg and Mostow. In some of my recent joint work, I use this approach in geometric group theory and dynamics, studying boundary rigidity for actions of hyperbolic groups on their Gromov boundary, and showing Anosov flows on 3-manifolds are determined by their periodic orbits. This talk will survey some of the history, philosophy and techniques related to proving rigidity results by looking out to infinity.

##### Abstract

Kazhdan’s property (T) for groups has a number of applications in pure and applied mathematics. I will report the recent development by several hands on the heavily computer assisted methods of proving property (T) (with math rigor), which eventually confirmed property (T) for $\operatorname{Aut}(F_n)$, $n>3$, thus solving a well-known problem in geometric group theory. I then talk about my recent human effort in coping with the computer assisted proof.

### 8 December, 2022

##### Abstract

In this talk I will explain how to use techniques from dynamical systems theory to study group theoretical properties of diffeomorphisms. A special discussion will be deserved to an old unsolved question raised by H. rosenberg, namely the path connectedness of the space of commuting circle diffeomorphisms. I will present several results on this, the most recent of which have been obtained in collaboration with Hélène Eynard.

##### Abstract

In the 70´s Coxeter considered the 4-dimensional regular polytopes and used the so-called Petrie Polygons to obtain quotients of the polytopes that, while having all possible rotational symmetry, lack reflectional symmetry. He called these objects Twisted Honeycombs. Nowadays, objects with such symmetry properties are often called chiral. In this talk I will review Coxeter´s twisted honeycombs and explore a natural way to extend Coxeter´s work to polytope-like structures, in particular, we shall see some properties and examples of so-called chiral skeletal polyhedra, and an application of them.

##### Abstract

Mathematics is a jungle: a jubilee of flowering plants and mysterious animals. The mathematician is a naturalist, describing the behaviours of whatever she encounters on her tours. With the advent of computers and modern technologies, the dispatches of the naturalist can now be illustrated in full colour. In my own travels, I’ve collected some illustrations that I hope will surprise and delight you. I’ll show you – in technicolour – some of the lesser-known antics of the inhabitants of the mathematical realm, including complex and rational numbers and the roots of polynomials.