Deep neural networks are a class of parameterized functions that have proven remarkably successful at making predictions about unseen data from finite labeled data sets. They do so even in settings when classical intuition suggests that they ought to be overfitting (aka memorizing) the data. I will begin by describing the structure of neural networks and how they learn. I will then advertise one of the theoretical questions animating the field: how does the relationship between the number of parameters and the size of the data set impact the dynamics of how they learn? Along the way I will emphasize the many ways in which topology, geometry, and combinatorics play a role in the field.

This talk will focus on the recent foundational paper by Bayer et al. on Bridgeland stability conditions in families.

The parameter space for any fixed architecture of neural networks serves as a proxy during training for the associated class of functions - but how faithful is this representation? For any fixed feedforward ReLU network architecture with at least one hidden layer, it is well-known that many different parameter settings can determine the same function. It is less well-known that the degree of this redundancy is inhomogeneous across parameter space. This inhomogeneity should impact the dynamics of training via gradient descent, especially when compared with recent work suggesting that gradient descent favors flat minima of the loss landscape. In this talk, I will carefully define the notion of the local functional dimension of a feedforward ReLU network function, discuss the relationship between local functional dimension of a parameter and the geometry of the underlying decomposition of the domain into linear regions, and present some preliminary experimental results on the probability distribution underlying functional dimension at network initialization. Some of this work is joint with Kathryn Lindsey, Rob Meyerhoff, and Chenxi Wu, and some is joint with Kathryn Lindsey and David Rolnick.

Following an eponymous survey by Mathieu Anel, I will describe how classical tangent spaces in algebraic geometry can be generalized to tangent complexes which recover the tangent space for smooth varieties but provide a better linearization for singular spaces (whose classical tangent spaces have too high a dimension). I will restrict my exposition to singular spaces that arise from taking non-transverse intersections or quotients of smooth spaces by non-free group actions. This is intended to serve as a concrete answer to the age-old question “Why should anyone care about derived algebraic geometry?”

I will continue explaining the exact WKB method; I will introduce spectral networks and explain how it governs the Borel summability of the formal WKB series, as well as the Stokes phenomena. Time permitting, I will also explain how the exact WKB method naturally gives an abelianization of the Stokes graded local system, and the cluster structure on the Betti moduli space.

We will discuss several recent results for aggregation-diffusion equations related to partial concentration of the density of particles. Nonlinear diffusions with homogeneous kernels will be reviewed quickly in the case of degenerate diffusions to have a full picture of the problem. Most of the talk will be devoted to discuss the less explored case of fast diffusion with homogeneous kernels with positive powers. We will first concentrate in the case of stationary solutions by looking at minimisers of the associated free energy showing that the minimiser must consist of a regular smooth solution with singularity at the origin plus possibly a partial concentration of the mass at the origin. We will give necessary conditions for this partial mass concentration to and not to happen. We will then look at the related evolution problem and show that for a given confinement potential this concentration happens in infinite time under certain conditions. We will briefly discuss the latest developments when we introduce the aggregation term. This talk is based on a series of works in collaboration with M. Delgadino, J. Dolbeault, A. Fernández, R. Frank, D. Gómez-Castro, F. Hoffmann, M. Lewin, and J. L. Vázquez.

What is this cool new shape that mathematicians are interested in? Is it a hat? A T-shirt? A pair of pants? In this talk I’ll give an introduction to the history behind aperiodic filings, some insight on the geometry of the hat polykite, and discuss the techniques used to prove the aperiodicity of its tiling. Deciding what object the shape most resembles will be left up to the audience.

Initial developments of factorization homology abstract the global observables of perturbative TQFTs, in which global observables are determined by point-observables/operators. In this talk, I will discuss upgrades of factorization homology tailored for a larger class of TQFTs, in which global observables are determined by point- and line-observables/operators. The discussion will be focused on low-dimensions, where, in a precise sense, factorization homology over a 3-manifold M derives the Skein module of M, thereby achieving a local-to-global expression for Skein modules. I will highlight two key technical ingredients: 1) the tangle hypothesis for 1-dimensional tangles; 2) an action of SO(3) on rigid braided-monoidal categories, with ribbon categories as the fixed points. This is all joint work with John Francis.

The cost of a probability measure preserving action of a countable group G on X is an invariant that generalizes the rank (minimal number of generators) of G and measures the “minimal average number of maps” needed to connect every pair of points of X in the same G orbit. This invariant rose to prominence after it was used by Gaboriau to distinguish the orbit equivalence relations of essentially free actions of two free groups on different number of generators. The fixed price conjecture predicts that any two essentially free p.m.p. actions of the group G have the same cost. In my talk I will report on a joint work with Sam Mellick and Amanda Wilkens in which we prove fixed price one for higher ranks lattices in semisimple real or p-adic groups. As a corollary we obtain that the number of generators of index n subgroup of such a group grows like o(n) which implies new state of the art results on the growth of mod-p homology groups. The proof is based on certain miraculous properties of Poisson-Voronoi tessellations of higher rank symmetric spaces that might be of independent interest.