Tensor categories have played an important role in areas as diverse as topology, mathematical physics, operator algebras, and representation theory.
This is an introductory talk. I will mostly talk about the classification of tensor categories with given tensor product rules and module categories for certain important examples.

Deep learning's success has revealed a number of phenomena that appear to conflict with classical intuitions in the fields of optimization and statistics.  First, the objective functions formulated in deep learning are highly nonconvex but are typically amenable to minimization with first-order optimization methods like gradient descent.  And second, neural networks trained by gradient descent are capable of 'benign overfitting': they can achieve zero training error on noisy training data and simultaneously generalize well to unseen data.  In this talk we go over our recent work towards understanding these phenomena. 

The Hilbert transform $H$ is a basic example of Fourier multipliers.  Its behaviour on Fourier series is the following:

$$
\sum_{n\in \mathbb{Z}}a_n e^{inx} \longmapsto \sum_{n\in \mathbb{Z}}m(n)a_n e^{inx},
$$
with $m(n)=-i\,{\rm sgn} (n)$.
Riesz proved that $H$ is a bounded operator on $L_p(\mathbb{T})$ for all $1<p<\infty$.
We study  Hilbert transform type Fourier multipliers on group algebras and their boundedness on corresponding non-commutative $L_p$ spaces.
The pioneering work in this direction is due to Mei and Ricard who proved $L_p$-boundedness of Hilbert transforms on free group von Neumann algebras using a Cotlar identity. In this talk, we introduce a generalised Cotlar identity and a new geometric form of Hilbert transform for groups acting on tree-like structures. This class of groups includes amalgamated free products, HNN extensions, left orderable groups and many others.

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Joint work with Adri\'an Gonz\'alez and Javier Parcet.

Differential forms are basic objects of manifolds and encode invariants. This talk will motivate the usefulness of considering pairs of differential forms together with a map linking them. We will show how this can lead to novel functionals and geometric flows. As an application, it leads to new notions of flat connections and Morse theory on symplectic manifolds. This talk is based on joint works with Jiawei Zhou, David Clausen and Xiang Tang.

The purpose of this talk is to illustrate an application of the powerful machinery of geometric measure theory to a conjecture in Gromov–Witten theory arising from physics. Very roughly speaking, the Gromov–Witten invariants of a symplectic manifold (X,ω) equipped with a tamed almost complex structure J are obtained by counting pseudo-holomorphic maps from mildly singular Riemann surfaces into (X,J). It turns out that Gromov–Witten invariants are quite complicated (or “have a rich internal structure”). This is true especially for if (X,ω) is a symplectic Calabi–Yau 3–fold (that is: dim X = 6, c_1(X,ω) = 0). In 1998, using arguments from M–theory, Gopakumar and Vafa argued that there are integer BPS invariants of symplectic Calabi–Yau 3–folds. Unfortunately, they did not give a direct mathematical definition of their BPS invariants, but they predicted that they are related to the Gromov–Witten invariants by a transformation of the generating series. The Gopakumar–Vafa conjecture asserts that if one defines the BPS invariants indirectly through this procedure, then they satisfy an integrality and a (genus) finiteness condition. The integrality conjecture has been resolved by Ionel and Parker. A key innovation of their proof is the introduction of the cluster formalism: an ingenious device to side-step questions regarding multiple covers and super-rigidity. Their argument could not resolve the finiteness conjecture, however. The reason for this is that it relies on Gromov’s compactness theorem for pseudo-holomorphic maps which requires an a priori genus bound. It turns out, however, that Gromov’s compactness theorem can (and should!) be replaced with the work of Federer–Flemming, Allard, and De Lellis–Spadaro–Spolaor. This upgrade of Ionel and Parker’s cluster formalism proves both the integrality and finiteness conjecture. This talk is based on joint work with Eleny Ionel and Aleksander Doan.

If we choose at random an integral binary form $f(x, z)$ of fixed degree $d$, what is the probability that the superelliptic curve with equation  $C \colon: y^m = f(x, z)$ has a $p$-adic point, or better, points everywhere locally? In joint work with Lea Beneish, we show that the proportion of forms $f(x, z)$ for which $C$ is everywhere locally soluble is positive, given by a product of local densities. By studying these local densities, we produce bounds which are suitable enough to pass to the large $d$ limit. In the specific case of curves of the form $y^3 = f(x, z)$ for a binary form of degree 6, we determine the probability of everywhere local solubility to be 96.94%, with the exact value given by an explicit infinite product of rational function expressions.

[pre-talk at 1:20PM]

One of the major tasks that a cell faces during its lifecycle is how to spatially localize its components. Correct spatial organization of various proteins (cellular polarity) is fundamental not only for the correct cell shape but also to carry out essential cellular functions, such as the spatial coordination of cell division. We present a mathematical model of the core mechanism responsible for the regulation of polarized growth dynamics in the model organism, fission yeast. The model is based on the competition of growth zones of polarity protein Cdc42 localized at the cell tips for a common substrate (inactive Cdc42) that diffuses in the cytosol. To explore the underlying mechanism for oscillations and the effect of diffusion and noise, we consider three model frameworks including a 1D deterministic model, a 2D deterministic model, and a stochastic model. We simulate and analyze these models using numerical bifurcation tools, PDEs, and stochastic simulation algorithms.

Many fundamental biophysical processes, from cell division to cellular motility, involve dynamics of thin structures immersed in a very viscous fluid. Various popular models have been developed to describe this interaction mathematically, but much of our understanding of these models is only at the level of numerics and formal asymptotics. Here we seek to develop the PDE theory of filament hydrodynamics.

First, we propose a PDE framework for analyzing the error introduced by slender body theory (SBT), a common approximation used to facilitate computational simulations of immersed filaments in 3D. Given data prescribed only along a 1D curve, we develop a novel type of boundary value problem and obtain an error estimate for SBT in terms of the fiber radius. This places slender body theory on firm theoretical footing.

Second, we consider a classical elastohydrodynamic model for the motion of an immersed inextensible filament. We highlight how the analysis can help to better understand undulatory swimming at a low Reynolds numbers. This includes the development of a novel numerical method to simulate inextensible swimmers.