In this talk, I will explain how to apply the Fock-Goncharov construction to the representation theory of a class of algebras introduced by Etingof, Oblomkov and Rains.

CANCELLED due to an unforeseen speaker emergency.

I will present a simple but curious observation on the zeros of centered stationary Gaussian processes (SGP) on $\mathbb{R}$. The object of interest is
$N_T$ which is the number of zeros in the interval $[0,T]$. We restrict our attention to SGP with compactly supported spectral measure $\mu$. Let $A > 0$ be the smallest number such that supp$(\mu) \subseteq [-A, A]$.

Our primary interest is in the probability of overcrowding (resp. under crowding) event, which is the event that there is an excess (resp. deficit) of zeros in $[0,T]$ compared to the expected number, which is proportional to $T$. Comparing a couple of known results, we observe that there is a change in the behaviour of the probability $\p(N_T \geq \eta T)$, as $\eta$ varies. We show that there is indeed a \textit{sharp transition}. That is, this probability is at least of the order of $\exp(-C_{\eta}T)$ for small $\eta$, and at most of order $\exp(-c_{\eta}T^2)$ for large $\eta$. We identify the critical $\eta$ where this transition happens to be $\eta_c = A/\pi$. We also prove a similar result for under crowding probability when supp$(\mu)$ has a gap at the origin.

This talk is based on a joint work with Naomi Feldheim $\&$ Ohad Feldheim.

Diverse equilibrium systems with heterogeneous interactions lie at the edge of stability. Such marginally stable states are dynamically selected as the most abundant ones or as those with the largest basins of attraction. On the other hand, systems with non-reciprocal (or asymmetric) interactions are inherently out of equilibrium, and exhibit a rich variety of steady states, including fixed points, limit cycles and chaotic trajectories. How are steady states dynamically selected away from equilibrium? We address this question in a simple neural network model, with a tunable level of non-reciprocity. Our study reveals different types of ordered phases and it shows how non-equilibrium steady states are selected in each phase. In the spin-glass region, the system exhibits marginally stable behaviour for reciprocal (or symmetric) interactions and it smoothly transitions to chaotic dynamics, as the non-reciprocity (or asymmetry) in the couplings increases. Such region, on the other hand, shrinks and eventually disappears when couplings become anti-symmetric.

Vlasov-Poisson type systems are well established kinetic models for dilute plasma. The precise structure of the model differs according to which species of charged particle (electrons or ions) it describes, with the most well known version of the system describing electrons. The model for ions, however, has been studied only more recently, owing to an additional exponential nonlinearity in the equation for the electrostatic potential that creates several mathematical difficulties not encountered in the electron case.

Quantitative stability estimates in Wasserstein distances have played a crucial role in the understanding of equations of Vlasov type. I will discuss recent developments in Wasserstein estimates in the context of the ionic Vlasov-Poisson system, including applications to the well-posedness theory, the quasineutral limit (in which the Debye length tends to zero) and the derivation of the equation from a particle system.

Based on joint works with Mikaela Iacobelli (ETH Zürich).

I will discuss a novel construction of field theories based on the idea that one has only a half boson degree of freedom per lattice site. Basically, instead of having a pair of canonical conjugate commuting variables at each site, one has only one degree of freedom and the non-trivial commutators arise from connections to the nearest neighbors. The construction is very similar to staggered fermions and naturally produces gapless systems with interesting topological properties. When considering gauging discrete translations on the phase space in one dimensional examples, one gets interesting critical spin chains, examples of which include the critical Ising model in a transverse magnetic field and the 3-state Potts model at criticality. I will explain how these staggered boson variables are very natural for describing non-invertible symmetries.

These non-invertible symmetries are useful to describe the critical properties of these non-trivial spin chains.

Models in higher dimensions obtained this way can automatically produce dynamical systems of gapless fractons.

I will outline the recent proof of the (global, unramified) geometric Langlands conjecture, obtained in collaboration with Arinkin, Chen, Gaitsgory, Faergeman, Lin, Raskin and Rozenblyum.

The talk is aimed at non-specialists: in particular, I will highlight some key geometric ingredients that might be useful in other situations.

CANCELLED due to an unforeseen speaker emergency.

We study the distribution of singular values for the product of random matrices related to the analysis of deep neural networks. The matrices are similar to the product of sample covariance matrices of statistics, but an important difference is that in statistics the population covariance matrices are assumed to be non-random or random but independent of the random data matrix, while now they are certain functions of the random data matrices (matrices of synaptic weights in the terminology of deep neural networks). The problem was treated recently by J. Pennington et al. assuming that the weight matrices are Gaussian and using the methods of free probability theory. Since, however, free probability theory deals with population covariance matrices that do not depend on data matrices, its applicability to this case must be justified. We use a version of the random matrix theory technique to prove the results of J. Pennington et al. in the general case where the entries of weight matrices are independent identically distributed random variables with zero mean and finite fourth
moment. This, in particular, extends the property of the so-called macroscopic universality to the random matrices in question.