L6

Conformal blocks appear in several areas of mathematical physics from random geometry to black hole physics. A probabilistic notion of conformal blocks using gaussian multiplicative chaos measures was recently formulated by Promit Ghosal, Guillaume Remy, Xin Sun, Yi Sun (arxiv:2003.03802). In this talk, I will show that the semiclassical limit of the probabilistic conformal blocks recovers a special case of the elliptic form of Painlevé VI equation, thereby proving a conjecture by Zamolodchikov. This talk is based on an upcoming paper with Promit Ghosal and Andrei Prokhorov.

I will present recent results on the growth of entanglement between two adjacent regions in a tripartite, one-dimensional many-body system after a quantum quench. Combining a replica trick with a space-time duality transformation a universal relation between the entanglement negativity and Renyi-1/2 mutual information can be derived, which holds at times shorter than the sizes of all subsystems. The proof is directly applicable to any local quantum circuit, i.e., any lattice system in discrete time characterised by local interactions, irrespective of the nature of its dynamics. The derivation indicates that such a relation can be directly extended to any system where information spreads with a finite maximal velocity. The talk is based on Phys. Rev. Lett. 129, 140503 (2022).

The Calogero-Painlevé systems were introduced in 2001 by K. Takasaki as a natural generalization of the classical Painlevé equations to the case of the several Painlevé “particles” coupled via the Calogero type interactions. In 2014, I. Rumanov discovered a remarkable fact that a particular case of the Calogero– Painlevé II equation describes the Tracy-Widom distribution function for the general $\beta$-ensembles with the even values of parameter $\beta$. in 2017 work of M. Bertola, M. Cafasso , and V. Rubtsov, it was proven that all Calogero-Painlevé systems are Lax integrable, and hence their solutions admit a Riemann-Hilbert representation. This important observation has opened the door to rigorous asymptotic analysis of the Calogero-Painlevé equations which in turn yields the possibility of rigorous evaluation of the asymptotic behavior of the Tracy-Widom distributions for the values of $\beta$ beyond the classical $\beta =1, 2, 4$. In the talk these recent developments will be outlined with a special focus on the Calogero-Painlevé system corresponding to $\beta = 6$. This is a joint work with Andrei Prokhorov.

Neural networks are the most practically successful class of models in modern machine learning, but there are considerable gaps in the current theoretical understanding of their properties and success. Several authors have applied models and tools from random matrix theory to shed light on a variety of aspects of neural network theory, however the genuine applicability and relevance of these results is in question. Most works rely on modelling assumptions to reduce large, complex matrices (such as the Hessians of neural networks) to something close to a well-understood canonical RMT ensemble to which all the sophisticated machinery of RMT can be applied to yield insights and results. There is experimental work, however, that appears to contradict these assumptions. In this talk, we will explore what can be derived about neural networks starting from RMT assumptions that are much more general than considered by prior work. Our main results start from justifiable assumptions on the local statistics of neural network Hessians and make predictions about their spectra than we can test experimentally on real-world neural networks. Overall, we will argue that familiar ideas from RMT universality are at work in the background, producing practical consequences for modern deep neural networks.

Harish-Chandra's Lefschetz principle suggests that representations of real and p-adic split reductive groups are closely related, even though the methods used to study these groups are quite different. The local Langlands correspondence (as formulated by Vogan) indicates that these representation theoretic relationships stem from geometric relationships between real and p-adic Langlands parameters. In this talk we will discuss how the geometric structure of real and p-adic Langlands parameters lead to functorial relationships between representations of real and p-adic groups. I will describe work in progress which applies this functoriality to the study of unitary representations and signatures of invariant hermitian forms for GL(n). The main result expresses signatures of invariant hermitian forms on graded affine Hecke algebra modules in terms of signature characters of Harish-Chandra modules, which are computable via the unitary algorithm for real reductive groups by Adams-van Leeuwen-Trapa-Vogan.

I will describe the interaction between a single soliton and a gas of solitons, providing for the first time a mathematical justification for the kinetic theory as posited by Zakharov in the 1970s.  Then I will explain how to use random matrix theory to introduce randomness into a large collection of solitons.

We show that the category of vector bundles over the vertices of a locally finite $\tilde{A}_{n-1}$ building $\Delta$, $Vec(\Delta)$, has the structure of a $Tilt(SL_{2k+1})$ module category. This module category is the $q$-analogue of the $Tilt(SL_{2k+1})$ action on vector bundles over the $sl_n$ weight lattice.  Our construction of the $Tilt(SL_{2k+1})$ action on $Vec(\Delta)$ extends to $Vec(\Delta)^{G}$, its equivariantization, which gives us a class of non-standard $Tilt(SL_{2k+1})$ module categories. When $G$ acts simply transitively, this recovers the fiber functors of Jones.

In this talk I explain the fertile relationship between the foundations of inference and learning and combinatorial geometry.

My presentation contains several powerful examples where famous theorems in discrete geometry answered natural  questions from machine learning and statistical inference:

In this tasting tour I will include the problem of deciding the existence of Maximum likelihood estimator in multiclass logistic regression, the variability of behavior of k-means algorithms with distinct random initializations and the shapes of the clusters, and the estimation of the number of samples in chance-constrained optimization models. These obviously only scratch the surface of what one could do with extra free time. Along the way we will see fascinating connections to the coupon collector problem, topological data analysis, measures of separability of data, and to the computation of Tukey centerpoints of data clouds (a high-dimensional generalization of median). All new theorems are joint work with subsets of the following wonderful folks: T. Hogan, D. Oliveros, E. Jaramillo-Rodriguez, and A. Torres-Hernandez.

Two relevant papers published/ to appear are

https://arxiv.org/abs/1907.09698https://arxiv.org/abs/1907.09698

https://arxiv.org/abs/2205.05743https://arxiv.org/abs/2205.05743

A finite W-algebra is a gadget associated to each nilpotent orbit in a complex semisimple Lie algebra g. There is a functor from W-modules to a full subcategory of g-modules, known as Skryabin’s equivalence, and every primitive ideals of the enveloping algebra U(g) as the annihilator of a module obtained in this way. This gives a convenient way of organising together primitive ideals in terms of nilpotent orbits, and this approach has led to a resurgence of interest in some hard open problems which lay dormant for some 20 years. The primitive ideals of U(g) which come from one-dimensional representations of W-algebras are especially nice, and we shall call them Losev—Premet ideals. The goal of this talk is to explain my recent work which seeks to: (1) describe the structure of the space of the dimensional representations of a finite W-algebra and (2) classify the Losev—Premet ideals.

After a brief introduction, I elucidate a technique, dubbed "bootstrap'', which generates an infinite family of D_p(G) theories, where for a given arbitrary group G and a parameter b, each theory in the same family has the same number of mass parameters, same number of marginal deformations, same 1-form symmetry, and same 2-group structure. This technique is utilized to establish the presence or absence of the 2-group symmetries in several classes of D_p(G) theories. I, then, argue that we found the presence of 2-group symmetries in a class of Argyres-Douglas theories, called D_p(USp(2N)), which can be realized by Z_2-twisted compactification of the 6d N=(2,0) of the D-type on a sphere with an irregular twisted puncture and a regular twisted full puncture. I will also discuss the 3d mirror theories of general D_p(USp(2N)) theories that serve as an important tool to study their flavor symmetry and Higgs branch.