This is a meeting of the COW seminar:

http://cow.alggeo.xyz/nextmeeting.html

1:00pm Laura Schaposnik (University of Illinois Chicago/Oxford): A map of Higgs bundles, generalized hyperpolygons, and more

2:30pm Hal Schenck (Auburn/Oxford): Dynamical systems meet algebraic geometry: From Kuramoto oscillators to Segre varieties

4:00pm Liana Heuberger (Bath): Combinatorial Reid's recipe for consistent dimer models

For a weakly nonlinear classical system, the kinetic equation for waves governs the evolution of the occupation number of a given wavevector. It is like the Boltzmann equation, but for waves instead of particles. As has been known for half a century, in addition to thermal equilibrium, the kinetic equation has another stationary solution: a turbulent state, describing a cascade of energy. Wave turbulence is observed in a wide range of physical contexts, most notably in surface gravity waves in the ocean. Higher order terms in the kinetic equation, going beyond leading order in the nonlinearity, have never been computed. We describe a method, based on quantum field theory, for computing such terms. We show that higher order terms can exhibit UV divergences. We sum the most divergent diagrams (bubble diagrams) to derive a kind of renormalized kinetic equation. Based on 2203.08168, 2212.02555, and work in progress with G. Falkovich.

If you are planning to attend, please send and email to pietro.benetti_genolini@kcl.ac.uk or alan.rios_fukelman@kcl.ac.uk so your name is added to the participants list in order to grant you access to the building.

Matrix models offer non-perturbative descriptions of quantum gravity in simple settings, allowing us to study large-N dualities between gauge and string theories. However, the large-N expansions of matrix models lead to divergent series, only defined as asymptotic series. By fine-tuning the couplings of the matrix model we obtain models of pure gravity coupled to minimal conformal field theories. The free energy for the simplest of these "minimal models" is 2d gravity also admits an asymptotic expansion which formally satisfies the Painlevé I equation.

These asymptotic properties are connected to the existence of exponentially small contributions not captured by a perturbative analysis and whose physical interpretation can be elusive. The emerging structure can be accurately described by means of a resurgent transseries, capturing this perturbative/non-perturbative connection and its consequences. This talk will focus on the essential role of this resurgent transseries for the cases of Painlevé I and the quartic matrix model: together with exponentially accurate numerical and summation methods, one can show how to go beyond the asymptotic results and obtain (analytically and numerically) non-perturbative data.

If you are planning to attend, please send and email to pietro.benetti_genolini@kcl.ac.uk or alan.rios_fukelman@kcl.ac.uk so your name is added to the participants list in order to grant you access to the building.

The study of the classical theta operator was key to Edixhoven's proof of the weight part of Serre's modularity conjecture. Since then, a lot of work has been devoted to extending the construction of this operator to other Shimura varieties, with an eye towards generalisations of Serre's conjecture.
My goal is to give an overview of a family of generalised theta operators, on certain unitary Shimura varieties, that I constructed in my thesis and studied in subsequent work.

TBA.
If you are planning to attend, please send and email to pietro.benetti_genolini@kcl.ac.uk or alan.rios_fukelman@kcl.ac.uk so your name is added to the participants list in order to grant you access to the building.

Dualities involving ensembles of theories represent a fascinating class of holographic correspondences. The inclusion of wormholes into a theory naturally motivates the study of ensembles, but doing so leads to many puzzles from the viewpoint of string theory. In this talk, I will discuss holographic dualities involving ensembles of 2D Narain conformal field theories. The bulk dual of such an ensemble is an Abelian Chern-Simons theory, but features a sum over 3D geometries. Generalizations of this correspondence lead to emergent ensemble symmetries – global symmetries that appear after averaging over the ensemble and are the vestiges of T-duality in the CFT. These symmetries are intimately related to the anyon data and 0-form symmetries of the bulk Chern-Simons theories. I will also discuss the relation of these emergent global symmetries with recent ideas in quantum gravity, and furthermore generally discuss the role of ensemble averaging in standard holography, the landscape, and the swampland.

If you are planning to attend, please send and email to pietro.benetti_genolini@kcl.ac.uk or alan.rios_fukelman@kcl.ac.uk so your name is added to the participants list in order to grant you access to the building.

The superconformal index of N=4 super Yang-Mills theory on a three-sphere is captured by a unitary matrix model with purely double trace operators in the action. The AdS/CFT correspondence predicts that this index should have exponential growth at large charges and large N, corresponding to the 1/16-BPS black hole (BH) in AdS5. I will show how the matrix model gives rise to this expected BH growth as well as an infinite number of new phases. In particular, I will introduce a deformation of the matrix model which allows us to solve it at large N. The deformation has interesting relations with the Bloch-Wigner dilogarithm, a function introduced by number theorists.
I will then show how this matrix model can be expressed in terms of a system of free fermions in a certain ensemble. Integrating out the fermions and averaging over the ensemble leads to a convergent expansion as a series of determinants, showing how giant gravitons in the dual AdS5 are encoded in the gauge theory.
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If you are planning to attend, please send and email to pietro.benetti_genolini@kcl.ac.uk or alan.rios_fukelman@kcl.ac.uk so your name is added to the participants list in order to grant you access to the building.