# Scientific

## Equivalences of Categories of Modules Over Quantum Groups and Vertex Algebras

Vertex operator algebras are the symmetry algebras of two dimensional conformal field theory. In a famous series of papers, Kazhdan and Lusztig proved an equivalence between particular semisimple categories of modules over affine Lie algebras and quantum groups, the former of which can also be realized as modules over a corresponding vertex operator algebra. Such equivalences between representation categories of vertex operator algebras and quantum groups are now broadly referred to as the Kazhdan-Lusztig correspondence. There has been substantial research interest over the last two decades in understanding the Kazhdan-Lusztig correspondence for vertex operator algebras with non-semisimple representation theory. In this talk I will present an overview of this research area and discuss recent results and future directions.

## Using data-driven stochastic lattice models to improve the representation of convection and clouds in Climate Models

Stochastic parameterizations (SMCM) are continuously providing promising simulations of unresolved atmospheric processes for global climate models (GCMs). One of the features of earlier SMCM is to mimic the life cycle of the three most common cloud types (congestus, deep, and stratiform) in tropical convective systems. In this present study, a new cloud type, namely shallow cloud, is included along with the existing three cloud types to make the model more realistic. Further, the cloud population statistics of four cloud types (shallow, congestus, deep, and stratiform) are taken from Indian (Mandhardev) radar observations. A Bayesian inference technique is used here to generate key time scale parameters required for the SMCM as SMCM is most sensitive to these time scale parameters as reported in many earlier studies. An attempt has been made here for better representing organized convection in GCMs, the SMCM parameterization is adopted in one of the state-of-art GCMs namely the Climate Forecast System version 2 (CFSv2) in lieu of the pre-existing simplified Arakawa–Schubert (default) cumulus scheme and has shown important improvements in key large-scale features of tropical convection such as intra-seasonal wave disturbances, cloud statistics, and rainfall variability. This study also shows the need for further calibration the SMCM with rigorous observations for the betterment of the model's performance in short term weather and climate scale predictions.

## Regularity of the cohomological equation for circle rotations

Given an smooth function h, this talk will focus on solving the equation \psi(Rz)-\psi(z) = h(z) for circle rotations. We will see how the Diophantine condition on the rotation implies smooth solutions.

## An invitation to the algebraic geometry over idempotent semirings - lecture 2

Idempotent semi-rings have been relevant in several branches of applied mathematics, like formal languages and combinatorial optimization.

They were brought recently to pure mathematics thanks to its link with tropical geometry, which is a relatively new branch of mathematics that has been useful in solving some problems and conjectures in classical algebraic geometry.

However, up to now we do not have a proper algebraic formalization of what could be called “Tropical Algebraic Geometry”, which is expected to be the geometry arising from idempotent semi-rings.

In this mini-course we aim to motivate the necessity for such theory, and we recast some old constructions in order theory in terms of commutative algebra of semi-rings and modules over them.

### Mini-Course

This lecture is the second part of a mini-course, please see also

## An invitation to the algebraic geometry over idempotent semirings - Lecture 1

Idempotent semi-rings have been relevant in several branches of applied mathematics, like formal languages and combinatorial optimization.

They were brought recently to pure mathematics thanks to its link with tropical geometry, which is a relatively new branch of mathematics that has been useful in solving some problems and conjectures in classical algebraic geometry.

However, up to now we do not have a proper algebraic formalization of what could be called “Tropical Algebraic Geometry”, which is expected to be the geometry arising from idempotent semi-rings.

In this mini-course we aim to motivate the necessity for such theory, and we recast some old constructions in order theory in terms of commutative algebra of semi-rings and modules over them.

### Mini-Course

This lecture is the first part of a mini-course, please see also

## Generalized valuations and idempotization of schemes

**Cristhian Garay (CIMAT Guanajuato, Mexico)**

Classical valuation theory has proved to be a valuable tool in number theory, algebraic geometry and singularity theory. For example, one can enrich spectra of rings with new points coming from valuations defined on them and taking values in totally ordered abelian groups.

Totally ordered groups are examples of idempotent semirings, and generalized valuations appear when we replace totally ordered abelian groups with more general idempotent semirings. An important example of idempotent semiring is the tropical semifield.

As an application of this set of ideas, we show how to associate an idempotent version of the structure sheaf of a scheme, which behaves particularly well with respect to idempotization of closed subschemes.

This is a joint work with Félix Baril Boudreau.

## Free boundary regularity for the obstacle problem

The classical obstacle problem consists of finding the equilibrium position of an elastic membrane whose boundary is held fixed and which is constrained to lie above a given obstacle. By classical results of Caffarelli, the free boundary is smooth outside a set of singular points. However, explicit examples show that the singular set could be, in general, as large as the regular set. This talk aims to introduce this beautiful problem and describe some classical and recent results on the regularity of the free boundary.

Speaker Biography: Alessio Figalli is a leading figure in the areas of Optimal Transport, partial differential equations and the calculus of variations. He received his Ph.D. from the Scuola Normale Superiore di Pisa and the Ecole Normale Superieur de Lyon and has held positions in Paris and Austin, Texas. He is currently a Professor at ETH Zurich. His work has been recognized with many awards including the Prize of the European Mathematical Society in 2012 and the Fields Medal in 2018.

## Opinion Dynamics and Spreading Processes on Networks

People interact with each other in social and communication networks, which affect the processes that occur on them. In this talk, I will give an introduction to dynamical proceses on networks. I will focus my discussion on opinion dynamics, and I will also discuss coupled opinion and disease dynamics on networks. Time-permitting, I may also briefly discuss a model of COVID-19 that centers on disabled people and their caregivers.

## Expansion, divisibility and parity

**Harald Andrés Helfgott University of Göttingen, Germany, and Institut de Mathématiques de Jussieu, France)**

We will discuss a graph that encodes the divisibility properties of integers by primes. We prove that this graph has a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the traditional parity barrier, by combining our result with Matomaki-Radziwill. For instance: for $\lambda$ the Liouville function (that is, the completely multiplicative function with $\lambda(p) = -1$ for every prime), $(1/\log x) \sum_{n\leq x} \lambda(n) \lambda(n+1)/n = O(1/\sqrt(\log \log x))$, which is stronger than well-known results by Tao and Tao-Teravainen. We also manage to prove, for example, that $\lambda(n+1)$ averages to $0$ at almost all scales when n restricted to have a specific number of prime divisors $\Omega(n)=k$, for any "popular" value of $k$ (that is, $k = \log \log N + O(\sqrt(\log \log N))$ for $n \leq N$).

For the Full abstract, please see: https://www.cs.uleth.ca/~nathanng/ntcoseminar/

## Forgotten conjectures of Andrews for Nahm-type sums

**Joshua Males (University of Manitoba, Canada)**

In his famous '86 paper, Andrews made several conjectures on the function σ(q) of Ramanujan, including that it has coefficients (which count certain partition-theoretic objects) whose sup grows in absolute value, and that it has infinitely many Fourier coefficients that vanish. These conjectures were famously proved by Andrews-Dyson-Hickerson in their '88 Invent. paper, and the function σ has been related to the arithmetic of Z[6–√]by Cohen (and extensions by Zwegers), and is an important first example of quantum modular forms introduced by Zagier.

A closer inspection of Andrews' '86 paper reveals several more functions that have been a little left in the shadow of their sibling σ , but which also exhibit extraordinary behaviour. In an ongoing project with Folsom, Rolen, and Storzer, we study the function v1(q) which is given by a Nahm-type sum and whose coefficients count certain differences of partition-theoretic objects. We give explanations of four conjectures made by Andrews on v1, which require a blend of novel and well-known techniques, and reveal that v1 should be intimately linked to the arithmetic of the imaginary quadratic field Q[−3−−−√]

.