# Mathematics

## Density functional theory and multi-marginal optimal transport: Introduction

Density functional theory (DFT) is one of the workhorses of quantum chemistry and material science. In principle, the joint probability of finding a specific electron configuration in a material is governed by a Schrödinger wave equation. But numerically computing this joint probability is computationally infeasible, due to the complexity scaling exponentially in the number of electrons. DFT aims to circumvent this difficulty by focusing on the marginal probability of one electron. In the last decade, a connection was found between DFT and a multi-marginal optimal transport problem with a repulsive cost. I will give a brief introduction to this topic, including some open problems, and recent progress.

## Bounds on the Number of Solutions to Thue Equations

In 1909, Thue proved that when $F(x,y)$ is an irreducible, homogeneous, polynomial with integer coefficients and degree at least 3, the inequality $\left\| F(x,y) \right\| \leq h$ has finitely many integer-pair solutions for any positive $h$. Because of this result, the inequality $\left\| F(x,y) \right\| \leq h$ is known as Thue’s Inequality. Much work has been done to find sharp bounds on the size and number of integer-pair solutions to Thue’s Inequality, with Mueller and Schmidt initiating the modern approach to this problem in the 1980s. In this talk, I will describe different techniques used by Akhtari and Bengoechea; Baker; Mueller and Schmidt; Saradha and Sharma; and Thomas to make progress on this general problem. After that, I will discuss some improvements that can be made to a counting technique used in association with “the gap principle” and how those improvements lead to better bounds on the number of solutions to Thue’s Inequality.

## Pro-p Iwahori Invariants

Let $F$ be the field of $p$-adic numbers (or, more generally, a non-

archimedean local field) and let $G$ be $\mathrm{GL}_n(F)$ (or, more generally,

the group of $F$-points of a split connected reductive group). In the

framework of the local Langlands program, one is interested in studying

certain classes of representations of $G$ (and hopefully in trying to match

them with certain classes of representations of local Galois groups).

In this talk, we are going to focus on the category of smooth representations

of $G$ over a field $k$. An important tool to investigate this category is

given by the functor that, to each smooth representation $V$, attaches its

subspace of invariant vectors $V^I$ with respect to a fixed compact open

subgroup $I$ of $G$. The output of this functor is actually not just a $k$-

vector space, but a module over a certain Hecke algebra. The question we are

going to attempt to answer is: how much information does this functor preserve

or, in other words, how far is it from being an equivalence of categories? We

are going to focus, in particular, on the case that the characteristic of $k$

is equal to the residue characteristic of $F$ and $I$ is a specific subgroup

called "pro-$p$ Iwahori subgroup".

- Read more about Pro-p Iwahori Invariants
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## Zeros of linear combinations of Dirichlet L-functions on the critical line

Fix $N\geq 1$ and let $L_1, L_2, \ldots, L_N$ be Dirichlet L-functions with distinct, primitive and even Dirichlet characters. We assume that these functions satisfy the same functional equation. Let $F(s)∶= c_1L_1(s)+c_2L_2(s)+\ldots+c_NL_N(s)$ be a linear combination of these functions ($c_j \in\mathbb{R}^*$ are distinct). $F$ is known to have two kinds of zeros: trivial ones, and non-trivial zeros which are confined in a vertical strip. We denote the number of non-trivial zeros $\rho$ with $\mathfrak{I}(\rho)\leq T$ by $N(T)$, and we let $N_\theta(T)$ be the number of these zeros that are on the critical line. At the end of the 90's, Selberg proved that this linear combination had a positive proportion of zeros on the critical line, by showing that $\kappa F∶=\lim \inf T (N_\theta(2T)−N_\theta(T))/(N(2T)−N(T))\geq c/N^2$ for some $c>0$. Our goal is to provide an explicit value for $c$, and also to improve the lower bound above by showing that $\kappa_F \geq 2.16\times 10^{-6}/(N \log N)$, for any large enough $N$.

## Zeros of linear combinations of Dirichlet L-functions on the critical line

Fix $N\geq 1$ and let $L_1, L_2, \ldots, L_N$ be Dirichlet L-functions with distinct, primitive and even Dirichlet characters. We assume that these functions satisfy the same functional equation. Let $F(s)∶= c_1L_1(s)+c_2L_2(s)+\ldots+c_NL_N(s)$ be a linear combination of these functions ($c_j \in\mathbb{R}^*$ are distinct). $F$ is known to have two kinds of zeros: trivial ones, and non-trivial zeros which are confined in a vertical strip. We denote the number of non-trivial zeros $\rho$ with $\frac{F}(\rho)$\leq T$ by $N(T)$, and we let $N_\theta(T)$ be the number of these zeros that are on the critical line. At the end of the 90's, Selberg proved that this linear combination had a positive proportion of zeros on the critical line, by showing that $\kappa F∶=\lim \inf T (N_\theta(2T)−N_\theta(T))/(N(2T)−N(T))\geq c/N^2$ for some $c>0$. Our goal is to provide an explicit value for $c$, and also to improve the lower bound above by showing that $\kappa F \geq 2.16\times 10^{-6}/(N \log N)$, for any large enough $N$.

## The fourth moment of quadratic Dirichlet L-functions

I will discuss the fourth moment of quadratic Dirichlet L-functions where we prove an asymptotic formula with four main terms unconditionally. Previously, the asymptotic formula was established with the leading main term under generalized Riemann hypothesis. This work is based on Li's recent work on the second moment of quadratic twists of modular L-functions. It is joint work with Joshua Stucky.

## Analogues of the Hilbert Irreducibility Theorem for integral points on surfaces

We will discuss conjectures and results regarding the Hilbert

Property, a generalization of Hilbert's irreducibility theorem to arbitrary

algebraic varieties. In particular, we will explain how to use conic fibrations

to prove the Hilbert Property for the integral points on certain surfaces,

such as affine cubic surfaces.

## On extremal orthogonal arrays

An orthogonal array with parameters \((N,n,q,t)\) (\(OA(N,n,q,t)\) for short) is an \(N\times n\) matrix with entries from the alphabet \(\{1,2,...,q\}\) such that in any of its \(t\) columns, all possible row vectors of length \(t\) occur equally often. Rao showed the following lower bound on \(N\) for \(OA(N,n,q,2e)\):

\[ N\geq \sum_{k=0}^e \binom{n}{k}(q-1)^k, \]

and an orthogonal array is said to be complete or tight if it achieves equality in this bound. It is known by Delsarte (1973) that for complete orthogonal arrays \(OA(N,n,q,2e)\), the number of Hamming distances between distinct two rows is \(e\). One of the classical problems is to classify complete orthogonal arrays.

We call an orthogonal array \(OA(N,n,q,2e-1)\) extremal if the number of Hamming distances between distinct two rows is \(e\). In this talk, we review the classification problem of complete orthogonal arrays with our contribution to the case \(t=4\) and show how to extend it to extremal orthogonal arrays. Moreover, we give a result for extremal orthogonal arrays which is a counterpart of a result in block designs by Ionin and Shrikhande in 1993.

## Interactions between topology and algebra: advances in algebraic K-theory

The field of algebraic topology has exposed deep connections between topology and algebra. One example of such a connection comes from algebraic K-theory. Algebraic K-theory is an invariant of rings, defined using tools from topology, that has important applications to algebraic geometry, number theory, and geometric topology. Algebraic K-groups are difficult to compute, but advances in algebraic topology have led to many recent computations which were previously intractable. In this talk I will introduce algebraic K-theory and its applications, and discuss recent advances in this field.

## Primes in arithmetic progressions to smooth moduli

The twin prime conjecture asserts that there are infinitely many primes p for which p+2 is also prime. This conjecture appears far out of reach of current mathematical techniques. However, in 2013 Zhang achieved a breakthrough, showing that there exists some positive integer h for which p and p+h are both prime infinitely often. Equidistribution estimates for primes in arithmetic progressions to smooth moduli were a key ingredient of his work. In this talk, I will sketch what role these estimates play in proofs of bounded gaps between primes. I will also show how a refinement of the q-van der Corput method can be used to improve on equidistribution estimates of the Polymath project for primes in APs to smooth moduli.