# Video Content by Date

**May, 23**: Introduction to unbalanced optimal transport and its efficient computational solutions

Speaker:

*Laetitia Chapel*

Optimal transport operates on empirical distributions which may contain acquisition artifacts, such as outliers or noise, thereby hindering a robust calculation of the OT map. Additionally, it necessitates equal mass between the two distributions, which can be overly restrictive in certain machine learning or computer vision applications where distributions may have arbitrary masses, or when only a fraction of the total mass needs to be transported. Unbalanced Optimal Transport addresses the issue of rebalancing or removing some mass from the problem by relaxing the marginal conditions. Consequently, it is often considered to be more robust, to some extent, against these artifacts compared to its standard balanced counterpart. In this presentation, I will review several divergences for relaxing the marginals, ranging from vertical divergences like the Kullback-Leibler or the L2-norm, which allow for the removal of some mass, to horizontal ones, enabling a more robust formulation by redistributing the mass between the source and target distributions. Additionally, I will discuss efficient algorithms that do not necessitate additional regularization on the OT plan.

**Apr, 18**: Generic Representations and ABV packets for p-adic Groups

Speaker:

*Sarah Dijols*

After a brief introduction on the theory of p-adic groups complex representations, I will explain why tempered and generic Langlands parameters are open. I will further derive a number of consequences, in particular for the enhanced genericity conjecture of Shahidi and its analogue in terms of ABV packets. This is a joint work with Clifton Cunningham, Andrew Fiori, and Qing Zhang.

**Apr, 11**: Hypergeometric functions through the arithmetic kaleidoscope

Speaker:

*Ling Long*

The classical theory of hypergeometric functions, developed by generations of mathematicians including Gauss, Kummer, and Riemann, has been used substantially in the ensuing years within number theory, geometry, and the intersection thereof. In more recent decades, these classical ideas have been translated from the complex setting into the finite field and p

-adic settings as well.

In this talk, we will give a friendly introduction to hypergeometric functions, especially in the context of number theory.

**Apr, 10**: Bounds on the Number of Solutions to Thue Equations

Speaker:

*Greg Knapp*

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.

**Apr, 4**: Density functional theory and multi-marginal optimal transport: Introduction

Speaker:

*Yair Shenfeld*

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.

**Mar, 25**: Zeros of linear combinations of Dirichlet L-functions on the critical line

Speaker:

*Jérémy Dousselin*

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 $N$.

**Mar, 25**: Zeros of linear combinations of Dirichlet L-functions on the critical line

Speaker:

*Jérémy Dousselin*

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 $N$.

**Mar, 21**: Pro-p Iwahori Invariants

Speaker:

*Emanuele Bodon*

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".

**Mar, 18**: The fourth moment of quadratic Dirichlet L-functions

Speaker:

*Quanli Shen*

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.

**Mar, 14**: Analogues of the Hilbert Irreducibility Theorem for integral points on surfaces

Speaker:

*Simone Coccia*

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.

**Mar, 13**: On extremal orthogonal arrays

Speaker:

*Sho Suda*

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.

**Mar, 8**: Interactions between topology and algebra: advances in algebraic K-theory

Speaker:

*Teena Gerhardt*

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.

**Mar, 4**: Primes in arithmetic progressions to smooth moduli

Speaker:

*Julia Stadlmann*

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.

**Feb, 29**: The Distribution of Logarithmic Derivatives of Quadratic L-functions in Positive Characteristic

Speaker:

*Félix Baril Boudreau*

To each square-free monic polynomial $D$ in a fixed polynomial ring $\mathbb{F}_q[t]$, we can associate a real quadratic character $\chi_D$, and then a Dirichlet $L$-function $L(s,\chi_D)$. We compute the limiting distribution of the family of values $L'(1,\chi_D)/L(1,\chi_D)$ as $D$ runs through the square-free monic polynomials of $\mathbb{F}_q[t]$ and establish that this distribution has a smooth density function. Time permitting, we discuss connections of this result with Euler-Kronecker constants and ideal class groups of quadratic extensions. This is joint work with Amir Akbary.

**Feb, 26**: L-functions in Analytic Number Theory: Biitu

Speaker:

*Bittu*

**Feb, 14**: Hilbert Class Fields and Embedding Problems

Speaker:

*Abbas Maarefparvar*

The class number one problem is one of the central subjects in algebraic number theory that turns back to the time of Gauss. This problem has led to the classical embedding problem which asks whether or not any number field $K$ can be embedded in a finite extension $L$ with class number one. Although Golod and Shafarevich gave a counterexample for the classical embedding problem, yet one may ask about the embedding in 'Polya fields', a special generalization of class number one number fields. The latter is the 'new embedding problem' investigated by Leriche in 2014. In this talk, I briefly review some well-known results in the literature on the embedding problems. Then, I will present the 'relativized' version of the new embedding problem studied in a joint work with Ali Rajaei.

**Feb, 9**: Consecutive sums of two squares in arithmetic progressions

Speaker:

*Vivian Kuperberg*

In 2000, Shiu proved that there are infinitely many primes whose last digit is 1 such that the next prime also ends in a 1. However, it is an open problem to show that there are infinitely many primes ending in 1 such that the next prime ends in 3. In this talk, we'll instead consider the sequence of sums of two squares in increasing order. In particular, we'll show that there are infinitely many sums of two squares ending in 1 such that the next sum of two squares ends in 3. We'll show further that all patterns of length 3 occur infinitely often: for any modulus q, every sequence (a mod q, b mod q, c mod q) appears infinitely often among consecutive sums of two squares. We'll discuss some of the proof techniques, and explain why they fail for primes. Joint work with Noam Kimmel.

**Feb, 8**: A Conjecture of Mazur predicting the growth of Mordell--Weil ranks in Z_p-extensions

Speaker:

*Debanjana Kundu*

Let \(p\) be an odd prime. We study Mazur's conjecture on the growth of the Mordell--Weil ranks of an elliptic curve \(E/\mathbb{Q}\) over an imaginary quadratic field in which \(p\) splits and \(E\) has good reduction at \(p\). In particular, we obtain criteria that may be checked through explicit calculation, thus allowing for the verification of Mazur's conjecture in specific examples. This is joint work with Rylan Gajek-Leonard, Jeffrey Hatley, and Antonio Lei.

**Feb, 8**: Gromov-Wasserstein Alignment: Statistical and Computational Advancements via Duality

Speaker:

*Ziv Goldfeld*

The Gromov-Wasserstein (GW) distance quantifies dissimilarity between metric measure (mm) spaces and provides a natural alignment between them. As such, it serves as a figure of merit for applications involving alignment of heterogeneous datasets, including object matching, single-cell genomics, and matching language models. While various heuristic methods for approximately evaluating the GW distance from data have been developed, formal guarantees for such approaches—both statistical and computational—remained elusive. This work closes these gaps for the quadratic GW distance between Euclidean mm spaces of different dimensions. At the core of our proofs is a novel dual representation of the GW problem as an infimum of certain optimal transportation problems. The dual form enables deriving, for the first time, sharp empirical convergence rates for the GW distance by providing matching upper and lower bounds. For computational tractability, we consider the entropically regularized GW distance. We derive bounds on the entropic approximation gap, establish sufficient conditions for convexity of the objective, and devise efficient algorithms with global convergence guarantees. These advancements facilitate principled estimation and inference methods for GW alignment problems, that are efficiently computable via the said algorithms.

**Feb, 7**: Moments of higher derivatives related to Dirichlet L-functions

Speaker:

*Samprit Ghosh*

The distribution of values of Dirichlet L-functions \(L(s, \chi)\) for variable \(χ\) has been studied extensively and has a vast literature. Moments of higher derivatives has been studied as well, by Soundarajan, Sono, Heath-Brown etc. However, the study of the same for the logarithmic derivative \(L'(s, \chi)/ L(s, \chi)\) is much more recent and was initiated by Ihara, Murty etc. In this talk we will discuss higher derivatives of the logarithmic derivative and present some new results related to their distribution and moments at s=1.

**Feb, 1**: Collision of orbits under the action of a Drinfeld module

Speaker:

*Dragos Ghioca*

We present various results and conjectures regarding unlikely intersections of orbits for families of Drinfeld modules. Our questions are motivated by the groundbreaking result of Masser and Zannier (from 15 years ago) regarding torsion points in algebraic families of elliptic curves.

**Jan, 31**: A discrete mean value of the Riemann zeta function and its derivatives

Speaker:

*Ertan Elma*

In this talk, we will discuss an estimate for a discrete mean value of the Riemann zeta function and its derivatives multiplied by Dirichlet polynomials. Assuming the Riemann Hypothesis, we obtain a lower bound for the 2kth moment of all the derivatives of the Riemann zeta function evaluated at its nontrivial zeros. This is based on a joint work with Kübra Benli and Nathan Ng.

**Jan, 29**: Fourier optimization and the least quadratic non-residue

Speaker:

*Emily Quesada-Herrera*

We will explore how a Fourier optimization framework may be used to study two classical problems in number theory involving Dirichlet characters: The problem of estimating the least character non-residue; and the problem of estimating the least prime in an arithmetic progression. In particular, we show how this Fourier framework leads to subtle, but conceptually interesting, improvements on the best current asymptotic bounds under the Generalized Riemann Hypothesis, given by Lamzouri, Li, and Soundararajan. Based on joint work with Emanuel Carneiro, Micah Milinovich, and Antonio Ramos.

**Jan, 25**: Sums of proper divisors with missing digits

Speaker:

*Kübra Benli*

Let $s(n)$ denote the sum of proper divisors of a positive integer $n$. In 1992, Erdös, Granville, Pomerance, and Spiro conjectured that if \(\square\) is a set of integers with asymptotic density zero then the preimage set \(s^{−1}(\square)\) also has asymptotic density zero. In this talk, we will discuss the verification of this conjecture when \(\square\) is the set of integers with missing digits (also known as ellipsephic integers) by giving a quantitative estimate on the size of the set \(s^{-1}(\square)\). This talk is based on the joint work with Giulia Cesana, C\'{e}cile Dartyge, Charlotte Dombrowsky and Lola Thompson.

**Jan, 24**: Projective Planes and Hadamard Matrices

Speaker:

*Hadi Kharaghani*

It is conjectured that there is no projective plane of order 12. Balanced splittable Hadamard matrices were introduced in 2018. In 2023, it was shown that a projective plane of order 12 is equivalent to a balanced multi-splittable Hadamard matrix of order 144. There will be an attempt to show the equivalence in a way that may require little background.

**Jan, 22**: Mean values of long Dirichlet polynomials

Speaker:

*Winston Heap*

We discuss the role of long Dirichlet polynomials in number theory. We first survey some applications of mean values of long Dirichlet polynomials over primes in the theory of the Riemann zeta function which includes central limit theorems and pair correlation of zeros. We then give some examples showing how, on assuming the Riemann Hypothesis, one can compute asymptotics for such mean values without using the Hardy-Littlewood conjectures for additive correlations of the von-Mangoldt functions.

**Jan, 18**: Equidistribution of some families of short exponential sums

Speaker:

*Théo Untrau*

Exponential sums play a role in many different problems in number theory. For instance, Gauss sums are at the heart of some early proofs of the quadratic reciprocity law, while Kloosterman sums are involved in the study of modular and automorphic forms. Another example of application of exponential sums is the circle method, an analytic approach to problems involving the enumeration of integer solutions to certain equations. In many cases, obtaining upper bounds on the modulus of these sums allow us to draw conclusions, but once the modulus has been bounded, it is natural to ask the question of the distribution of exponential sums in the region of the complex plane in which they live. After a brief overview of the motivations mentioned above, I will present some results obtained with Emmanuel Kowalski on the equidistribution of exponential sums indexed by the roots modulo p of a polynomial with integer coefficients.

**Jan, 16**: Explicit bounds for $\zeta$ and a new zero free region

Speaker:

*Chiara Bellotti*

In this talk, we prove that |ζ(σ+it)|≤ 70.7 |t|4.438(1-σ)^{3/2} log2/3|t| for 1/2≤ σ ≤ 1 and |t| ≥ 3, combining new explicit bounds for the Vinogradov integral with exponential sum estimates. As a consequence, we improve the explicit zero-free region for ζ(s), showing that ζ(σ+it) has no zeros in the region σ ≥ 1-1/(53.989 (log|t|)2/3(log log|t|)1/3) for |t| ≥ 3.