Scientific

As a refinement of Goldfeld’s conjecture, there is a conjecture of Keating–Snaith asserting that $\log L(1/2,E_d)$ for certain quadratic twists $E_d$ of an elliptic curve $E$ behaves like a normal random variable. In light of this, Radziwill and Soundararajan conjectured that the distribution of $\log(|Sha(E_d)|/\sqrt{|d|}$ is approximately Gaussian for these $E_d$, and proved that the conjectures of Keating–Snaith and theirs are both valid “from above”. More recently, under GRH, they further established a lower bound for the involving distribution towards Keating–Snaith’s conjecture. In this talk, we shall discuss the joint distribution of central values and orders of Sha groups of $E_d$ and how to adapt Radziwill–Soundararajan’s methods to study upper bound and lower bounds for such a joint distribution if time allows.

Let $\pi(x; q, a)$ be the number of primes $p\leq x$ such that $p \equiv a (\mod q)$. The classical Shanks–Rényi prime number race problem asks, given positive integers $q \geq 3$ and $2 \leq r \leq \phi(q)$ and distinct reduced residue classes $a_1, a_2, . . . , a_r$ modulo $q$, whether there are infinitely many integers $n$ such that $\pi (n; q, a1) > \pi(n; q, a2) > \cdots > \pi(n; q, ar)$. In this talk, I will describe what is known on this problem when the number of competitors $r \geq 3$, and how this compares to the Chebyshev’s bias case which corresponds to $r = 2$.

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.

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.

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.