# Mathematics

## L-functions in Analytic Number Theory: Biitu

## Hilbert Class Fields and Embedding Problems

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.

## Moments of higher derivatives related to Dirichlet L-functions

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.

## Consecutive sums of two squares in arithmetic progressions

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.

## A Conjecture of Mazur predicting the growth of Mordell--Weil ranks in Z_p-extensions

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.

## Collision of orbits under the action of a Drinfeld module

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.

## Gromov-Wasserstein Alignment: Statistical and Computational Advancements via Duality

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.

## A discrete mean value of the Riemann zeta function and its derivatives

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.

## Projective Planes and Hadamard Matrices

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.

## Fourier optimization and the least quadratic non-residue

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.