Video Content by Date

Abstract:

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.

Abstract:

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
Abstract:

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.

Abstract:

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.

Abstract:

If A and B are two geometric progressions, we characterize all 3-
term arithmetic progressions in the sumset A+B. Somewhat surprisingly, while
mostly elementary, this appears to require quite deep machinery from
Diophantine Approximation.

Mar, 4: Primes in arithmetic progressions to smooth moduli
Speaker: Julia Stadlmann
Abstract:

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: A shifted convolution problem arising from physics
Speaker: Kim Klinger-Logan
Abstract:

Certain eigenvalue problems involving the invariant Laplacian on
moduli spaces have potential applications to scattering problems in physics.
Green, Russo, Vanhove, et al., discovered the behavior of gravitons
(hypothetical particles of gravity represented by massless string states) is
also closely related to eigenvalue problems for the Laplace-Beltrami operator
on various moduli spaces. In this talk we will examine applications and
results related to solutions $(\Delta - \lambda) f = E_aE_b$ on
$SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})/SO_2(\mathbb{R})$, where $E_s$ is
a non-holomorphic Eisenstein series on $GL(2)$ and $\Delta = y^2(\partial_x^2+
\partial_y^2)$. One such interesting finding from this work is a family of
identities relating convolution sums of divisor functions to Fourier
coefficients on modular forms. This work is in collaboration with Ksenia
Fedosova, Stephen D. Miller, Danylo Radchenko, and Don Zagier.

Abstract:

The Farey sequence FQ of order Q is an ascending sequence of fractions a/b in the unit interval (0,1] such that (a,b)=1 and 0

Abstract:

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.

Feb, 14: Hilbert Class Fields and Embedding Problems
Speaker: Abbas Maarefparvar
Abstract:

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.

Abstract:

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.

Abstract:

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.

Abstract:

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.

Abstract:

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.

Abstract:

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.

Abstract:

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
Abstract:

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
Abstract:

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
Abstract:

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
Abstract:

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.

Abstract:

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
Abstract:

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.