Scientific

The fourth moment of quadratic Dirichlet L-functions

Speaker: 
Quanli Shen
Date: 
Mon, Mar 18, 2024 to Thu, Apr 18, 2024
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
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.

Class: 

Analogues of the Hilbert Irreducibility Theorem for integral points on surfaces

Speaker: 
Simone Coccia
Date: 
Thu, Mar 14, 2024
Location: 
PIMS, University of British Columbia
Zoom
Online
Conference: 
UBC Number Theory Seminar
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.

Class: 

Arithmetic progressions in sumsets of geometric progressions

Speaker: 
Michael Bennett
Date: 
Thu, Mar 7, 2024
Location: 
PIMS, University of British Columbia
Online
Zoom
Conference: 
UBC Number Theory Seminar
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.

Class: 

A shifted convolution problem arising from physics

Speaker: 
Kim Klinger-Logan
Date: 
Thu, Feb 29, 2024
Location: 
PIMS, University of British Columbia
Online
Zoom
Conference: 
UBC Number Theory Seminar
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.

Class: 

A Discrete Mean Value of the Riemann Zeta Function and its Derivatives

Speaker: 
Ertan Elma
Date: 
Thu, Feb 15, 2024
Location: 
PIMS, University of British Columbia
Online
Zoom
Conference: 
UBC Number Theory Seminar
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.

Class: 

On extremal orthogonal arrays

Speaker: 
Sho Suda
Date: 
Wed, Mar 13, 2024
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
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.

Class: 

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

Speaker: 
Teena Gerhardt
Date: 
Fri, Mar 8, 2024
Location: 
PIMS, University of Regina
Zoom
Conference: 
University of Regina PIMS Distinguished Lecture
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.

Class: 
Subject: 

Primes in arithmetic progressions to smooth moduli

Speaker: 
Julia Stadlmann
Date: 
Mon, Mar 4, 2024
Location: 
PIMS, University of Lethbridge
Zoom
Online
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
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.

Class: 

L-functions in Analytic Number Theory: Biitu

Speaker: 
Bittu
Date: 
Mon, Feb 26, 2024
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
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

Class: 
Subject: 

Hilbert Class Fields and Embedding Problems

Speaker: 
Abbas Maarefparvar
Date: 
Wed, Feb 14, 2024
Location: 
PIMS, University of Lethbridge
Zoom
Online
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
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.

Class: 

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