Southern California Number Theory Day

Speaker: 

Francesc Castella, Nadia Heninger, Kyle Pratt, Carl Wang-Erickson

Institution: 

UCSB, UCSD, BYU, University of Pittsburgh

Time: 

Saturday, October 7, 2023 - 10:00am to 5:15pm

Location: 

NS II 1201

 

 

Southern California Number Theory Day at UC Irvine

        Saturday, October 7th, 2023

SPEAKERS:

Francesc Castella (UCSB)

Nadia Heninger (UCSD)

Kyle Pratt (BYU)

Carl Wang-Erickson (University of Pittsburgh) 

LOCATION: UC Irvine, Natural Sciences II room 1201

The first lecture will begin at 10:00, and the last will end around 5:15.  There will be a dinner after the lectures, details TBA.  More information is available on the conference web page:

 https://www.math.uci.edu/~nckaplan/scntd23.html

which will be updated when the schedule information and talk titles are available.

LIGHTNING TALKS: We are planning a session with LIGHTNING TALKS where number theory graduate students and postdocs are invited to present their research. These talks will be approximately 5-10 minutes.  If you would like to give a lightning talk, please contact Nathan Kaplan by September 15th. Please include your name, affiliation, advisor's name (if you are a graduate student), talk title, and brief abstract.

There are no fees (except for the dinner), but we need to know how many people to plan for, so please register using the link on the conference web page. Please email Nathan Kaplan if you have any questions.

Optimization problems in analytic number theory: Low-lying zeros of L-functions

Speaker: 

Andres Chirre

Institution: 

University of Rochester

Time: 

Thursday, June 1, 2023 - 3:00pm to 4:00pm

Location: 

https://uci.zoom.us/j/98672180147

In this talk, we will talk about some optimization problems related to the Riemann zeta function and $L$-functions. In particular, we will talk about the distribution of the low-lying zeros of families of $L$-functions. We will see how we can use the one-level density theorems in the literature to estimate the proportion of non-vanishing of $L$-functions in a family at a low-lying height on the critical line. This is based on joint work with E. Carneiro and M. B. Milinovich. 

Number of points of algebraic sets over finite fields

Speaker: 

Sudhir Ghorpade

Institution: 

Indian Institute of Technology, Bombay

Time: 

Thursday, April 20, 2023 - 3:00pm to 4:00pm

Location: 

RH 306

Let F be a finite field with q elements. A (projective) algebraic set over F is the set of common zeros in the projective m-space over F of a bunch of homogeneous polynomials in m+1 variables with coefficients in F. Fix positive integers r, m and with d < q.  We consider the following question:

What is the maximum number of points in an algebraic set in the projective m-s[space over given by the vanishing of linearly independent homogeneous polynomials of degree with coefficients in F?

The case of a single homogeneous polynomial (or in geometric terms, a projective hypersurface) corresponds to a classical inequality proved by Serre in 1989. For the general case, an elaborate conjecture was made by Tsfasman and Boguslavsky, which was open for almost two decades. Recently significant progress in this direction has been made, and it is shown that while the Tsfasman-Boguslavsky Conjecture is true in certain cases, it can be false in general. Some new conjectures have also been proposed. We will give a motivated outline of these developments. If there is time and interest, we will also explain the close connections of these questions to the problem of counting points of sections of Veronese varieties by linear subvarieties of a fixed dimension, and also to coding theory.

This talk is mainly based on joint works with Mrinmoy Datta and with Peter Beelen and Mrinmoy Datta.

Configuration spaces and applications in arithmetic statistics

Speaker: 

Anh Hoang Trong Nam

Institution: 

University of Minnesota

Time: 

Tuesday, April 4, 2023 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

In the last dozen years, topological methods have been shown to produce a new pathway to study arithmetic statistics over function fields, most notably in Ellenberg-Venkatesh-Westerland's work on the Cohen-Lenstra conjecture. More recently, Ellenberg, Tran and Westerland proved the upper bound in Malle's conjecture over function fields by studying the twisted homology of configuration spaces. In this talk, we will give an overview of their framework and extend their techniques to study other questions in arithmetic statistics. As an example, we will demonstrate how this extension can be used to study the asymptotic average of the quadratic character of the resultant of polynomials over finite fields, answering a question of Ellenberg-Shusterman.

About the joint moments of the Riemann zeta function and its logarithmic derivative

Speaker: 

Alessandro Fazzari

Institution: 

American institute of Math

Time: 

Thursday, March 2, 2023 - 10:00am to 11:00am

Location: 

RH 510R

Abstract:  We will discuss classical statistics for the Riemann zeta function when the averages are tilted by powers of zeta on the critical line. In particular, we will focus on the weighted statistics for the non-trivial zeros of the Riemann zeta function, blending together the theory of moments and that of n-th level density. This weighted approach allows for a better understanding of the interplay between zeros and large values of zeta.

Moments of the Hurwitz zeta function on the critical line

Speaker: 

Anurag Sahay

Institution: 

University of Rochester

Time: 

Thursday, February 2, 2023 - 10:00am to 11:00am

Location: 

RH 306

The Hurwitz zeta function is a shifted integer analogue of the Riemann zeta function, for shift parameters $0<\alpha\leqslant 1$. We consider the integral moments of the Hurwitz zeta function on the critical line $\Re(s)=\tfrac12$. We will focus on rational shift parameters. In this case, the Hurwitz zeta function decomposes as a linear combination of Dirichlet $L$-functions, which leads us into investigating moments of products of $L$-functions. Using heuristics from random matrix theory, we conjecture an asymptotic of the same form as the moments of the Riemann zeta function. If time permits, we will discuss the case of irrational shift parameters, which will include some joint work with Winston Heap and Trevor Wooley and some ongoing work with Heap.

Counting numerical semigroups by Frobenius number, multiplicity, and depth

Speaker: 

Sean Li

Institution: 

MIT

Time: 

Thursday, January 26, 2023 - 10:00am to 11:00am

Location: 

Zoom: https://uci.zoom.us/j/94729574163

A numerical semigroup Λ is a subset of the nonnegative integers which contains 0, has finite complement, and is closed under addition. We characterize Λ by a number of invariants: the genus g = |N0 \ Λ|, the multiplicity m = min(Λ \ {0}), and the Frobenius number f = max(N0 \ Λ). Recently, Eliahou and Fromentin introduced the notion of depth q = ⌈(f+1)/q⌉. In 1990, Backelin showed that the number of numerical semigroups with Frobenius number f approaches Ci · 2^(f/2) for constants C0 and C1 depending on the parity of f. In this talk, we use Kunz words and graph homomorphisms to generalize Backelin’s result to numerical semigroups of arbitrary Frobenius number, multiplicity, and depth, in particular showing that there are ⌊(q+1)^2/4⌋^(f/(2q-2)+o(f)) semigroups with Frobenius number f and depth q.

Local solubility in families of superelliptic curves

Speaker: 

Christopher Keyes

Institution: 

Emory University

Time: 

Thursday, January 19, 2023 - 3:30pm to 4:30pm

Location: 

Zoom: https://uci.zoom.us/j/95668199292
If we choose at random an integral binary form $f(x, z)$ of fixed degree $d$, what is the probability that the superelliptic curve with equation $C \colon: y^m = f(x, z)$ has a $p$-adic point, or better, points everywhere locally? In joint work with Lea Beneish, we show that the proportion of forms $f(x, z)$ for which $C$ is everywhere locally soluble is positive, given by a product of local densities. By studying these local densities, we produce bounds which are suitable enough to pass to the large $d$ limit. In the specific case of curves of the form $y^3 = f(x, z)$ for a binary form of degree 6, we determine the probability of everywhere local solubility to be 96.94\%, with the exact value given by an explicit infinite product of rational function expressions.

Twisted $2k$th moments of primitive Dirichlet $L$-functions: beyond the diagonal

Speaker: 

Siegfred Baluyot

Institution: 

AIM

Time: 

Thursday, December 1, 2022 - 3:00pm to 4:00pm

Location: 

RH 306

In this joint work with Caroline Turnage-Butterbaugh, we study the family of Dirichlet $L$-functions of all even primitive characters of conductor at most $Q$, where $Q$ is a parameter tending to infinity. We approximate the twisted $2k$th moment of this family using Dirichlet polynomials of length between $Q$ and $Q^2$. Assuming the Generalized Lindelof Hypothesis, we prove an asymptotic formula for these approximations. Our result agrees with the prediction of Conrey, Farmer, Keating, Rubinstein, and Snaith, and provides the first rigorous evidence beyond the diagonal terms for their conjectured asymptotic formula for the general $2k$th moment of this family. The main device we use in our proof is the asymptotic large sieve developed by Conrey, Iwaniec, and Soundararajan. 

Quadratic twists of modular L-functions

Speaker: 

Xiannan Li

Institution: 

Kansas State University

Time: 

Thursday, January 12, 2023 - 10:00am to 11:00am

Location: 

https://uci.zoom.us/j/95304920920

The behavior of quadratic twists of modular L-functions is at the critical point is related both to coefficients of half integer weight modular forms and data on elliptic curves.  Here we describe a proof of an asymptotic for the second moment of this family of L-functions, previously available conditionally on the Generalized Riemann Hypothesis by the work of Soundararajan and Young.  Our proof depends on deriving an optimal large sieve type bound.

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