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.

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 d 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 F given by the vanishing of r linearly independent homogeneous polynomials of degree d 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.

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.

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.

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.