Seminars and Colloquia by Series

Stability of the Prime-Omega Function in Shrinking Sectors of Gaussian Integers by Alex Burgin

Series
Number Theory
Time
Wednesday, September 23, 2026 - 15:30 for
Location
Skiles 005
Speaker
Alex Burgin – Georgia Institute of Technology –
We study the number $\Omega(n)$ of Gaussian prime factors of $n$, counted with multiplicity, when $n$ ranges over an angular sector of the Gaussian integers. We prove that, uniformly over the initial angle and over sector widths $\gamma\geq\gamma_N$, where $\gamma_N^{-1}=N^{o(1)}$, the distribution of $\Omega(n)$ is asymptotically shift-invariant. In concrete terms, the total variation between the proportions of Gaussian integers having $k$ and $k+1$ prime factors tends to zero, within a shrinking sector. If time permits, I'll mention some ergodic consequences. Joint work with Christina Giannitsi (Virginia Tech).
 

Some integrals of dynamical Green's functions

Series
Number Theory
Time
Wednesday, April 15, 2026 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Matt Olechnowicz – Concordia University –

Let $f$ be an endomorphism of projective space defined over a number field.  When counting rational points ordered by a certain "canonical" height function attached to $f$, we encounter a mysterious asymptotic constant in the main term.  This constant is a product of local factors over the primes of bad reduction of $f$; and these local factors (which take the form of $v$-adic integrals) are rather difficult to calculate explicitly.  In this talk I will present my partial progress towards evaluating these integrals.  No knowledge of arithmetic dynamics will be assumed.

Remarks on Siegel zeros

Series
Number Theory
Time
Wednesday, April 1, 2026 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jesse Thorner – University of Illinois Urbana-Champaign –

I will present some recent work with Debmalya Basak and Alexandru Zaharescu on potential improvements to the Siegel—Walfisz upper bound on the greatest real zero of a Dirichlet $L$-function.

Smooth forms on graphs and Berkovich curves

Series
Number Theory
Time
Wednesday, March 11, 2026 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Joe Rabinoff – Duke University –

Chambert-Loir and Ducros have introduced a theory of real-valued smooth differential forms on Berkovich spaces that play the role of smooth forms on complex varieties.  We compute the associated Dolbeault cohomology groups of curves by reducing to the case of metric graphs.  I'll introduce smooth forms on graphs, and explain how the theory in CLD has to be modified in order to get finite-dimensional cohomology groups.

Inverse Sieve Problems

Series
Number Theory
Time
Wednesday, March 4, 2026 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Chi Hoi (Kyle) Yip – Georgia Tech –

Many problems in number theory boil down to bounding the size of a set contained in a certain set of residue classes mod $p$ for various sets of primes $p$; and then sieve methods are the primary tools for doing so. Motivated by the inverse Goldbach problem, Green–Harper, Helfgott–Venkatesh, Shao, and Walsh have explored the inverse sieve problem: if we let $S \subseteq [N]$ be a maximal set of integers in this interval where the residue classes mod $p$ occupied by $S$ have some particular pattern for many primes $p$, what can one say about the  structure of the set $S$ beyond just its size? In this talk, I will give a gentle introduction to inverse sieve problems, and present some progress we made when $S$ mod $p$ has rich additive structure for many primes $p$. In particular, in this setting, we provide several improvements on the larger sieve bound for $|S|$, parallel to the work of Green–Harper and Shao for improvements on the large sieve. Joint work with Ernie Croot and Junzhe Mao.

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