Pittsburgh Number Theory Day

 Spring 2024 number theory day

Spring 2024 number theory day: Thursday, April 18, 2024

Speakers: Katherine Stange (Colorado-Boulder) and Frank Thorne (South Carolina)

Spring 2024 seminar details: location, schedule, abstracts

Thursday, April 18, 2024

Talks and coffee in room 7218, Wean Hall, Carnegie-Mellon University campus

Speakers: Katherine Stange (Colorado-Boulder) and Frank Thorne (South Carolina)

10:00am-11:00am: Katherine Stange's talk

Title: The local-global conjecture for Apollonian circle packings is false

Abstract: Primitive integral Apollonian circle packings are fractal arrangements of tangent circles with integer curvatures.  The curvatures form an orbit of a 'thin group,' a subgroup of an algebraic group having infinite index in its Zariski closure.  The curvatures that appear must fall into one of six or eight residue classes modulo 24. The twenty-year old local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture.  This is joint work with Summer Haag, Clyde Kertzer, and James Rickards.  Time permitting, I will discuss some new results, joint with Rickards, that extend these phenomena to certain settings in the study of continued fractions.

11:00am-11:30am: Coffee break

11:30am-12:30pm: Frank Thorne's talk

Title: Fourier analysis in arithmetic statistics

Abstract: "Arithmetic statistics" is all about counting arithmetic objects, and it's been the subject of a great deal of recent and ongoing work.

I will give an overview of some of what's been happening in this field recently. Most talks on the subject emphasize the underlying algebraic constructions or the use of the "geometry of numbers", and I will approach the topic from a different angle – and explain how Fourier analysis can be incredibly helpful.

Past number theory days

 Fall 2023 number theory day

 Fall 2023 number theory day: Thursday, September 14, 2023

Speakers: John Bergdall (University of Arkansas) and Ila Varma (University of Toronto)

 Fall 2023 seminar details: location, schedule, abstracts

 Thursday, September 14, 2023 

Talks and coffee in room 703 (7th floor), Thackeray Hall, University of Pittsburgh campus

1:30pm-2:30pm: Ila Varma's talk

Title: Counting number fields and predicting asymptotics

Abstract: A guiding question in number theory, specifically in arithmetic statistics, is: Fix a degree n and a Galois group G in Sn. How does the count of number fields of degree n whose normal closure has Galois group G grow as their discriminants tend to infinity? In this talk, we will discuss the history of this question and take a closer look at the story in the case that n = 4, i.e. the counts of quartic fields. 

2:30pm-3:00pm: Coffee break

3:00pm-4:00pm: John Bergdall's talk

Title: P-adic slope distributions for modular forms

Abstract: The Sato--Tate conjecture predicts statistical properties of point counts on elliptic curves. Its proof over the rationals relies on elliptic curves being modular. In the modular world, there is a wider class of distributions to investigate. For instance, Serre and Conrey--Duke--Farmer established (c. 2000) the distribution of fixed Hecke eigenvalues in modular weights growing to infinity.

This seminar will survey non-Archimedean analogues of these distributions. We first describe the p-adic slope problem for modular eigenforms. Then, we survey the past 20-30 years of research on slopes. One aim is Liu, Truong, Xiao, and Zhao's recent results on the "ghost conjecture" of myself and Pollack. Among its by-products, we learn about p-adic distributions in weight aspects, as previously predicted by Gouvêa.