Mathematics Department - Number Theory Seminar - Spring 2017

Number Theory Seminar - Spring 2017


Stephen D. Miller, Alex V. Kontorovich, Pedro H Pontes


Past Talks

Tuesday, March 28th

Max Ehrman, Yale University

"Almost Prime Coordinates in Thin Pythagorean Triangles"

Time: 2:00 PM
Location: Hill 525
Abstract: The affine sieve is a technique first developed by Bourgain, Gamburd, and Sarnak in 2006 and later completed by Salehi Golsefidy and Sarnak in 2010 which proves that in a broad class of affine linear actions, polynomial functions of the coordinates saturate. While this works in large generality, the bounds it produces on the saturation number are often far from optimal. Thin orbits of Pythagorean triangles have been of particular interest since the outset of the affine sieve, and I will discuss recent progress on improving bounds for saturation numbers of their hypotenuse, area, and the product of all three coordinates by using Archimedean sieve theory and the dispersion method.

Tuesday, March 21st

Arthur Baragar , UNLV

"Apollonian packings in higher dimensions and ample cones for K3 surfaces"

Time: 2:00 PM
Location: Hill 525
Abstract: The Apollonian circle and sphere packings are well known classical objects. The four dimensional analog, though, has long been thought to not exist, owing to a perceived obstruction. In this talk, we describe the four dimensional analog and the obstruction. Most of the talk will be geometrical, requiring little background, and with only a little emphasis on the connection to number theory. Near the end of the talk (after a brief intermission or Q&A?), we will describe how all these objects can be thought of as ample cones for elliptic K3 surfaces.

Tuesday, March 7th

Chris Skinner, Princeton University

"Elliptic curves of ranks 0 and 1"

Time: 2:00 PM
Location: Hill 525
Abstract: I will describe some of the recent work establishing criteria for an elliptic curve to have rank and analytic rank equal and both either 0 or 1. I will also explain how the proportion of curves satisfying these criteria can be counted.

Tuesday, February 28th

Liyang Zhang , Yale University

"Quantum Unique Ergodicity of degenerate Eisenstein Series on GL(n)"

Time: 2:00 PM
Location: Hill 525
Abstract: We prove quantum unique ergodicity for a subspace of the continuous spectrum spanned by the degenerate Eisenstein Series on GL(n)

Tuesday, February 21st

Shamgar Gurevitch , University of Wisconsin

" `Small' Representations of Finite Groups""

Time: 2:00 PM
Location: Hill 525
Abstract: Suppose you have a finite group G and you want to study certain related structures (e.g., random walks, Cayley graphs, word maps, etc.). In some cases, this might be done using sums over the characters of G. It seems that, in some cases, an obstacle in applying these formulas is lack of control over the low-dimensional representations of G. In fact, numerics shows that the “small" representations tend to contribute the largest terms to these sums, so a systematic knowledge of them might assists in the solution of some interesting problems.

This talk discusses a joint project with Roger Howe (Yale). We introduce a language to speak about “size” of a representation, and we develop a method for systematically construct (conjecturally all the) ''small'' representations of finite classical groups.

We will illustrate our theory with concrete motivations and numerical data obtained with John Cannon (MAGMA, Sydney) and Steve Goldstein (Scientific computing, Madison).

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