No Boundaries Seminar
Description
The No Boundaries Seminar aims to be a venue for people to share insights and understanding from across the mathematical spectrum. We encourage speakers to give predominantly (or even completely) expository talks on a beautiful topic.
 Talks are 50 minutes, followed by a question/answer session.
 Many in the audience will likely be nonexperts.
 Principles are preferred over detailed proofs.
Where and When
 The No Boundaries Seminar will be held on Wednesday at 2pm CDT for the Fall 2021 quarter (location TBD).
 If you would like to join the mailing list, please email tghyde (at) uchicago.edu.
Fall 2021
 October 6th: TBD
 October 13th: Exceptional Lie groups and some related geometry
Bruce Hunt
Abstract: The classification of Lie groups was already given in the 1880s (KillingCartan classification). In additiion to the the known classical types (unitary, orthogonal, symplectic) there are 5 exceptional groups. The interpretation of these is based on the existence of a division algebra of dimension 8 over the reals, the CayleyGraves octionion algebra. The groups occur in many different parts of mathemetics, from the classification of rational surface singularities to compactifications of supergravity. After a quick introduction to the exceptional groups, this talk considers one tiny aspect: some geometry arising from the Weyl group of E_6. In addition to the connection with the 27 lines on a cubic surface some beautiful algebraic varieties occur, and a new discovery will be briefly reported.
 October 20th: TBD
 October 27th: TBD
 November 3rd: TBD
Michael Klug (University of Chicago)
 November 10th: TBA
Laura DeMarco (Harvard University)
 December 1st: TBD
 December 8th: TBD
Organizers
Spring 2021

June 11th: Codes from varieties over finite fields
Nathan Kaplan (University of California, Irvine)
Abstract: There are q^{20} homogeneous cubic polynomials in four variables with coefficients in the finite field F_q. How many of them define a cubic surface with q^2+7q+1 F_qrational points? What about other numbers of rational points? How many of the q^{20} pairs of homogeneous cubic polynomials in three variables define cubic curves that intersect in 9 F_qrational points?
The goal of this talk is to explain how ideas from the theory of errorcorrecting codes can be used to study families of varieties over a fixed finite field. We will not assume any previous familiarity with coding theory. We will start from the basics and emphasize examples.
Video

June 4th: Lehmer's number in geometry and dynamics
Eriko Hironaka (Florida State University, AMS)
Abstract: Lehmer's number is conjectured to be the smallest Salem number, that is,
the smallest algebraic integer all of whose algebraic conjugates lie on or within the
unit circle (with at least one on the unit circle). The number, approximately 1.17628,
appears evocatively in geometric topology, for example, it is the smallest growth rate
of hyperbolic Coxeter reflection groups on the plane, and is the smallest expansion
factor of pseudoAnosov mapping classes on a genus 2 surface. In this talk, we will
give some background on Lehmer's question from 1933 and the special role the number
has played in questions around a possible "gap" between chaos vs periodicity, and
hyperbolicity vs flatness.
Video

May 28th: Products of matrices
Ron Donagi (University of Pennsylvania)
Abstract: Given an ordered set of conjugacy classes C_i, i=1,…,N, of nxn matrices, the DeligneSimpson problem asks whether there are representatives A_i \in C_i whose product is 1. This is equivalent to asking whether there is a rank n local system on P^1 minus N points with monodromy C_i at the ith point. The answer is known if one of the classes C_i is regular, and in various other cases. We will discuss some of what is known, using nonAbelian Hodge theory and middle convolution. We will also describe some unexpected connections between the DeligneSimpson problem and Hitchin systems, motivated by work on super conformal field theories of Class S and 3D mirror symmetry.
Video

May 21st: Cluster transformations
Sergey Fomin (University of Michigan)
Abstract: Cluster transformations are a surprisingly ubiquitous family of algebraic recurrences. They arise in diverse mathematical contexts, from representation theory and enumerative combinatorics to mathematical physics and classical geometry. I will present some of the most basic and concrete examples of cluster transformations, and discuss their remarkable properties such as periodicity, Laurentness, and positivity.
Video

May 7th: Bounding the number of rational points on curves
Joseph H. Silverman (Brown University)
Abstract: The Mordell conjecture, famously proven by Faltings, says that an
algebraic curve of genus at least 2 has only finitely many rational
points. A subsequent alternative proof by Vojta allows one to give an
explicit upper bound for the number of points. In this expository talk I
will explain the many threads that go into Vojta's proof and describe
some very recent advances that give a uniform bound for the number of
rational points that is "moreorless" independent of the curve and
depends only on its genus.
Video

April 30th: Hyperbolic groups and generalizations
Carolyn Abbott (Columbia University)
Abstract: The algebra of a finitely generated group can be encoded in the geometry of its Cayley graph. In this talk, I will introduce the class of hyperbolic groups, which are groups whose Cayley graphs are negatively curved. This class includes free groups and fundamental groups of surfaces of sufficient complexity and hyperbolic 3—manifolds, among many others. In fact, hyperbolic groups are ubiquitous among finitely
presented groups: a random finitely presented group is hyperbolic. I will describe how to use the geometry
of the Cayley graph to derive various nice algebraic properties of these groups, with a focus on algorithmic
properties. I will also discuss various ways to generalize this class of groups.
Video

April 9th: Lie algebras and group theory
Andrew Putman (University of Notre Dame)
Abstract: I will discuss a circle of ideas going back to Magnus and continued by Witt, Malcev, Quillen, Sullivan, and others that show how to find nilpotent Lie groups and algebras that encode deep structural information about finitely generated groups.
Video