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
 We have somewhat sporadic talks planned for the coming semester, so check below for the specific time/place.
 If you would like to join the mailing list, please email michaelklug (at) uchicago.edu.
Organizers
Fall 2023
 October 19th: The Torus Trick
Robion Kirby (UC Berkeley)
(3:00PM  4:00PM in Eckhart 312)
Abstract: The torus trick was the key to existence and uniqueness of triangulations of manifolds long long ago in a place far far away. Little background is needed (less than what can be found in Wikipediasee Alexander's trick).
 October 24th: Vector fields, mapping class groups, and abelian differentials
Aaron Calderon (U Chicago)
(3:30PM  4:30PM in Eckhart 202)
Abstract: Given a vector field on a surface, one would like to understand which mapping classes preserve it(s isotopy class). Despite the fundamental nature of this question, little is known about these “framed mapping class groups.” In this talk I will describe some joint work with Nick Salter in which we gave explicit, finite generating sets for framed mapping class groups, as well as highlight an application to the topology of moduli spaces of abelian differentials.
 October 28th: Vector fields, mapping class groups, and abelian differentials
Corey Bregman (University of Southern Maine)
(2:00PM  3:00PM in Eckhart 206)
Abstract: Bartels, Lück, and Weinberger showed that a torsionfree hyperbolic group G whose boundary is the nsphere (n≥5) is the fundamental group of a closed aspherical manifold. When n=1, this follows from Eckmann's classification of PD(2) groups, and when n=2, this is the Cannon conjecture. Building on work of Markovic, Haïssinsky has shown that the Cannon conjecture does hold when G has sufficiently many quasiconvex codimension1 subgroups. In this talk, we examine the structure of quasiconvex codimension1 subgroups when the boundary of G is the 3sphere, in order to generalize Haïssinsky's result. We show that if G contains a quasiconvex, codimension1 subgroup, then G contains a quasiconvex subgroup whose boundary is either the 2sphere, or a space with prescribed Cech cohomology, which we conjecture is the Pontryagin sphere. This is joint work with M. IncertiMedici.
 November 29th: Resolvent degree for arithmetic groups and variations of Hodge structures
Jesse Wolfson (UC Irvine)
(3:30PM  4:30PM in Ryerson 352)
Abstract: In the 13th of his list of mathematical problems, Hilbert conjectured that the general degree 7 polynomial cannot be solved using only arithmeticoperations and algebraic functions of 2 or fewer variables. In the language of resolvent degree, Hilbert conjectures that RD(S_7) = 3. Reichstein has recently extended the notion of resolvent degree to general algebraic groups G. In this context, a conjecture of Tits asserts that RD(G) = 1 for any connected complex linear algebraic group. Reichstein proves unconditionally that RD(G)\le 5 for such G, and he offers this as possible evidence against Hilbert's conjecture. The goal of this talk is to offer analogous evidence *for* Hilbert's conjecture by extending Reichstein's definition to a notion of resolvent degree for arithmetic groups, variations of Hodge structure, and related moduli problems. We then use geometric techniques to give examples of problems F with RD(F) arbitrarily large. From this perspective, one can paraphrase Hilbert's 13th as asking which is a finite group more like: a connected complex linear algebraic group or an arithmetic lattice? This is joint work with Benson Farb and Mark Kisin.
Spring 2022
 April 12th: LefshetzSmith theory and the fundamental group
Shmuel Weinberger (University of Chicago)
(in Eckhart 312)
Abstract: Any Z/n action on a finite contractible complex has a fixed point, and if n is a prime power the same is true for actions on finite dimensional simplicial complexes. These are classical theorems of Lefshetz and Smith.
I will start by exploring a bit of the icebergs beneath these tips, and continue by investigating the role of the fundamental group in such generalizations. I was surprised by the results, but perhaps you won't be.
 May 10th: Rational points and spaces of holomorphic maps
Phil Tosteson (University of Chicago)
(in Eckhart 312)
Abstract: I'll talk about the analogy between rational points an algebraic variety (over Q), and the space of holomorphic maps from a curve to an algebraic variety (over C). In particular, I'll discuss conjectures due to Manin, Batyrev, and Peyre on the asymptotic number of rational points on Fano varieties and how these conjectures are related to the asymptotic topology of the space of holomorphic maps.
 May 17th: The Weil Conjectures and A1homotopy theory
Kirsten Wickelgren (Duke University)
(via Zoom)
Abstract: In a celebrated and beautiful paper from 1948, André Weil proposed a striking connection between algebraic topology and the number of solutions to equations over finite fields. This talk will introduce the Weil Conjectures; introduce A1homotopy theory; and then discuss a connection between the two from joint work with Margaret Bilu, Wei Ho, Padma Srinivasan, and Isabel Vogt.
Video
 May 24th: Ranks of elliptic curves
Bjorn Poonen (MIT)
(via Zoom)
Abstract: The simplest algebraic varieties whose rational points are not fully understood are the elliptic curves. I will discuss some things that are known and conjectured about them, including in particular a heuristic due to myself, Jennifer Park, John Voight, and Melanie Matchett Wood that controverts the earlier conventional wisdom.
Video
Preprint
Winter 2022
 March 9th: From braid groups to Artin groups
Ruth Charney (Brandeis University)
(via Zoom)
Abstract: Braid groups can be approached from many different viewpoints, geometrically as mapping class groups or as fundamental groups of hyperplane arrangements, and algebraically as Garside groups. Braid groups belong to a much larger, but less well understood, class of groups known as Artin groups. Recently there has been progress on extending several of these different viewpoints to more general Artin groups. This will be a (mostly) expository talk surveying some of these ideas.
Video
 March 2nd: Bott periodicity, algebrogeometrically
Ravi Vakil (Stanford University)
(in Eckhart 206 *Note change of location*)
Abstract: I will report on joint work with Hannah Larson, and joint work in progress with Jim Bryan, in which we try to make sense of Bott periodicity from a naively algebrogeometric point of view.
 February 23rd: Cohomology of arithmetic groups and geometric cycles
Bena Tshishiku (Brown University)
(in Eckhart 207A)
Abstract: I will discuss a geometric method for constructing nontrivial elements in the cohomology of arithmetic groups like SL(n,Z) that originates in the work of Millson and MillsonRaghunathan in the 1970s. I will also mention some recent results in this area, including an application to the mapping class group of the K3 surface.
 February 16th: On the powers of the Euler class for pure mapping class groups
Rita Jiménez Rolland (Institute of Mathematics of UNAM)
(via Zoom)
Abstract: The mapping class group of an orientable closed surface with one marked point can be identified, by the Nielsen action, with a subgroup of the group of orientationpreserving homeomorphisms of the circle. This inclusion pulls back the “discrete universal Euler class” producing a nonzero class in the second integral cohomology of the mapping class group. In this talk I will present an overview of what is known about the vanishing and nonvanishing behaviour of the powers of this class.
Video
 February 2nd: Local systems on the braid group and arithmetic averages
Jordan Ellenberg (University of WisconsinMadison)
Abstract: I will give a pretty loosely structured talk with lots of questions and no theorems involving topology and arithmetic. For example: if I take two random squarefree coprime polynomials f,g over F_q[t], of degree m and n, one may well expect their resultant Res(f,g) to be a square in F_q^* half the time; how big is the deviation from that expectation? I'll talk about how to set this up as a question about the cohomology of a finiteindex subgroup of the braid group, tell you what Ishan B. has proved about this, explain what it has to do with moments of special values of Lfunctions, and maybe gesture wildly in the direction of perverse sheaves and double covers of FI (these are two different directions.) This will be a sequel to the short talk I gave at a Banff meeting in October, for those who may have been there.
Video
Fall 2021
 December 8th: Complex dynamics and mapping class groups
Dan Margalit (Georgia Tech)
(via Zoom, link will be sent to mailing list)
Abstract: I will discuss recent work with James Belk, Justin Lanier, Lily Li,
Caleb Partin, and Rebecca Winarski on topological aspects of complex
polynomials. A basic problem in the area is the recognition problem,
which is the problem of identifying a polynomial from its topological
description as a branched cover of the complex plane. We will discuss
Hubbard's twisted rabbit problem, as a special case of this, and explain
a new inductive procedure for understanding generalizations of Hubbard's
problem.
Video
 November 10th: Arithmetic intersection and measures of maximal entropy
Laura DeMarco (Harvard University)
(in Eckhart 207A)
Abstract: About 10 years ago, Xinyi Yuan and Shouwu Zhang proved that if two holomorphic maps f and g on P^N have the same sets of preperiodic points (or if the intersection of Preper(f) and Preper(g) is Zariski dense in P^N), then they must have the same measure of maximal entropy. This was new even in dimension N=1. I will describe some ingredients in their proof, while emphasizing the dynamical history behind this result. If there's time, I will also sketch the proof of a theorem of Levin and Przytycki from the 1990s, in dimension N=1, that two (nonexceptional) maps have the same measure of maximal entropy if and only if they "essentially" share an iterate.
 November 3rd: Z/2invariantology in lowdimensional topology
Michael Klug (University of Chicago)
(in Eckhart 207A)
Abstract: I will discuss a connection between several different Z/2 invariants in lowdimensional topology (the Arf invariant of a knot, the Arf of a surface with a spin structure, the Rochlin invariant of a homology 3sphere, and the Arf invariant of a characteristic surface in a 4manifold, the KirbySiebenmann invariant of a 4manifold). We will unify these invariants and discuss how the results generalize to links and general compact 4manifolds.
 October 13th: Exceptional Lie groups and some related geometry
Bruce Hunt
(via Zoom, link will be sent to mailing list)
Abstract: The classification of Lie groups was already given in the 1880s (KillingCartan classification). In addition 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 octonion algebra. The groups occur in many different parts of mathematics, 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.
Video
Slides
Further details
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