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 non-experts.
- Principles are preferred over detailed proofs.

- The No Boundaries Seminar will be held on
**Tuesday at 3:30pm CDT**for the Spring 2022 quarter.

Talks will either be held in Eckhart 312 or via Zoom, depending on the speaker. - If you would like to join the mailing list, please email
**tghyde (at) uchicago.edu**.

**April 12th:**Lefshetz-Smith 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 A1-homotopy 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 A1-homotopy 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**- Benson Farb (bensonfarb (at) gmail.com)
- Trevor Hyde (tghyde (at) uchicago.edu)
- Phil Tosteson (ptoste (at) math.uchicago.edu)
**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, algebro-geometrically**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 algebro-geometric 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 Millson-Raghunathan 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 orientation-preserving homeomorphisms of the circle. This inclusion pulls back the “discrete universal Euler class” producing a non-zero 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 non-vanishing behaviour of the powers of this class.**Video****February 2nd:**Local systems on the braid group and arithmetic averages**Jordan Ellenberg (University of Wisconsin-Madison)**

*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 finite-index 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 L-functions, 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****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 (non-exceptional) maps have the same measure of maximal entropy if and only if they "essentially" share an iterate.**November 3rd:**Z/2-invariantology in low-dimensional topology**Michael Klug (University of Chicago)**

(in Eckhart 207A)

*Abstract:*I will discuss a connection between several different Z/2 invariants in low-dimensional topology (the Arf invariant of a knot, the Arf of a surface with a spin structure, the Rochlin invariant of a homology 3-sphere, and the Arf invariant of a characteristic surface in a 4-manifold, the Kirby-Siebenmann invariant of a 4-manifold). We will unify these invariants and discuss how the results generalize to links and general compact 4-manifolds.**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 (Killing-Cartan 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 Cayley-Graves 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**-
**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_q-rational 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_q-rational points? The goal of this talk is to explain how ideas from the theory of error-correcting 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 pseudo-Anosov 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 Deligne-Simpson 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 i-th 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 non-Abelian Hodge theory and middle convolution. We will also describe some unexpected connections between the Deligne-Simpson 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 "more-or-less" 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**