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.

1. Talks are 50 minutes, followed by a question/answer session.
2. Many in the audience will likely be non-experts.
3. 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

• Benson Farb (bensonfarb (at) gmail.com)
• Michael Klug (michaelklug (at) math.uchicago.edu)

Fall 2023

• December 6th: Simplicial "badness" arguments and the Solomon--Tits theorem
  Jenny Wilson (University of Michigan)
  (2:00 - 3:00 PM in Eckhart 206)
  Abstract: I'll survey a technique for proving that a simplicial complex is homotopy equivalent to a wedge of spheres. As an application we'll prove this result for the Tits buildings associated to the special linear groups.


• November 29th: Formality and dominant maps
  Aleksandar Milivojevic
  (2:00 - 3:00 PM in Eckhart 206)
  Abstract: In the mid 70’s, Deligne-Griffiths-Morgan-Sullivan demonstrated a strong topological condition a closed manifold would have to satisfy if it were to carry a Kähler complex structure. Namely, the manifold would have to be formal, in the sense of its de Rham algebra of forms being weakly equivalent to its cohomology. In particular, there can be no non-trivial Massey products on a compact Kähler manifold. The salient underlying property of compact Kähler manifolds which implies formality is preserved under surjective holomorphic maps (non-zero degree maps in the holomorphic setting). It turns out (joint work with Jonas Stelzig and Leopold Zoller), formality itself is preserved under non-zero degree continuous maps of spaces satisfying Poincaré duality on their rational cohomology. I will explain the key components of this argument and show how one can then apply it to various situations. For example, we obtain that the presence of a non-negative Ricci curvature metric on a closed manifold with large first Betti number implies formality.


• November 15th: Introduction to Enriched Enumerative Geometry
  Candace Bethea
  (2:00 - 3:00 PM in Eckhart 206)
  Abstract: At its core, enumerative geometry asks for integral solutions to geometric questions, such as how many lines are contained in a smooth cubic surface defined over the complex numbers. Enumerative results are particularly exciting when they can be extended to other settings, for example in exploring how many of the lines on a smooth cubic surface are defined over the real numbers. This talk will introduce enriched enumerative geometry, which seeks to extend classical enumerative results using ideas from motivic and equivariant topology. We will discuss examples, intuition behind the form that enriched enumerative results should take, and potential future directions in equivariant enumerative geometry specifically. This talk will be widely accessible.


• November 8th: Orbifold uniformization of varieties
  Matt Stover (Temple University)
  (2:00 - 3:00 PM in Eckhart 206)
  Abstract: Classifying smooth projective curves by their genus is as old as salt, and by the uniformization theorem (one of the most fundamental theorems in mathematics) they all admit a Riemannian metric of constant curvature. A variant I quite like is that they all admit an orbifold metric of constant curvature -1, and hence are dominated by a hyperbolic surface. Amazingly the classification of smooth projective surfaces remains open: the analogues of "genus 0" and "genus 1" are enumerated, but the "higher genus" box may as well be a¯\_(ツ)_/¯ with tons of examples. A question due independently to Farb and Gromov could actually make the "higher genus" box a primitive recursive enumeration problem using orbifolds of holomorphic sectional curvature -1, succinctly generalizing orbifold uniformization of curves. My talk will be dedicated to setting the stage for their question and using simple topological arguments (and a giant uniformization hammer) to give some evidence for it, spinning examples due to Deligne-Mostow and Hirzebruch (and perhaps my own) in this light.


• October 18th: Coxeter groups, Euler characteristics, aspherical manifolds and γ-numbers
  Karim Adiprasito (Hebrew University)
  (3:00 - 4:00 PM in Eckhart 206)
  Abstract: The Chern-Hopf-Thurston conjecture states that the Euler characteristic of an aspherical closed and compact manifold should be of a sign dependent only on its dimension, and it remains mysterious to this day. Even when asphericity is more manageable, and the manifold nonpositively curved (which implies that it is aspherical) the conjecture, in that form emphasized by Chern and Thurston, is elusive. And even more puzzling is a strengthening of this particular case of the conjecture due to Gal, who, based on work of the combinatorialists Foata, Schützenberger and Strehl, introduced several other numbers, called γ-numbers, which generalize the signed Euler characteristic and are equally conjectured to be nonnegative. A topological or algebraic interpretation has eluded us so far. I will provide a gentle introduction for everyone, provide such an interpretation that implies nonnegativity, and relate it to work of Gromov on hyperkähler manifolds. The talk will be very basic and understandable to everyone.


• September 27th: Configuration spaces and applications in arithmetic statistics
  Anh Hoang Trong Nam (University of Minnesota)
  (3:00 - 4:00 PM in Eckhart 202)
  Abstract: In the last dozen years, topological methods have been shown to produce a new pathway to study arithmetic statistics over function fields. Recently, Ellenberg, Tran and Westerland proved the upper bound in Malle’s conjecture for the enumeration of function fields by studying the homology of braid groups with certain exponential coefficients. In this talk, we will give an overview of their framework and extend their techniques to study other subjects in arithmetic statistics. In particular, we will give a bound on character sums of the resultant of monic squarefree polynomials over finite fields, answering a question posed by Jordan Ellenberg at this very seminar, and propose a proof for a refined version of Malle's conjecture for function fields. Some ideas in this talk will feel eerily familiar from Aaron Landesman's talk last winter.

Spring 2023

• May 16th: Plane curves, holomorphic maps of moduli spaces and isometric embeddings
  Frederik Benirschke (University of Chicago)
  (3:30 PM - 4:30 PM in Ryerson 358)
  Abstract: In classical algebraic geometry a wealth of geometric constructions start with some algebraic variety, for example a curve in the plane, and associate to it a different algebro-geometric object. The hope is that geometric properties are reflected in the new construction. Some examples are the dual curve of a plane curve, the Jacobian of a curve, intermediate Jacobians, etc… Reinterpreted in modern language, these constructions often lead to holomorphic maps of moduli spaces. In this talk we will focus on the moduli space of curves and discuss the question: What are natural constructions that start with a Riemann surface and produce from it a different Riemann surface? Pretty much all constructions we know are related to branched coverings of surfaces and one wonders if that is the only way to produce maps between moduli spaces of curves. We will discuss some recent work with Carlos Serván showing that, at least if we add some assumptions on the metric geometry, indeed all holomorphic maps arise from branched coverings of surfaces.


• May 9th: Introduction to nilpotent geometry
  Moon Duchin (Tufts)
  (3:30 PM - 4:30 PM in Ryerson 358)
  Abstract: First I will try to convince you why to care about the geometry of nilpotent groups, then I will tell some stories in that direction. My goal is to help build up some intuition for how it looks to walk around in nilpotent spaces. No particular background required.


• April 18th: From sphere packings and energy minimization to Fourier interpolation and modular forms
  Danylo Radchenko (University of Lille)
  (1:00 PM - 2:00 PM in Ryerson 352 (the barn))
  Abstract: I will talk about a few recent results in the sphere packing and energy minimization problems and their connection to questions about Fourier interpolation formulas -- formulas that allow to reconstruct a function from discrete samples of the function itself and its Fourier transform. I will then discuss how in some exceptional cases such interpolation formulas with very good properties can be constructed with the help of modular forms, and how this helps with sphere packing and energy minimization problems in 8 and 24 dimensions.

Winter 2023

• February 21st: Questions between quantitative topology and combinatorics
  Karim Adiprasito (Hebrew University)
  (3:30 PM - 4:30 PM in Eckhart 206)
  Abstract: I will survey some recent results on complexity of triangulations for given manifolds, and present some interesting open questions related to comparison theorems for scalar curvature, among others.


• February 2nd: Identities for 1/pi and 1/pi^2, and special hypergeometric functions
  John Voight (Dartmouth)
  (12:30PM - 1:30PM in Eckhart 202)
  Abstract: More than a century ago, Ramanujan discovered remarkable formulas for 1/pi. Inspired by these discoveries, similar Ramanujan-like expressions for 1/pi^2 have been uncovered recently by Guillera. We explain the provenance of these formulas, in terms of hypergeometric motives with a special property. This is joint work with Lassina Dembele, Alexei Panchishkin, and Wadim Zudilin.


• January 19th: Applying topology to arithmetic statistics
  Aaron Landesman (Harvard)
  (12:30PM - 1:30PM in Eckhart 202)
  Abstract: We will explain a method for applying ideas in topology to answer questions in number theory. We will focus on an application to the Cohen Lenstra heuristics, which predict the average number of \ell-torsion line bundles on hyperelliptic curves over finite fields. We will explain why verifying these heuristics boils down to understanding the topology of a certain covering space of the configuration space of n points in the complex plane.

Fall 2022

• 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.


• 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 torsion-free hyperbolic group G whose boundary is the n-sphere (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 quasi-convex codimension-1 subgroups. In this talk, we examine the structure of quasi-convex codimension-1 subgroups when the boundary of G is the 3-sphere, in order to generalize Haïssinsky's result. We show that if G contains a quasi-convex, codimension-1 subgroup, then G contains a quasi-convex subgroup whose boundary is either the 2-sphere, or a space with prescribed Cech cohomology, which we conjecture is the Pontryagin sphere. This is joint work with M. Incerti-Medici.


• 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 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 Wikipedia--see Alexander's trick).

Spring 2022

• 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

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, 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

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 (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

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_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 25th: Lie algebras and group theory
  Andy Putman (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