# Seminars and Colloquia

## Colloquium

Sep 26

#### Mathematics Colloquium

04:20 pm

Exley Science Center Tower ESC 121

Speaker: Mark Ellingham, Vanderbilt University Title: Toughness and spanning structures Abstract:Toughness is a parameter for graphs that measures how hard it is to break the graph up into many pieces. A well-known conjecture due to Chv\'{a}tal proposes that graphs with sufficiently high toughness have a hamilton cycle, a cycle using all of the vertices. This is known to be true for certain classes of graphs, such as chordal graphs, but is still open for general graphs. For general graphs, sufficiently high toughness can guarantee the existence of various generalizations of a hamilton cycle or hamilton path, such as a spanning walk that uses each vertex at most $k$ times ($k \ge 2$), or a spanning tree of maximum degree at most $k$ ($k \ge 3$). Stronger results, with weaker toughness conditions, can be obtained for special graph classes. We discuss some recent results of this kind. We also discuss the algorithmic difficulty of computing toughness; although this is NP-hard in general, we have recently shown that it can be done in polynomial time for graphs of bounded treewidth. The work discussed is joint work with a number of people, including Guantao Chen, Akira Saito, Pouria Salehi Nowbandegani, Songling Shan, Dong Ye and Xiaoya Zha.

Nov 14

#### Virtual combinatorics colloquium by Vic Reiner, Univ. of Minnesota

04:00 pm

Exley Science Center Tower ESC 339

Speaker: Vic Reiner, University of Minnesota Time and Date: 4:00-5:00 p.m., November 14, 2018 Location: 339 Exley (also available on Zoom) Title: Cyclic Sieving: Old and New Abstract: Cyclic sieving, identified in work with Dennis Stanton and Dennis White, is a happy situation, where counting how many among some objects enjoy cyclic symmetry is as easy as q-counting all of the objects. We will illustrate this with two kinds of examples: old ones that still plague us with only uninsightful proofs, and new ones that have joined our list of favorites. Zoom link: https://smcvt.zoom.us/j/446360634 Zoom tip and further information: https://sites.google.com/view/northeastcombinatoricsnetwork/events/virtual-combinatorics-colloquium

Apr 19

#### Mathematics Colloquium

04:20 pm

Exley Science Center Tower ESC 121

Rachel Roberts, Washington University Introducing meridional cusps Abstract: I will define taut foliations and branched surfaces in 3-manifolds, and describe a search for co-oriented taut foliations using branched surfaces as the primary tool. This search is inspired by results from Heegaard Floer homology. This work is joint with Charles Delman.

Apr 12

#### Mathematics Colloquium

04:20 pm

Exley Science Center Tower ESC 121

Aaron Brown, University of Chicago Recent progress in the Zimmer program Abstract: The Zimmer program refers to a number of questions and conjectures about actions of certain discrete groups, namely, lattices in higher-rank simple Lie groups. The primary example example of a such a group is SL(n,R). In the past few years, there has been significant progress in the Zimmer program. In my talk, I will discuss a recent proof of Zimmer's conjecture which shows that (cocompact and certain non-uniform) higher-rank lattices do not act on manifolds with low dimension. I will also discuss recent results and work in progress that classify all possible non-trivial actions under certain dynamical or dimension assumptions.

Dec 7

#### Mathematics Colloquium

04:20 pm

Exley Science Center Tower ESC 121

Vincent Guingona, Towson Title: Machine learning theory and model theory. Abstract: We discuss some basic concepts in machine learning theory, including concept classes, VC-dimension, VC-density, PAC-learning, and sequence compressions. We then explore the relationship between these concepts and model theory.

Nov 9

#### Mathematics Colloquium

04:20 pm

Exley Science Center Tower ESC 121

Speaker: Nadja Hempel, UCLA Mekler constructions in generalized stability Abstract: Given a so called nice graph (no triangles, no squares, for any choice of two distinct vertices there is a third vertex which is connected to one and not theother), Mekler considered the 2-nilpotent subgroup generated by the vertices of the graph in which two elements given by vertices commute if and only if there is an edge between them. These groups form an interesting collection of examples from a model theoretic point of view. It was shown that such a group is stable if and only if the corresponding graph is stable and Baudisch generalized this fact to the simple theory context. In a joint work with Chernikov, we were able to verify this result for k-dependent and NTP_2 theories. This leads totheexistence of groups which are (k+1)-dependent but not k-dependent, providing the first algebraic objects witnessing the strictness of thesehierarchy. This is joint work with Artem Chernikov.

Nov 2

#### Mathematics Colloquium

04:20 pm

Exley Science Center Tower ESC 121

Jonathan Huang, Wes Title:Zeta Functions, Witt rings, and a Classical Formula of MacDonald Abstract: A remarkable formula of MacDonald provides a closed expression for the generating series of the Poincar polynomial of the symmetric powers Sym n X of a space X . We show that this formula takes a very nice form when rewritten in the big ring of Witt vectors W ( [ z ]) of the polynomial ring [ z ]. We then provide some motivation for similarly viewing the Hasse-Weil zeta function of varieties over finite fields as elements in the big Witt ring W ( ). In this setting, the zeta function Z ( X , t ) takes the form of an Euler-Poincar characteristic.

Oct 19

#### Mathematics Colloquium

04:20 pm

Exley Science Center Tower ESC 121

Speaker: Alex Kruckman, Indiana University-Bloomington Title:First-order logic and cologic over a category Abstract: In ordinary first-order logic, each formula comes with a finite variable context. In order to assign a truth value to the formula, we need an interpretation of its context: an assignment of the variables to elements of a structure. I will describe a categorical generalization of first-order logic, obtained by replacing the category of finite sets (variable contexts) with any small category C with finite colimits, and replacing arbitrary sets (domains of structures) with formal directed colimits from C. I will present a deductive system and completeness theorem for this logic, which is related to hyperdoctrines, a notion from categorical logic. Once this categorical framework is in place, it is easily dualizable. The result is a first-order "cologic", which is well-suited for studying profinite structures in terms of their finite quotients; indeed, this was the original motivation. As particular examples, I will explain how the framework includes the "cologic" of profinite groups due to Cherlin, Macintyre, and van den Dries, and the theories of projective Fraisse limits due to Solecki and Irwin.

Oct 12

#### Mathematics Colloquium

04:20 pm

Exley Science Center Tower ESC 121

Speaker: James Freitag, University of Illinois-Chicago Title: Algebraic relations between solutions of Painlev equations Abstract: Painlev equations are families of second order nonlinear differential equations which were first discovered in the late 19th century, in connection with problems in analysis around analytic continuation. Interest in the equations has increased in large part because of connections to numerous other subjects including random matrix theory, monodromy of linear differential equations, and diophantine geometry. In this talk, we will describe recent interactions with model theory which have resulted in the proof of several conjectures related to transcendence of solutions of Painlev equations.

Oct 5

#### Mathematics Colloquium

04:20 pm

Exley Science Center Tower ESC 121

Speaker: Shelly Harvey, Rice University Title: Corank of 3-manifold groups G with H_2(G)=0 Abstract: The corank of a group G, c(G), is the maximal r such that there is a surjective homomorphism from G to a non-abelian free group of rank r. We note that for any group G, c(G) is bounded above by b_1(G), the rank of the abelianization of G. For closed surface groups S, we have a further relationship between these two complexities, namely b_1(S) = 2 c(S). It was asked whether such a relationship exists for 3-manifold groups. In a previous paper, I showed that there were closed 3-manifold groups G with b_1(G) arbitrarily large but with c(G)=1. It was asked by Michael Freedman whether such a statement was known when the group was the group of a 3-dimensional homology handlebody. These groups are much more subtle and have properties that make them look like a free group so the question becomes much more difficult. In fact, all of the previous techniques used by the author fail. The complete answer to the question is still unknown. However, we show that there are groups G_m (for all m \geq 2) which are the fundamental group of a 3-dimensional handlebody (in particular, H_2(G_m)=0) and satisfy the following: b_1(G_m)= m and c(G_m)=f(m) where f(m)=m/2 for m even and f(m)=(m+1)/2 for m odd. This is joint work with Eamonn Tweedy.

Mar 9

#### Mathematics Colloquium

04:20 pm

Exley Science Center Tower ESC 121

Kevin Tucker, University of Illinois - Chicago: An Introduction to F-Signature Abstract: In positive characteristic p > 0, Frobenius splitting methods have long been used to measure singularities. Although these techniques originally found applications in commutative algebraandrepresentation theory, in recent years they have increased in importance following the discovery of surprising connections to the singularities of the minimal model program in complex algebraic geometry. In this talk, I will discuss an invariant governing the asymptotic behavior of F-splittings called the F-signature, together with numerous examples.

Dec 8

#### Mathematics Colloquium

04:20 pm

Exley Science Center Tower ESC 121

Moon Duchin, Tufts: Sprawland other geometric statistics Abstract: I'll define a statistic called the "sprawl" of a metric measure space which quantifies the degree of rapid, homogeneous spreading out that is characteristic of trees. Related statistics come up across geometry, in group theory, in category theory, and in applications from biodiversity to gerrymandering. In this talk I'll spend some time on examples from convex geometry and will try to get to voting applications by the end.

Dec 1

#### Mathematics Colloquium

04:20 pm

Exley Science Center Tower ESC 121

Karen Melnick, University of Maryland: Limits of local autommorphisms of geometric structures Abstract: The automorphism group of a rigid geometric structure is a Lie group. In fact, the local automorphisms form a Lie pseudogroup; this property is often taken as an informal definition of rigid geometric structure. In which topology is this the case? The classical theorems of Myers and Steenrod say that $C^0$ convergence of local isometries of a smooth Riemannian metric implies $C^\infty$ convergence; in particular, the compact-open and $C^\infty$ topologies coincide on the isometry group. I will present joint results with C. Frances in which we prove the same result for local automorphisms of smooth parabolic geometries, a rich class of geometric structures including conformal and projective structures.