# Seminars and Colloquia

## Colloquium

Dec 7

#### Mathematics Colloquium

04:20 pm

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

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

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

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

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

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

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

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

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.

Apr 28

#### Math CS Colloquium, Olga Kharlampovich (Hunter College, CUNY):"Tarski-type questions for group rings"

04:15 pm

Abstract: We consider some fundamental model-theoretic questions that can be asked about a given algebraic structure (a group, a ring, etc.), or a class of structures, to understand its principal algebraic and logical properties. These Tarski type questions include: elementary classification and decidability of the first-order theory. We describe solutions to Tarski's problems in the class of group algebras of free groups. We will show that unlike free groups, two groups algebras of free groups over infinite fields are elementarily equivalent if and only if the groups are isomorphic and the fields are equivalent in the weak second order logic. We will also show that for any field, the theory of a group algebra of a torsion free hyperbolic group is undecidable and for a field of zero characteristic even the diophantine problem is undecidable. (These are joint results with A. Miasnikov)

Mar 24

#### Math CS Colloquium, Peter Maceli (Wes): "Graphs and Algorithms"

04:15 pm

Abstract: Graph theory is a young and exciting area of discrete mathematics. Visually, a graph is just a collection of dots together with lines joining certain pairs of these dots. Though at first glance graphs may seem like simple objects to study, the field of graph theory contains some of the deepest and most beautiful mathematics of the last fifty years. Being an extremely vi- sual field, many questions and problems in graph theory are easily stated, yet have complex solutions with far reaching implications and applications. In this talk, we will explore the close relationship shared between graphs and algorithms. Describing how certain families of graphs look and can be built, and how, in turn, this allows one to efficiently solve certain important combinatorial problems.

Mar 3

#### Math CS Colloquium, Andrei S. Rapinchuk (University of Virginia):"Hearing the shape of a locally symmetric space, and arithmetic groups"

04:15 pm

Abstract: I will discuss the famous question of Mark Kac Can one hear the shape of a drum? in the context of (compact) locally symmetric spaces. In a joint work with G. Prasad, we were able to resolve this question in many situations using our analysis of weakly commensurable arithmetic subgroups of algebraic groups. The notion of weak commensurability makes sense for arbitrary Zariski-dense subgroups and time permitting I will report on the ongoing project (joint with V. Chernousov and I. Rapinchuk) to develop a new form rigidity (called the eigenvalue rigidity) based on this concept. This work involves problems in the theory of algebraic groups of independent interest.

Feb 11

#### Math CS Colloquium, Alex Lubotzky (Hebrew University of Jerusalem and Yale University):"Ramanujan complexes and topological expanders"

04:15 pm

Abstract: Expander graphs in general, and Ramanujan graphs , in particular, have played a major role in computer science in the last 4 decades and more recently also in pure math. In recent years a high dimensional theory of expanders is emerging. A notion of topological expanders was defined by Gromov who proved that the complete d-dimensional simplical complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1. This question was answered recently (by T. Kaufman, D. Kazdhan and A. Lubotzky for d=2 and by T. Kaufman and S. Evra for general d) by showing that the d-skelton of (d+1)-dimensional Ramanujan complexes provide such topological expanders. We will describe these developments and the general area of high dimensional expanders.

Jan 26

#### Math CS Colloquium, Peter Sarnak (Princeton University and Institute for Advanced Study):"Strong approximation ,Ramanujan Graphs and Universal Quantum Gates"

04:15 pm

Abstract : We discuss some recent developments concerning the diameters of the number theoretic Ramanujan Graphs and of related number theoretic continued fraction algorithms in PU(2) and their applications to optimal universal quantum gates.

Jan 21

#### Math CS Colloquium, Brendan Hassetts (Brown University): "New perspectives on obstructions to rationality"

04:15 pm

A fundamental problem in algebraic geometry is to find rational parametrizations for the solutions of a polynomial equation--or demonstrate that such a parametrization is impossible. Such parametrizations are useful in many fields, from mapmaking and computer-aided design to number theory. The goal of this talk is to summarize very recent progress on these questions due to Voisin and others, while highlighting remaining outstanding open questions.

Dec 10

#### Math CS Colloquium, Anthony Vrilly-Alvarado (Rice University):"Elliptic curves, torsion subgroups, and uniform bounds for Brauer groups of K3 surfaces"

04:15 pm

Abstract: Elliptic curves are smooth plane curves defined by a homogeneous equation of degree three that come with a marked point. Results on elliptic integrals going back to Euler show that one can endow such a curve with an abelian group structure, making the marked point the origin of this group. Mordell showed in 1922 that if E is an elliptic curve defined by an equation over the rational numbers Q, then the group of points E(Q) is finitely generated. Surprisingly, there are only 15 possibilities for the torsion subgroup of E(Q). This is a spectacular theorem of Mazur from 1977. I will explore this circle of ideas for a higher dimensional analogue of elliptic curves: K3 surfaces. Unlike ``abelian surfaces'', K3 surfaces have no group structure, so even understanding what the analogue of E(Q) should be is tricky. I will explain how the Brauer group of K3 surface comes to the rescue, argue for a conjecture along the lines of Mazur's theorem, and explain the impact this would have in our understanding of K3 surfaces.

Dec 3

#### Math CS Colloquium, Michael Kelly (University of Michigan): "Mathematical Crystals and Quasicrystals: Solid-to-Solid Phase Transitions"

04:15 pm

Abstract: In the early 1980's Dan Schectman made the Nobel Prize wining discovery of quasicrystals. These objects posses strikingly similar properties, especially long range order, to physical crystals (which are defined by a periodic molecular structure) but have a distinctive non-periodic molecular structure. An almost universal mathematical model for quasicrystals are the so called cut-and-project sets. The vertices of the Penrose tiling, for instance, is an example of such a set. It is a fundamental question to determine whether a given quasicrystal can be obtained by a displacive, as opposed to a diffusive, phase transition from a crystal. That is, can a quasicrystal be obtained by taking a crystal and applying a perturbation to it which moves each atom a uniformly bounded distance? We will show that in most moduli spaces of cut-and-project quasicrystals that (1) a quasicrystal can almost surely obtained from a crystal via a displacive phase transition, and (2) there is always a topologically large (i.e. residual) subset of quasicrystals that cannot be realized in this way. The results are obtained by relating cut-and-project sets as return times to a section for linear toral flows and employing cutting edge techniques from Fourier analysis, dynamics, and Diophantine approximation. This is joint work with Alan Haynes and Barak Weiss.

Nov 19

#### Math CS Colloquium, Vincent Guingona (WES):"On Generalized Notions of Dimension"

04:15 pm

Abstract: In many areas of mathematics, there are various notions of dimension, like the dimension of a vector space, or the dimension of an algebraic variety over the complex field, or the dimension of a semi-algebraic set over the reals. What can we say about dimension in a general setting when looking at an arbitrary structure? In this talk, I discuss several notions of dimension for abstract structures, including dp-rank, o-minimal dimension, and Morley rank. Tying all of these dimension notions together is the notion of VC-density, which is a measure of the combinatorial complexity of set systems. I define VC-density, discuss how it relates (or conjecturally relates) to the other notions of dimension, and give open problems and partial solutions about computing VC-density in certain classes of structures.

Nov 18

#### Math CS Colloquium, Wen-Ching Winnie Li (the Pennsylvania State University), "Isospectrality in number theory, geometry and combinatorics"

04:15 pm

Abstract: In 1966, Marc Kac posed the question ``Can one hear the shape of a drum?'' It can be rephrased as ``Does the spectrum of the Laplacian on a compact Riemannian manifold determine the manifold up to isometry?'' This problem had attracted many people in geometry. To this day, interesting pairs of isospectral but nonisometric manifolds, graphs and complexes have been constructed. Some constructions are based on Sunada's algebraic criterion published in 1985. In this talk we shall discuss isospectrality in the context of number theory, geometry and combinatorics, as well as the role played by Sunada's criterion.

Nov 5

#### Math CS Colloquium, Cynthia Vinzant (North Carolina State University):"Determinants, polynomials, and matroids"

04:15 pm

Abstract: Writing a multivariate polynomial as the determinant of a matrix of linear forms is a classical problem in algebraic geometry and complexity theory. Requiring that this matrix is Hermitian and positive definite at some point puts topological and algebraic restrictions on the polynomials that appear as the determinant and its minors. In particular the real zero sets of these polynomials are hyperbolic (or real stable) and interlace. Such polynomials appear in many areas of mathematics, including optimization, combinatorics and differential equations. Recently, tight connections have been developed with combinatorial objects called matroids. I will give an introduction to some of these objects and the fascinating connections between them.

Oct 29

#### Math CS Colloquium, Alyson Hildum (Wes):"Four-manifolds with right-angled Artin fundamental groups"

04:15 pm

Abstract: It is well known in low-dimensional topology that given a finitely presented group G, one can always find a 4-manifold M with fundamental group G. (There is a standard construction, which I will describe.) One of the items on Kirby's problem list is to find the minimum Euler characteristic of 4-manifolds with prescribed fundamental group G. This is a subproblem of the more general "geography problem" for a group G, in which one hopes to determine all possible values of the signature and Euler characteristic of M with fundamental group G. In this talk I will focus on 4-manifolds with fundamental groups belonging to a certain class of groups called right-angled Artin groups (or RAAGs, as they are often called). RAAGs are a popular study among geometric group theorists today as they have rich subgroup properties. But they are also very easily presented. RAAGs are also known as graph groups because their presentations can uniquely be defined by graphs, where each vertex represents a generator and each edge between vertices represents a commutator relation between the associated generators. It is not difficult to construct a 4-manifold which has a particular RAAG as its fundamental group, however for most RAAGs, the "standard" construction is not minimal (i.e. the Euler characteristic is not minimal). I will give upper and lower bounds on the minimal Euler characteristic, and will then focus on tools for constructing minimal 4-manifolds. (This was the subject of my PhD thesis.)

Oct 22

#### Math CS Colloquium, Amir Mohammadi (The University of Texas at Austin): "Geodesic planes in hyperbolic 3-manifolds"

04:15 pm

Abstract: In this talk we discuss the possible closures of geodesic planes in a hyperbolic 3-manifold M. When M has finite volume Shah and Ratner (independently) showed that a very strong rigidity phenomenon holds, and in particular such closures are always properly immersed submanifolds of M with finite area. We show that a similar rigidity phenomenon holds for a class of infinite volume manifolds. The proof uses elements from hyperbolic geometry and Margulis' approach in the proof of the Oppenheim conjecture. This is a joint work with C. McMullen and H. Oh.

Oct 15

#### Math CS Colloquium, Alex Eskin (University of Chicago): "The SL(2,R) action on Moduli space"

04:15 pm

Abstract: There is a natural action of the group SL(2,R) of 2x2 matrices on the moduli space of compact Riemann surfaces. This action can be visualized using flat geometry models. I will survey some recent developments in the area, and give some applications to the study of billiards in polygons and other problems.

Oct 8

#### Math CS Colloquium, Hanfeng Li (SUNY at Buffalo):"Sofic mean length"

04:15 pm

Abstract: For a unital ring R, a length function on left R-modules assigns a (possibly infinite) nonnegative number to each module being additive for short exact sequences of modules. For any unital ring R and any group G, one can form the group ring RG of G with coefficients in R. The modules of RG are exactly R-modules equipped with a G-action. I will discuss the question of how to define a length function for RG-modules, given a length function for R-modules. An application will be given to the question of direct finiteness of RG, i.e. whether every one-sided invertible element of RG is two-sided invertible. This is based on joint work with Bingbing Liang.

Sep 24

#### Math CS Colloquium, Dmitry Kleinbock (Brandeis University): 'Intrinsic Diophantine approximation on quadrics via homogeneous dynamics'

04:15 pm

Abstract: The goal of the talk is to quantify the density of rational points on quadric hypersurfaces using a connection between the distribution of rational points on those manifolds and dynamics on homogeneous spaces. This improves on recent results of Ghosh, Gorodnik and Nevo. Most of the talk will be about spheres; no background in dynamics or number theory is needed. Joint work with Lior Fishman,Keith Merrill and David Simmons.

Sep 18

#### Math CS Colloquium, Dorian Goldfeld (Columbia University): 'Prime number theorem for GL(n)'

01:10 pm

Abstract: The classical prime number theorem states that the number of primes less than x is asymptotic to x/log x as x tends to infinity. This result is obtained by studying the Riemann zeta function. In this talk I will discuss generalizations of the prime number theorem in the case where the Riemann zeta function is replaced by higher rank L-functions on GL(n) with n > 1.

Sep 10

#### Math CS Colloquium, Roger Howe (Yale University): 'Small representations of finite groups'

04:15 pm

Abstract: Finite group theorists have established many formulas that express interestingproperties of a finite group in terms of sums of characters of the group.An obstacle to applying these formulas is lack of control over the dimensionsof representations of the group. In particular, the representationsof small dimension tend to contribute the largest terms to these sums,so a systematic knowledge of these small representations couldlead to proofs of some of these facts. This talk will discuss a conjecturalmethod for constructing the small representations of finite classical groups.The method is closely related to the theory of theta series,and has an analog over local fields.

Apr 2

#### Math Colloquium, Jeremy Rouse (Wake Forest): 'Integers represented by quadratic forms'

04:15 pm

Abstract: We say a quadratic form represents a positive integer if the equation has a solution where is a vector with integer entries.Given a set S of positive integers, how does one classify all quadratic formsthat represent all elements of? This talk will survey recent results on such questions, including Bhargava and Hanke's '290-theorem'. The notion of escalator latticesreduces the problem to studying a finite collection of quadratic forms to answer such a question.The theory of modular forms provides the right tool to prove that a quadratic form representsall the elements.