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Geometry, Groups and
Dynamics/GEAR Seminar

The Geometry, Groups and Dynamics/GEAR Seminar, held at the Illinois hub of GEAR, features speakers of interest to network members. The talks are live streamed on the Illinois Department of Mathematics Youtube channel playlist. Seminars postings can also be found on the Illinois Mathematics Department Events Calendar.

Archive of past lectures.

Current Semester

12:00 pm, Tuesday, March 13, 2018, 243 Altgeld Hall
Vicky Hoskins (Freie Universität Berlin)
Group actions on quiver varieties and applications

Abstract: We study two types of actions on King's moduli spaces of quiver representations over a field k, and we decompose their fixed loci using group cohomology in order to give modular interpretations of the components. The first type of action arises by considering finite groups of quiver automorphisms. The second is the absolute Galois group of a perfect field k acting on the points of this quiver moduli space valued in an algebraic closure of k; the fixed locus is the set of k-rational points, which we decompose using the Brauer group of k, and we describe the rational points as quiver representations over central division algebras over k. Over the field of complex numbers, we describe the symplectic and holomorphic geometry of these fixed loci in hyperkaehler quiver varieties using the language of branes. This is joint work with Florent Schaffhauser.

12:00 pm, Tuesday, March 6, 2018, Room 214 Ceramics Building
Heejoung Kim (Illinois Math)
Stable subgroups and Morse subgroups of mapping class groups

Abstract: The notion of a "quasiconvex" subgroup plays of a word-hyperbolic group G plays an important role in the theory of hyperbolic groups. This notion has several equivalent characterizations in that context, in terms of being "undistorted", in terms of the action on the boundary, in terms of being "rational" with respect to automatic structures on G, in terms of the contracting properties of the projection maps, etc. For an arbitrary finitely generated group G, there are two recent generalizations of the notion of a quasiconvex subgroup: a "stable" subgroup and a "Morse" subgroup. In this talk, we will discuss these two notions and their different properties. We prove that the properties of being Morse and being stable coincide for a subgroup of infinite index in the mapping class group of an oriented, connected, finite type surface with negative Euler characteristic.

12:00 pm, Tuesday, February 20, 2018, 243 Altgeld Hall
Yair Hartman (Northwestern University)
Which groups have bounded harmonic functions?

Abstract: Bounded harmonic functions on groups are closely related to random walks on groups. It has long been known that all abelian groups, and more generally, virtually nilpotent groups are "Choquet-Deny groups": these groups cannot support non-trivial bounded harmonic functions. Equivalently, their Furstenberg-Poisson boundary is trivial, for any random walk. I will present a very recent result where we complete the classification of discrete countable Choquet-Deny groups. In particular, we show that any finitely generated group which is not virtually nilpotent, is not Choquet-Deny. Surprisingly, the key is not the growth rate of the group, but rather the algebraic infinite conjugacy class property (ICC). This is joint work with Joshua Frisch, Omer Tamuz and Pooya Vahidi Ferdowsi.

12:00 pm, Thursday, February 15, 2018, 243 Altgeld Hall
Merriman Claire (Illinois Math)
Coding geodesic flows and various continued fractions

Abstract: Continued fractions are frequently studied in number theory, but they can also be described geometrically. I will give both pictorial and algebraic descriptions of the flows that describe continued fraction expansions. This talk will focus on continued fractions of the form $a_1\pm\frac{1}{a_2\pm\frac{1}{a_3\pm\ddots}}$, where the $a_i$ are odd. I will show how to describe these continued fractions as geodesic flows on a modular surface, and compare it to the modular surface needed when $a_i$ are even.

2:00 pm, Monday, February 5, 2018, 241 Altgeld Hall
Christopher Connell (Indiana University)
Lower hyperbolic rank rigidity of quarter-pinched manifolds
Abstract: A Riemannian manifold M has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature -1 with the geodesic. If in addition, the sectional curvatures of M lie in the interval [−1,−1/4], and M is closed, we show that M is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial converse to Hamenstädt's hyperbolic rank rigidity result for sectional curvatures at most −1, and complements well-known results on Euclidean and spherical rank rigidity. This is joint work with Thang Nguyen and Ralf Spatzier.

12:00 pm, Tuesday, January 30, 2018, 243 Altgeld Hall
Ilya Kapovich (Illinois Math)
Counting conjugacy classes of fully irreducibles in Out(Fr)
Abstract: Inspired by results of Eskin and Mirzakhani counting closed geodesics of length $\le L$ in the moduli space of a closed surface $\Sigma_g$ of genus $g\ge 2$, we consider a similar question in the $Out(F_r)$ setting. Let $h=6g-6$. The Eskin-Mirzakhani result, giving the asymptotics of $\frac{e^{hL}}{hL}$, can be equivalently stated in terms of counting the number of $MCG(\Sigma_g)$-conjugacy classes of pseudo-Anosovs $\phi\in MCG(\Sigma_g)$ with dilatation $\lambda(\phi)$ satisfying $\log\lambda(\phi)\le L$. For $L\ge 0$ let $\mathfrak N_r(L)$ denote the number of $Out(F_r)$-conjugacy classes of fully irreducibles $\phi\in Out(F_r)$ with dilatation $\lambda(\phi)$ satisfying $\log\lambda(\phi)\le L$. In a joint result with Catherine Pfaff, we prove for $r\ge 3$ that as $L\to\infty$, the number $\mathfrak N_r(L)$ has double exponential (in $L$) lower and upper bounds. We also obtain a companion result, joint with Michael Hull, and show that of distinct $Out(F_r)$-conjugacy classes of fully irreducibles $\phi$ from an $L$-ball in the Cayley graph of $Out(F_r)$ with $\log\lambda(\phi)$ on the order of $L$ grows exponentially in $L$.

12:00 pm, Tuesday, January 23, 2018, 243 Altgeld Hall
Tullio Ceccerini-Silberstein (University of Sannio)
A Garden of Eden Theorem for Abelian Harmonic Models

Abstract: In this talk, completely self contained (I'll recall the basic of Pontryagin duality), I would like to introduce the audience to the beautiful theory of algebraic actions (in the sense of K. Schmidt) and present a Garden of Edent type theorem for the class of weakly-expansive principal algebraic actions of Abelina groups, which includes, as a particular case, transient Abelian Harmonic Models. This is a recent result obtained in collaboration with Michel Coornaert and Hanfeng Li.

12:00 pm, Tuesday, January 16, 2018, 243 Altgeld Hall
Georgios Kydonakis (Illinois)
A Higgs bundle construction for representations in exceptional components of Sp(4,R) -character varieties

Abstract: For a compact Riemann surface of genus g > 2, the components of the moduli space of Sp(4,R)-Higgs bundles, or equivalently the Sp(4,R) character varieties, are partially labeled by an integer d known as the Toledo invariant. The subspace for which this integer attains a maximum has been shown to have 3 . 22g + 2g−4 many components. A gluing construction between parabolic Higgs bundles over a connected sum of Riemann surfaces provides model Higgs bundles in a subfamily of particular significance. This construction is formulated in terms of solutions to the Hitchin equations, using the linearization of a relevant elliptic operator.
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