Department of

Mathematics


Seminar Calendar
for Geometry, Groups and Dynamics/GEAR events the year of Tuesday, March 13, 2018.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
    February 2018            March 2018             April 2018     
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              1  2  3                1  2  3    1  2  3  4  5  6  7
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Tuesday, January 16, 2018

12:00 pm in 243 Altgeld Hall,Tuesday, January 16, 2018

A Higgs bundle construction for representations in exceptional components of $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-character varieties.

Georgios Kydonakis (UIUC)

Abstract: For a compact Riemann surface of genus $g\ge 2$, the components of the moduli space of $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-Higgs bundles, or equivalently the $\text{Sp(4}\text{,}\mathbb{R}\text{)}$ 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 $3\cdot {{2}^{2g}}+2g-4$ 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.

Tuesday, January 23, 2018

12:00 pm in 243 Altgeld Hall,Tuesday, January 23, 2018

A Garden of Eden Theorem for Abelian Harmonic Models

Tullio Ceccherini-Silberstein (University of Sannio)

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.

Tuesday, January 30, 2018

12:00 pm in 243 Altgeld Hall,Tuesday, January 30, 2018

Counting conjugacy classes of fully irreducibles in $Out(F_r)$

Ilya Kapovich   [email] (Illinois Math)

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 $L$.

Monday, February 5, 2018

2:00 pm in 241 Altgeld Hall,Monday, February 5, 2018

Lower hyperbolic rank rigidity of quarter-pinched manifolds

Christopher Connell (Indiana University)

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.

Thursday, February 15, 2018

12:00 pm in 243 Altgeld Hall,Thursday, February 15, 2018

Coding geodesic flows and various continued fractions

Merriman Claire (Illinois Math)

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.

Tuesday, February 20, 2018

12:00 pm in 243 Altgeld Hall,Tuesday, February 20, 2018

Which groups have bounded harmonic functions?

Yair Hartman (Northwestern University)

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.

Tuesday, March 6, 2018

12:00 pm in Room 214 Ceramics Building,Tuesday, March 6, 2018

Stable subgroups and Morse subgroups of mapping class groups

Heejoung Kim (Illinois Math)

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.

Tuesday, March 13, 2018

12:00 pm in 243 Altgeld Hall,Tuesday, March 13, 2018

Group actions on quiver varieties and applications

Vicky Hoskins (Freie Universitšt Berlin)

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.

Tuesday, March 27, 2018

12:00 pm in 243 Altgeld Hall,Tuesday, March 27, 2018

Random Walks on Out(F_r)

Catherine Pfaff (University of California at Santa-Barbara)

Abstract: While many mathematicians hypothesized for years as to which elements of mapping class groups and the Out(F_r) are generic, there has only in the past decade been an explosion of results on the topic. This explosion began with Maher and Rivin proving that pseudo-Anosovs are indeed generic within the mapping class group. Rivin further gave that fully irrreducibles (specifically fully irreducibles not induced by surface homeomorphisms) are generic. Many results followed by Sisto, Calegari-Maher, Maher-Tiozzo, Karlsson, Horbez, and Dahmani-Horbez. Kapovich-Pfaff gave a refinement of this work in a particular Out(F_r) setting by proving specific invariant values along a ``train track directed'' random walk. Answering a question of that paper, Gadre-Maher proved that pseudo-Anosovs are generically ``principal.'' Inspired by the work of Gadre-Maher, we are expanding the ` `train track directed'' random walk work to a full random walk on Out(F_r). This is joint work in progress.

Tuesday, April 10, 2018

12:00 pm in 243 Altgeld Hall,Tuesday, April 10, 2018

Central Limit Theorem for odometers and B-free integers

Francesco Cellarosi (Queens University Math)

Abstract: Odometers (or von Neumann–Kakutani adding machines) are classical examples of dynamical systems of low complexity, much alike irrational rotations of the circle. We consider generalized adding machines. In spite of their rigid behaviour (zero entropy, not weakly mixing), we are able to prove a Central Limit Theorem for the ergodic sums corresponding to certain (randomly chosen) observables, generalizing the work of M.B. Levin and E. Merzbach.
  Time permitting, I will describe the connections of odometers to the dynamical systems naturally arising when studying the statistical properties of B-free integers and explain why it would be interesting to obtain a central limit theorem for these systems. Joint work with Maria Avdeeva.

Tuesday, April 24, 2018

12:00 pm in 243 Altgeld Hall,Tuesday, April 24, 2018

Genus bounds in right-angled Artin groups

Jing Tao (University of Oklahoma)

Abstract: In this talk, I will describe an elementary and topological argument that gives bounds for the stable commutator lengths in right-angled Artin groups.

Tuesday, May 1, 2018

12:00 pm in 243 Altgeld Hall,Tuesday, May 1, 2018

Polygonal billiards, Liouville currents, and rigidity

Chris Leininger (Illinois Math)

Abstract: A particle bouncing around inside a Euclidean polygon gives rise to a biinfinite "bounce sequence" (or "cutting sequence") recording the (labeled) sides encountered by the particle. In this talk, I will describe recent work with Duchin, Erlandsson, and Sadanand, where we prove that the set of all bounce sequences---the "bounce spectrum"---essentially determines the shape of the polygon. This is consequence of a technical result about Liouville currents associated to nonpositively curved Euclidean cone metrics on surfaces. In the talk I will explain the objects mentioned above, how they relate to each other, and give some idea of how one determines the shape of the polygon from its bounce spectrum.

Tuesday, May 29, 2018

12:00 pm in 243 Altgeld Hall ,Tuesday, May 29, 2018

Fibering of 3-manifolds and related groups

Dawid Kielak (University of Bielefeld )

Abstract: We will take a new look at Thurston's results on the structure of fibrings of 3-manifolds from a more algebraic perspective. This will allow us to generalise these results to (many) Poincare duality groups in dimension 3, and to (all) free-by-cyclic groups.