Department of

Mathematics


Seminar Calendar
for Geometry, Groups and Dynamics/GEAR Seminar events the year of Friday, September 14, 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.
     August 2018           September 2018          October 2018    
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           1  2  3  4                      1       1  2  3  4  5  6
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                        30                                         

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 Universität 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.

Tuesday, September 11, 2018

12:00 pm in 243 Altgeld Hall,Tuesday, September 11, 2018

From the dynamics of surface automorphisms to the computational complexity of 3-manifolds

Eric Samperton (UCSB)

Abstract: Every 3-manifold admits a Heegaard splitting, and many 3-manifold invariants admit formulas using Heegaard splittings. These facts are one starting point for a common theme in the study of 3-manifolds: one can relate various topological or geometric properties of 3-manifolds to dynamical systems in 1 or 2 dimensions. We’ll explore this theme in the context of computational complexity. I’ll start with two examples (coloring invariants and the Jones polynomial) that translate dynamical properties of mapping class group actions into complexity-theoretic hardness properties of 3-manifold invariants. I’ll conclude with some brainstorming about future directions. I will introduce all of the necessary complexity theory as we go.

Thursday, September 20, 2018

12:00 pm in 243 Altgeld Hall,Thursday, September 20, 2018

The generalization of the Goldman bracket to three manifold and its relation to Geometrization

Moira Chas (Stony Brook)

Abstract: In the eighties, Bill Goldman discovered a Lie algebra structure on the free abelian group with basis the free homotopy classes of closed oriented curves on an oriented surface S. In the nineties, jointly with Dennis Sullivan, we generalized this Lie algebra structure to families of loops (defining the equivariant homology of the free loop space of a manifold). This Lie algebra, together with other operations in spaces of loops is now known as String Topology. The talk will start with a discussion of the Goldman Lie bracket in surfaces, and how it "captures" the geometric intersection number between curves. It will continue with the description of the string bracket, which generalizes of the Goldman bracket to oriented manifolds of dimension larger than two, and the space of families of loops where the string bracket is defined. The second part of the lecture describes how this structure in degrees zero and one plus the power operations in degree zero recognizes key features of the Geometrization, the above mentioned joint work. The lions share of effort concerns the torus decomposition of three manifolds which carry mixed geometry. This is joint work with Siddhartha Gadgil and Dennis Sullivan.

Tuesday, September 25, 2018

12:00 pm in 243 Altgeld Hall,Tuesday, September 25, 2018

Normal generators for mapping class groups are abundant

Justin Lanier (Georgia Tech Math)

Abstract: For mapping class groups of surfaces, we provide a number of simple criteria that ensure that a mapping class is a normal generator, with normal closure equal to the whole group. We then apply these criteria to show that every nontrivial periodic mapping class that is not a hyperelliptic involution is a normal generator whenever genus is at least 3. We also show that every pseudo-Anosov mapping class with stretch factor less than √2 is a normal generator. Showing that pseudo-Anosov normal generators exist at all answers a question of Darren Long from 1986. In addition to discussing these results on normal generators, we will describe several ways in which they can be leveraged to answer other questions about mapping class groups. This is joint work with Dan Margalit.

Tuesday, October 2, 2018

12:00 pm in 243 Altgeld Hall,Tuesday, October 2, 2018

Generalized square knots and the 4-dimensional Poincare Conjecture

Alex Zupan (University of Nebraska-Lincoln)

Abstract: The smooth version of the 4-dimensional Poincare Conjecture (S4PC) states that every homotopy 4-sphere is diffeomorphic to the standard 4-sphere. One way to attack the S4PC is to examine a restricted class of 4-manifolds. For example, Gabai's proof of Property R implies that every homotopy 4-sphere built with one 2-handle and one 3-handle is standard. In this talk, we consider homotopy 4-spheres X built with two 2-handles and two 3-handles, which are uniquely determined by the attaching link L for the 2-handles in the 3-sphere. We prove that if one of the components of L is the connected sum of a torus knot T(p,2) and its mirror (a generalized square knot), then X is diffeomorphic to the standard 4-sphere. This is joint work with Jeffrey Meier.

Tuesday, November 13, 2018

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

Taut sutured handlebodies as twisted homology products

Margaret Nichols (University of Chicago)

Abstract: A basic problem in the study of 3-manifolds is to determine when geometric objects are of ‘minimal complexity’. We are interested in this question in the setting of sutured manifolds, where minimal complexity is called ‘tautness’. One method for certifying that a sutured manifold is taut is to show that it is homologically simple - a so-called ‘rational homology product’. Most sutured manifolds do not have this form, but do always take the more general form of a ‘twisted homology product’, which incorporates a representation of the fundamental group. The question then becomes, how complicated of a representation is needed to realize a given sutured manifold as such? We explore some classes of relatively simple sutured manifolds, and see one class is always a rational homology product, but that the next natural class contains examples which require twisting. We also find examples that require twisting by a representation which cannot be ‘too simple’.