Department of


Seminar Calendar
for Geometry, Groups and Dynamics/GEAR events the year of Saturday, December 16, 2017.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, January 19, 2017

12:00 pm in 243 Altgeld Hall,Thursday, January 19, 2017

Orders from $\widetilde{PSL_2(\mathbb{R})}$ Representations and Non-examples.

Xinghua Gao   [email] (University of Illinois)

Abstract: Let $M$ be an integer homology 3-sphere. One way to study left-orderability of $\pi_1(M)$ is to construct a non-trivial representation from $\pi_1(M)$ to $\widetilde{PSL_2(\mathbb{R})}$. However this method does not always work. In this talk, I will give examples of non L-space irreducible integer homology 3-spheres whose fundamental groups do not have nontrivial $\widetilde{PSL_2(\mathbb{R})}$ representations.

Tuesday, January 24, 2017

12:00 pm in 243 Altgeld Hall,Tuesday, January 24, 2017

Boundary maps for some hierarchically hyperbolic spaces

Sarah Mousley (Illinois Math)

Abstract: There are natural embeddings of right-angled Artin groups $G$ into the mapping class group $Mod(S)$ of a surface $S$. The groups $G$ and $Mod(S)$ can each be equipped with a geometric structure called a hierarchically hyperbolic space (HHS) structure, and there is a notion of a boundary for such spaces. In this talk, we will explore the following question: does an embedding $\phi: G \rightarrow Mod(S)$ extend continuously to a boundary map $\partial G \rightarrow \partial Mod(S)$? That is, given two sequences $(g_n)$ and $(h_n)$ in $G$ that limit to the same point in $\partial G$, do $(\phi(g_n))$ and $(\phi(h_n))$ limit to the same point in $\partial Mod(S)$? No background in HHS structures is needed.

Thursday, February 2, 2017

12:00 pm in 243 Altgeld Hall,Thursday, February 2, 2017

Algebraic and topological properties of big mapping class groups

Nick Vlamis (U Michigan Math)

Abstract: There has been a recent surge in studying surfaces of infinite type, i.e. surfaces with infinitely-generated fundamental groups. In this talk, we will focus on their mapping class groups, often called big mapping class groups. In contrast to the finite-type case, there are many open questions regarding the basic algebraic and topological properties of big mapping class groups. I will discuss several such questions and provide some answers. In particular, I will focus on automorphisms of pure mapping class groups and topological generating sets. This work is joint with Priyam Patel.

Tuesday, February 7, 2017

12:00 pm in 243 Altgeld Hall,Tuesday, February 7, 2017

The generation problem in Thompson group F

Gili Golan (Vanderbilt)

Abstract: We show that the generation problem in Thompson group F is decidable, i.e., there is an algorithm which decides if a finite set of elements of F generates the whole F. The algorithm makes use of the Stallings 2-core of subgroups of F, which can be defined in an analogue way to the Stallings core of subgroups of a free group. An application of the algorithm shows that F is a cyclic extension of a group K which has a maximal elementary amenable subgroup B. The group B is a copy of a subgroup of F constructed by Brin and Navas.

Tuesday, February 21, 2017

12:00 pm in 243 Altgeld Hall,Tuesday, February 21, 2017

Groups Associated with Rational Proper Maps

John P. D'Angelo (Illinois Math)

Abstract: Given a rational proper map $f$ between balls of typically different dimensions, we define a subgroup $\Gamma_f$ of the source automorphism group. We prove that this group is noncompact if and only if $f$ is linear. We show how these groups behave under certain constructions such as juxtaposition and partial tensor products. We then sketch a proof of the following result. If $G$ is an arbitrary finite subgroup of the source automorphism group, then there is a rational map $f$ for which $\Gamma_f = G$. We provide many examples and, if time permits, discuss the degree estimate conjecture. This work is joint with Ming Xiao.

Thursday, March 2, 2017

12:00 pm in 243 Altgeld Hall,Thursday, March 2, 2017

Hyperbolic volumes of random links

Malik Obeidin   [email] (University of Illinois)

Abstract: What does a random link look like? There have been a few different proposed models for sampling from the set of links -- in this talk, I will describe a model based on random link diagrams in the plane. Such diagrams can be sampled uniformly on a computer due to the work of Gilles Schaeffer, so one can experiment with various invariants of links with the topology software SnapPy. I will present data showing what happens with some of the different invariants SnapPy can compute, and I will outline a proof that the hyperbolic volume of the complement of a random alternating link diagram is asymptotically a linear function of the number of crossings. In contrast, for nonalternating links, I will show why the diagrams we get generically represent satellite (and hence nonhyperbolic) links.

Thursday, March 9, 2017

12:00 pm in 243 Altgeld Hall,Thursday, March 9, 2017

Polynomial-time curve reduction

Mark Bell (Illinois Math)

Abstract: A pair of curves on a surface can appear extremely complicated and so it can be difficult to determine properties such as their intersection number. We will discuss a new argument that, when the curve is given by its intersections with the edges of an ideal triangulation, there is always a "reduction" to a simpler configuration in which such calculations are straightforward. This relies on finding an edge flip or a (power of a) Dehn twist that decreases the complexity of a curve by a definite fraction.

Tuesday, March 14, 2017

12:00 pm in 243 Altgeld Hall,Tuesday, March 14, 2017

Flat submaps in CAT(0) $(p,q)$-maps and maps with angles

Mark Sapir (Vanderbilt and Illinois Math)

Abstract: This is a joint work with A. Olshanskii. Let $p, q$ be positive integers with $1/p+1/q=1/2$. We prove that if a $(p,q)$-map $M$ does not contain flat submaps of radius $\ge r$, then its area does not exceed $c(r+1)n$ where $n$ is the perimeter of $M$ and $c$ is an absolute constant. Earlier Ivanov and Schupp proved an exponential bound in terms of $r$. We prove an estimate similar to Ivanov and Schupp for much more general ``maps with angles" which include, for example, van Kampen diagrams over the presentation of the Baumslag-Solitar group $BS(1,2)$ and many groups corresponding to $S$-machines. We also show that a $(p,q)$ map $M$ tessellating a plane ${\mathbb R}^2$ has path metric quasi-isometric to the Euclidean metric on the plane if and only if $M$ has only finitely many non-flat vertices and faces.

Tuesday, March 28, 2017

12:00 pm in 243 Altgeld Hall,Tuesday, March 28, 2017

Alternating knots and Montesinos knots satisfy the L-space knot conjecture. Joint work with Rachel Roberts

Charles Delman (EIU Math)

Abstract: An L-space is a homology \(3\)-sphere whose Heegard-Floer homology has minimal rank; lens spaces are examples (hence the name). Results of Ozsváth - Szabó, Eliashberg -Thurston, and Kazez - Roberts show that a manifold admitting a taut, co-orientable foliation cannot be an L-space. Let us call such a manifold foliar. Ozsváth and Szabó have asked whether or not the converse is true for irreducible \(3\)-manifolds; Juhasz has conjectured that it is. Restricting attention to manifolds obtained by Dehn surgery on knots in \(S^3\), we posit the following: L-space Knot Conjecture. Suppose \( \kappa \subset S^3\) is a knot in the 3-sphere. Then a manifold obtained by Dehn filling along \(\kappa\) is foliar if and only if it is irreducible and not an L-space. Using generalized surface decomposition techniques that build on earlier work of Gabai, Menasco, Oertel, and the authors, we prove that both alternating knots and Montesinos Knots satisfy the L-space Knot Conjecture. We believe these techniques will prove fruitful in the further study of taut foliations in \(3\)-manifolds.

Thursday, April 6, 2017

12:00 pm in 243 Altgeld Hall,Thursday, April 6, 2017

Crash course in convex cocompactness.

Autumn Kent (Wisconsin Math)

Abstract: Farb and Mosher introduced the notion of convex cocompactness from Kleinian groups to the theory of mapping class groups, with the beautiful application of potentially producing atoroidal non-hyperbolic groups of finite type, which would provide counterexamples to Gromov's coarse-hyperbolization conjecture for infinite groups. I'll give an overview of the foundations of the theory and the current state of things.

Tuesday, April 11, 2017

12:00 pm in 243 Altgeld Hall,Tuesday, April 11, 2017

The conjugacy problem for Mod(S)

Mark Bell (Illinois Math)

Abstract: We will discuss a new approach for tackling the conjugacy problem for the mapping class group of a surface. This relies on recently developed tools for finding tight geodesics in the curve complex. This is joint work with Richard Webb.

Thursday, April 13, 2017

12:00 pm in 243 Altgeld Hall,Thursday, April 13, 2017

Stable Strata of Geodesics in Outer Space

Catherine Pfaff (UC Santa Barbara)

Abstract: Outer automorphisms of free groups are understood via their action on Culler-Vogtmann Outer space. We study the behavior of geodesics in Outer Space by focusing on a class of geodesics that have strong stability properties. We further show that pathologies exist among these geodesics that don't occur, for example, in the Teichmuller space setting. This is joint work with Yael Algom-Kfir and Ilya Kapovich.

Tuesday, April 18, 2017

12:00 pm in 243 Altgeld Hall,Tuesday, April 18, 2017

Compatibility of length functions and the geometry of their deformation spaces

Edgar Bering (UIC Math)

Abstract: Given two length functions l, m of minimal irreducible G actions on R-trees A, B, when is l + m again the length function of a minimal irreducible G action on an R-tree? We will show that additivity is characterized by the geometry of the Guirardel core of A and B, and also by a combinatorial compatibility condition generalizing the condition given by Behrstock, Bestvina, and Clay for Fn actions on simplicial trees. This compatibility condition allows us to characterize the PL-geometry of common deformation spaces of R-trees, such as the closure of Culler-Vogtmann Outer Space or the space of small actions of a hyperbolic group G.

Thursday, April 20, 2017

12:00 pm in 243 Altgeld Hall,Thursday, April 20, 2017

Special subgroups of Bianchi groups

Michelle Chu (UT Austin Math)

Abstract: The study of (virtually) special cube complexes and groups played a key role in the resolution of the virtual Haken and virtual fibering conjectures. Recently there has been interest in determining the index of special subgroups of virtually special groups. In this talk, we answer this question in the case of the Bianchi groups. We find special congruence subgroups for each Bianchi group $PSL(2,O_d)$ with uniformly bounded index independent of d.

Thursday, May 4, 2017

12:00 pm in 243 Altgeld Hall,Thursday, May 4, 2017

The geometry of maximal SO(2,n) representations

Brian Collier   [email] (University of Maryland)

Abstract: Like Hitchin representations, the set of maximal representations of a surface group into a Lie group of Hermitian type form connected components of representations with Labourie's Anosov property. For such representations, Guichard and Wienhard have constructed geometric structures modeled on certain flag varieties. In this talk we will use Higgs bundle techniques to construct a unique maximal space-like surface in the pseudo-Riemannian hyperbolic space H2,n-1 associated to any maximal SO(2,n) representation. Using the geometry of this surface, we construct geometric structures on certain homogeneous bundles on the surface and prove that they agree with those of Guichard-Wienhard. As a corollary, we also prove (a generalization of) a conjecture of Labourie for all maximal representations into rank 2 Hermitian Lie groups.This is based on joint work with Nicholas Tholozan and Jérémy Toulisse.

Tuesday, August 29, 2017

12:00 pm in 243 Altgeld Hall,Tuesday, August 29, 2017

The twisted rabbit problem via the arc complex.

Rebecca Winarski (UW Milwaukee)

Abstract: The twisted rabbit problem is a celebrated problem in complex dynamics. Work of Thurston proves that up to equivalence, there are exactly three branched coverings of the sphere to itself satisfying certain conditions. When one of these branched coverings is modified by a mapping class, a map equivalent to one of the three coverings results. Which one? After remaining open for 25 years, this problem was solved by Bartholdi—Nekyrashevych using iterated monodromy groups. In joint work with Lanier and Margalit, we formulate the problem topologically and solve the problem using the arc complex.

Thursday, September 14, 2017

12:00 pm in Altgeld Hall,Thursday, September 14, 2017

Self-Similar Interval Exchange Transformations

Kelly Yancey   [email] (Institute for Defense Analyses)

Abstract: During this talk we will discuss the class of self-similar 3-IETs and show that they satisfy Sarnak's conjecture. We will do this by appealing to the theory of joinings. Specifically we will show how to prove the property of minimal self-joinings for substitution systems (self-similar IETs can be thought of in this context).

Thursday, September 21, 2017

12:00 pm in Altgeld Hall 243,Thursday, September 21, 2017

Pair correlation in Apollonian circle packings

Xin Zhang (University of Illinois at Urbana-Champaign)

Abstract: Consider four mutually tangent circles, one containing the other three. An Apollonian circle packing is formed when the remaining curvilinear triangular regions are recursively filled with tangent circles. The extensive study of this object in the last fifteen years has led to many beautiful theorems in number theory, graph theory, and homogeneous dynamics. In this talk I will discuss a new type of problems, which concern the fine scale structure of Apollonian circle packings. In particular, I will show that the limiting pair correlation of circles exists. A critical tool we use is an extended version of a theorem of Mohammadi-Oh on the equidistribution of expanding horospheres in infinite volume hyperbolic spaces. This work is motivated by an IGL project that I mentored in Spring 2017.

Tuesday, September 26, 2017

12:00 pm in 243 Altgeld Hall,Tuesday, September 26, 2017

Limits of cubic differentials and projective structures

David Dumas (University of Illinois at Chicago)

Abstract: A construction due independently to Labourie and Loftin identifies the moduli space of convex RP^2 structures on a compact surface S with the bundle of holomorphic cubic differentials over the Teichmueller space of S. We study pointed geometric limits of sequences that go to infinity in this moduli space while remaining over a compact set in Teichmueller space. For such a sequence, we construct a local limit polynomial (in one complex variable) which describes the rate and direction of accumulation of zeros of the cubic differentials about the sequence of base points. We then show that this polynomial determines the convex polygon in RP^2 that is the geometric limit of the images of the developing maps of the projective structures. This is joint work with Michael Wolf.

Thursday, September 28, 2017

12:00 pm in 243 Altgeld Hall,Thursday, September 28, 2017

Effective Twisted Conjugacy Separability of Nilpotent Groups

Mark Pengitore (Purdue Math)

Abstract: There has been a recent interest in providing effective proofs of separability properties such as residual finiteness and conjugacy separability. Unlike residual finiteness, conjugacy separability does not respect finite extensions. Thus, we introduce twisted conjugacy separability, originally defined by Fel'shtyn, in order to study effective conjugacy separability of finite extensions of conjugacy separable groups. In joint work with Jonas Dere, we provide an effective proof of twisted conjugacy separability of finitely generated nilpotent groups. That, in turn, provides an effective bound for conjugacy separability of all finite extensions of a fixed nilpotent group in terms of the asymptotic behavior of conjugacy separability of the base nilpotent group.

Tuesday, October 10, 2017

12:00 pm in 243 Altgeld Hall,Tuesday, October 10, 2017

Dynamics, geometry, and the moduli space of Riemann surfaces

Alex Wright (Stanford)

Abstract: The moduli space of Riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in Teichmuller dynamics lead to connections with homogeneous spaces, algebraic geometry, dynamics, and other areas. The talk will mention joint works with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.

Thursday, October 19, 2017

12:00 pm in 243 Altgeld Hall,Thursday, October 19, 2017

The Topology of Representation Varieties

Maxime Bergeron (University of Chicago)

Abstract: Let H be a finitely generated group, let G be a complex reductive algebraic group (e.g. a special linear group) and let K be a maximal compact subgroup of G (e.g. a special unitary group). I will discuss exceptional classes of groups H for which there is a deformation retraction of Hom(H,G) onto Hom(H,K), thereby allowing us to obtain otherwise inaccessible topological invariants of these representation spaces.

Thursday, October 26, 2017

12:00 pm in 243 Altgeld Hall,Thursday, October 26, 2017

Siegel-Veech transforms are in L2.

Jayadev Athreya (University of Washington)

Abstract: Let H denote a connected component of a stratum of translation surfaces. We show that the Siegel-Veech transform of a bounded compactly supported function on R2 is in L2(H,μ), where μ is the Masur-Veech measure on H, and give applications to bounding error terms for counting problems for saddle connections. We will review classical results in the Geometry of Numbers which anticipate this result. This is joint work with Yitwah Cheung and Howard Masur.

Monday, November 13, 2017

4:00 pm in 343 Altgeld Hall,Monday, November 13, 2017

Stability in the homology of configuration spaces

Jenny Wilson (Stanford)

Abstract: This talk will illustrate some patterns in the homology of the space F_k(M) of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of these configuration spaces to become increasingly complicated. Church and others showed, however, that when M is connected and open, there is a representation-theoretic sense in which the homology groups of these spaces stabilize. In this talk I will explain these stability patterns, and describe higher-order stability phenomena – relationships between unstable homology classes in different degrees – established in recent work joint with Jeremy Miller. This project was inspired by work-in-progress of Galatius–Kupers–Randal-Williams.

Tuesday, November 14, 2017

12:00 pm in 243 Altgeld Hall,Tuesday, November 14, 2017

Skeins and Characters

Charles Frohman (U. Iowa)

Abstract: Skein theory is a $K$-theoretic like construction. Think of the underlying three- manifold as a ring, and a link in that manifold as a projective module. Crossings correspond to extensions of one module by another, and the skein relation says that the extension is equivalent to the direct sum of the two links that it extends. The skein module is the K-group from this relation. If the underlying three manifold is a cylinder over a surface, the links act like a category of bimodules, and the skein module is an algebra. In the talk, I will define the Kauffman bracket skein algebra and describe its properties.

Tuesday, November 28, 2017

12:00 pm in 243 Altgeld Hall,Tuesday, November 28, 2017

On Cayley and Langlands type correspondences for Higgs bundles

Laura Schaposnik   [email] (UIC)

Abstract: The Hitchin fibration is a natural tool through which one can understand the moduli space of Higgs bundles and its interesting subspaces (branes). After reviewing the type of questions and methods considered in the area, we shall dedicate this talk to the study of certain branes which lie completely inside the singular fibres of the Hitchin fibrations. Through Cayleyand Langlands type correspondences, we shall provide a geometric description of these objects, and consider the implications of our methods in the context of representation theory, Langlands duality, and within a more generic study of symmetries on moduli spaces.

Thursday, November 30, 2017

12:00 pm in 243 Altgeld Hall,Thursday, November 30, 2017

Green metric, Ancona inequalities and Martin boundary for relatively hyperbolic groups

Ilya Gekhtman (Yale University)

Abstract: Generalizing results of Ancona for hyperbolic groups, we prove that a random path between two points in a relatively hyperbolic group (e.g. a nonuniform lattice in hyperbolic space) has a uniformly high probability of passing any point on a word metric geodesic between them that is not inside a long subsegment close to a translate of a parabolic subgroup. We use this to relate three compactifications of the group: the Martin boundary associated with the random walk, the Bowditch boundary, associated to an action of the group on a proper hyperbolic space, and the Floyd boundary, obtained by a certain rescaling of the word metric. We demonstrate some dynamical consequences of these seemingly combinatorial results. For example, for a nonuniform lattice G in hyperbolic space H^n, we prove that the harmonic (exit) measure on the boundary associated to any finite support random walk on G is singular to the Lebesgue measure. Moreover, we construct a geodesic flow and G invariant measure on the unit tangent bundle of hyperbolic space projecting to a finite measure on T^1H^n/G whose geodesic current is equivalent to the square of the harmonic measure. The axes of random loxodromic elements in G equidistribute with respect to this measure. Analogous results hold for any geometrically finite subgroups of isometry groups of manifolds of pinched negative curvature, or even proper delta-hyperbolic metric spaces.

Tuesday, December 5, 2017

12:00 pm in 243 Altgeld Hall,Tuesday, December 5, 2017

Characteristic random subgroups and their applications

Rostyslav Kravchenko (Northwestern University)

Abstract: The invariant random subgroups (IRS) were implicitly used by Stuck and Zimmer in 1994 and defined explicitly by Abert, Glasner and Virag in 2012. They were actively studied since then. We define the notion of characteristic random subgroups (CRS) which are a natural analog of IRSs for the case of the group of all automorphisms. We determine CRS for free abelian groups and for free groups of finite rank. Using our results on CRS of free groups we show that for groups of geometrical nature (like hyperbolic groups, mapping class groups and outer automorphisms groups) there are infinitely many continuous ergodic IRS. This is a joint work with R. Grigorchuk and L. Bowen

Thursday, December 7, 2017

12:00 pm in 243 Altgeld Hall,Thursday, December 7, 2017

Least Dilatation of Pure Surface Braids

Marissa Loving (Illinois Math)

Abstract: The n-stranded pure surface braid group of a genus g surface can be described as the subgroup of the pure mapping class group of a surface of genus g with n-punctures which becomes trivial on the closed surface. For the n=1 case, much is known about this group including upper and lower bounds on the least dilatation of its pseudo-Anosovs due to Dowdall and Aougab—Taylor. I am interested in the least dilatation of pseudo-Anosov pure surface braids for n>1 punctures. In this talk, I will describe the upper and lower bounds I have proved as a function of g and n