Geometry/Topology Seminar

Spring 2018

Thursdays (and sometimes Tuesdays) 3-4pm, in Ryerson 358

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• 04/03/18
• 04/10/18
• 04/26/18
• 05/03/18
• 05/08/18
• 05/09/18
• 05/10/18
• 05/15/18
• 05/17/18
• 05/24/18
• 05/29/18
• 07/12/18
• 10/11/18

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• contacts

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• previous quarters
• W2018
• F2017
• S2017
• W2017
• F2016
• S2016
• W2016
• F2015
• S2015
• W2015
• F2014
• S2014
• W2014
• F2013
• S2013
• W2013
• F2012
• S2012
• W2012
• F2011
• S2011
• W2011
• F2010
• S2010
• W2010
• F2009
• S2009
• W2009
• F2008
• S2008
• W2008
• F2007
• S2007
• W2007
• F2006
• S2006
• W2006

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• • Tuesday April 03 at 3-4pm in Ry 358
  • Manuel Krannich, Copenhagen
  • Topological moduli spaces, E₂-algebras, and homological stability
  • Abstract: Since the seventies, many families of topological moduli spaces have been proven to stabilize homologically, including moduli spaces of Riemann surfaces (Harer), unordered configuration spaces (McDuff, Segal), and moduli spaces of higher-dimensional manifolds (Galatius, Randal-Williams). From the perspective of homotopy theory, a common structure these examples share is that of an E₂-algebra, or at least of a module over such an algebra. In this talk, I will introduce a framework which provides a uniform treatment of classical and new (twisted) homological stability results from this perspective. If time permits, I will also discuss how these results imply representation stability for related moduli spaces. (This talk is joint with Topology Seminar.)
• • Tuesday April 10 at 3-4pm in Eck 203
  • Ben Knudsen, Harvard
  • Edge stabilization in the homology of graph braid groups.
  • Abstract: We discuss a novel type of stabilization map on the configuration spaces of a graph, which increases the number of particles occupying an edge. Through these maps, the homology of the configuration spaces forms a module over the polynomial ring generated by the edges of the graph, and we show that this module is finitely generated, implying eventual polynomial growth of Betti numbers over any field. Moreover, the action lifts to an action at the level of singular chains, which contains strictly more information; indeed, we show that this differential graded module is almost never formal over the ring of edges. These results, along with numerous calculations, arise from consideration of an explicit chain complex, which is a structured enhancement of a cellular model first considered by Swiatkowski. We arrive at this model through a local-to-global approach combining ideas from factorization homology and discrete Morse theory. (This talk is joint with Topology Seminar. THERE WILL BE A PRETALK AT 1:30 IN ECKHART 203)
• • Thursday April 26 at 3-4pm in Ry 358
  • Ramanujan Santharoubane, University of Virginia
  • Quantum representations of surface groups.
  • Abstract: I will show how we can produce exotic representations of surface groups from the Witten-Reshetikhin-Turaev TQFT. These representations have infinite images and give points on character varieties that are fixed by the action of the mapping. Moreover we can approximate these representations by representations into finite groups in order to build exotic regular finite covers of surfaces. These covers have the following property: the integral homology is not generated by pullbacks of simple closed curves on the base. This is joint work with Thomas Koberda.
• • Thursday May 03 at 3-4pm in Ry 358
  • Richard Canary, Michigan
  • Dynamics and the Hitchin component
  • Abstract: Hitchin discovered a component of the space of representations of a surface group into PSL(n,R), which bears many resemblances to the Teichmuller space of Fuchsian representations of the surface group into PSL(2,R). Labourie introduced dynamical techniques to show that these Hitchin representations are discrete, faithful quasi-isometric embeddings. Sambarino associated Anosov flows to Hitchin representations whose periods record the spectral data of the representation. In this talk, we will see how to use these flows to attach and study dynamical quantities to Hitchin representations, e.g. entropies, Liouville currents and associated Liouville volumes. We will also discuss rigidity results and natural Riemannian metrics on the Hitchin component. (These results are joint work with Martin Bridgeman, Francois Labourite and Andres Sambarino.)
• • Tuesday May 08 at 3-4pm in Ry 358
  • Miklós Abért, MTA Alfréd Rényi Institute of Mathematics
  • Limits of eigenfunctions in negative curvature.
  • Abstract: Abstract: Let M be a compact manifold with negative curvature. The Quantum Unique Ergodicity conjecture of Rudnick and Sarnak says that eigenfunctions of the Laplacian on M get equidistributed as the eigenvalue tends to infinity. The Quantum Ergodicity theorem by Shnirelman, Colin de Verdière, and Zelditch says that this is true for a density 1 subsequence of eigenvalues. On the other hand, a conjecture of Berry, that has not been formulated in a mathematically precise way, says that high energy eigenfunctions behave like Gaussian random Euclidean waves. Using the framework of Benjamini-Schramm convergence of Riemannian manifolds, we give a mathematically exact formulation of Berry’s conjecture. This allows us to establish a relation to Quantum Unique Ergodicity and to prove Berry’s conjecture in a specific setting. We also investigate the case when the limit lives on a negatively curved symmetric space, like covering towers of hyperbolic surfaces. Our approach is related to the recent result of Backhausz and Szegedy on almost eigenfunctions of random regular graphs. A major difference is that instead of expander graphs, here one deals with a very non-expander (hyperfinite) sequence of manifolds. In particular, opposed to the Backhausz-Szegedy theorem, Berry's conjecture will not hold for almost eigenfunctions.
• • Wednesday May 09 at 3-4pm in Ryerson 358
  • Dave Witte Morris, Lethbridge
  • Efficient bounded generation
  • Abstract: A subset X "boundedly generates" a group G if every element g of G is the product of a bounded number of elements of X. This is a very powerful notion in abstract group theory, but geometric group theorists may need the bounded generation to be "efficient", which means that each g can be written as a bounded number of elements of X whose sizes are bounded by a constant times the word length of g. Twenty-five years ago, Lubotzky, Mozes, and Raghunathan observed that SL(n,Z) is efficiently boundedly generated by the elements of its natural SL(2,Z) subgroups. We will explain the proof of this result, and discuss a recent generalization to other arithmetic groups.
• • Thursday May 10 at 3-4pm in Ry 358
  • Mikolaj Fraczyk, MTA Alfréd Rényi Institute of Mathematics
  • Growth of mod-2 homology of higher rank locally symmetric spaces.
  • Abstract: We study the geometry of length-minimizing representatives of mod-2 homology classes for a locally symmetric space M modeled after a fixed higher rank symmetric space X . Using the presence of large 2-dimensional local flats in M one can show that every first mod-2 homology class in M is represented by a cycle of total length at most o(volume(M)). In this talk I will sketch a short proof of this bound and explain how it implies that the first mod-2 Betti number of M is o(volume(M)). Because of our geometric approach those results are valid even for sequences of non-commensurable locally symmetric spaces - a case that wasn't treated before in the context of mod-p homology groups. If time permits I will discuss the difficulties encountered in adapting our argument to the mod-p setting for p odd and describe the link between this problem and certain invariant random patterns on higher rank symmetric spaces.
• • Tuesday May 15 at 3-4pm in Ry 358
  • Eduard Looijenga, Tsinghua University
  • Representations of mapping class groups associated with surfaces endowed with a fixed automorphism group. (joint work with Marco Boggi)
  • Abstract: Riemann surfaces endowed with a faithful action of a finite group of a given topological type have their own moduli space. These have monodromy representations which commute with the action of the finite group and give rise to virtual representations of the mapping class group of the orbit surface. With the help of Hodge theory we obtain some properties of these (topologically defined) representations. NOTE: These two talks will be in an informal style, where the audience is particularly encouraged to ask questions.
• • Thursday May 17 at 3-4pm in Ry 358
  • Eduard Looijenga, Tsinghua University
  • Representations of mapping class groups associated with surfaces endowed with a fixed automorphism group. (joint work with Marco Boggi)
  • Abstract: Riemann surfaces endowed with a faithful action of a finite group of a given topological type have their own moduli space. These have monodromy representations which commute with the action of the finite group and give rise to virtual representations of the mapping class group of the orbit surface. With the help of Hodge theory we obtain some properties of these (topologically defined) representations. NOTE: These two talks will be in an informal style, where the audience is particularly encouraged to ask questions.
• • Thursday May 24 at 3-4pm in Ry 358
  • Dmitri Gekhtman, Caltech
  • Holomorphic retracts of Teichmuller space
  • Abstract: The Teichmuller space of a closed surface carries a natural complex structure, whose analytic properties reflect the topology and geometry of the surface. In this talk, we discuss the problem of classifying the holomorphic retracts of Teichmuller space. Our approach hinges on the analysis of two dynamical flows - one in the moduli space of half-translation surfaces, and the other in the space of bounded holomorphic functions on the polydisk. We will also discuss connections to the mapping class group.
• • Tuesday May 29 at 3-4pm (pretalk 1:30-2:30) in Eck 203
  • Sander Kupers, Harvard
  • Cellular E₂-algebras and the unstable homology of mapping class groups
  • Abstract: We discuss joint work with Soren Galatius and Oscar Randal-Williams on the application of higher-algebraic techniques to classical questions about the homology of mapping class groups. This uses a new "multiplicative" approach to homological stability -- in contrast to the "additive" one due to Quillen -- which has the advantage of providing information outside of the stable range.
• • Thursday July 12 at 3-4pm in Ry 358
  • Christin Bibby, University of Michigan
  • Combinatorics of orbit configuration spaces
  • Abstract: From a group action on a space, define a variant of the configuration space by insisting that no two points inhabit the same orbit. When the action is almost free, this "orbit configuration space” is the complement of an arrangement of subvarieties inside the cartesian product, and we use this structure to study its topology. We give an abstract combinatorial description of its poset of layers (connected components of intersections from the arrangement) which turns out to be of much independent interest as a generalization of partition and Dowling lattices. The close relationship to these classical posets is then exploited to give explicit cohomological calculations akin to those of (Totaro '96)
• • Thursday October 11 at 3-4pm in Ry 358
  • Rita Gitik, University of Michigan
  • On Tame Subgroups of Finitely Presented Groups
  • Abstract: We describe several examples of tame subgroups of finitely presented groups and prove that the fundamental groups of certain finite graphs of groups are locally tame.

Due to the high number of requests, we are no longer accepting speakers via self-invitations.

For questions, contact

• Benson Farb, farb at math.uchicago.edu
• Alex Eskin, eskin at math.uchicago.edu
• Shmuel Weinberger, shmuel at math.uchicago.edu
• Howard Masur, masur at math.uchicago.edu
• Danny Calegari, dannyc at math.uchicago.edu
• Maxime Bergeron, mbergeron at math.uchicago.edu