Geometry/Topology Seminar

Fall 2019

Thursdays (and sometimes Tuesdays) 3:40-4:30pm, in Ryerson 358

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• 9/26/19
• 10/3/19
• 10/10/19
• 10/15/19
• 10/21/19
• 10/24/19
• 11/21/19
• 12/05/19

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• contacts

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• previous quarters
• S2019
• W2019
• F2018
• S2018
• 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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• • Thursday September 26 at 4:00-5:00pm in Ry 358
  • Daniel Woodhouse, University of Oxford
  • Action rigidity of free products of hyperbolic manifold groups.
  • Abstract: Gromov's program for understanding finitely generated groups up to their large scale geometry considers three possible relations: quasi-isometry, abstract commensurability, and acting geometrically on the same proper geodesic metric space. A *common model geometry* for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is *action rigid* if any group G' that has a common model geometry with G is abstractly commensurable to G. We show that free products of closed hyperbolic manifold groups are action rigid. As a corollary, we obtain torsion-free, Gromov hyperbolic groups that are quasi-isometric, but do not even virtually act on the same proper geodesic metric space. This is joint work with Emily Stark.
• • Thursday October 3 at 3:40-4:30pm in Ry 358
  • Lvzhou Chen, University of Chicago
  • Big mapping class groups and rigidity of the simple circle.
  • Abstract: Surfaces of infinite type, such as the plane minus a Cantor set, occur naturally in dynamics. However, their mapping class groups are much less studied and understood compared to the mapping class groups of surfaces of finite type. Many fundamental questions remain open. We will discuss the mapping class group G of the plane minus a Cantor set, and show that any nontrivial G-action on the circle is semi-conjugate to its action on the so-called simple circle. Along the way, we will discuss some structural results of G to address the following questions: What are some interesting subgroups of G? Is G generated by torsion elements? This is joint work with Danny Calegari.
• • Thursday October 10 at 3:40-4:30pm in Ry 358
  • Antoni Rangachev, UChicago
  • Lipschitz geometry and Zariski equisingularity
  • Abstract: After briefly reviewing the history and importance of various equisingularity theories, I will describe an approach to showing that Zariski equisingular families of hypersurfaces are Lipschitz equisingular in all dimensions. It is based on the well-known observation that Zariski equisingular families are birational to families of quasi-ordinary singularities having the same characteristic exponents. A singularity is called quasi-ordinary if there exists a finite map from it to an affine space whose discriminant is contained in a normal crossing divisor. I will show that two germs of quasi-ordinary singularities are biLipschitz equivalent if and only if they have the same normalized characteristic exponents. This is joint work with H. Mourtada and B. Teissier.
• • Tuesday October 15 at 3:40-4:30pm in Ry 358
  • Gavril Farkas, Humbolt Universitat zu Berlin
  • Topological invariants of groups via Koszul modules.
  • Abstract: I will discuss the deep connection between the structure of the equations of certain algebraic varieties and Alexander invariants of groups. On the topological side this produced a universal bound on the nilpotency index of Torelli groups or fundamental group of non-fibred Kähler groups, whereas on the algebro-geometric side it has led to a simple proof of Green's Conjecture on syzygies of canonical curves. Joint work with Aprodu, Papadima, Raicu and Weyman.
• • Monday October 21 at 1:30-2:30pm in Eckhart 203
  • Margaret Bilu, Courant Institute of Mathematical Sciences (NYU)
  • Motivic Euler products in motivic statistics.
  • Abstract: -- NOTE SPECIAL TIME AND PLACE-- The Grothendieck group of varieties over a field k is the quotient of the free abelian group of isomorphism classes of varieties over k by the so-called cut-and-paste relations. It moreover has a ring structure coming from the product of varieties. Many problems in number theory have a natural, more geometric counterpart involving elements of this ring. Thus, Poonen's Bertini theorem over finite fields has a motivic analog due to Vakil and Wood, which expresses the motivic density of smooth hypersurface sections as the degree goes to infinity in terms of a special value of Kapranov's zeta function. I will present a broad generalization of Vakil and Wood's result, which implies in particular a motivic analog of Poonen's Bertini theorem with Taylor conditions, as well as motivic analogs of many generalizations and variants of Poonen's theorem. A key ingredient for this is a notion of motivic Euler product which allows us to write down candidate motivic probabilities.
• • Monday October 21 at 2:30-3:30pm in Eckhart 203
  • Sean Howe, University of Utah
  • A(nother) conjecture about zeta functions, or, "it's zeta functions all the way down."
  • Abstract: -- NOTE SPECIAL TIME AND PLACE-- We conjecture a unification of arithmetic and motivic statistics over finite fields through a natural analytic topology on the ring of zeta functions. A key step will be to explain exactly what it means to evaluate the zeta function of a zeta function at a zeta function.
• • Thursday October 24 at 3:40-4:30pm in Ry 358
  • Khashayar Filom, Northwestern
  • On the topology of dynamical moduli spaces of rational maps
  • Abstract: The dynamical moduli space of rational maps of degree d, defined as the space of Möbius conjugacy classes of degree d holomorphic self-maps of the Riemann sphere, is a ubiquitous object in complex and arithmetic dynamics. Using the techniques of Geometric Invariant Theory, Silverman constructs this orbit space as an affine variety of dimension 2d-2 which admits a model over the rationals. In the case of degree two, Milnor identifies this space with the affine plane. I will present the results of a joint work with Maxime Bergeron and Sam Nariman regarding the topology of these moduli spaces. We compute the fundamental group of the dynamical moduli space and show that the space is rationally acyclic while its cohomology groups with finite coefficients could be non-trivial. As an application, the ranks of certain rational homotopy groups of the parameter space of rational maps (within the unstable range) will be computed.
• • Thursday November 21 at 3:40-4:30pm in Ry 358
  • Michelle Chu, University of Illinois at Chicago
  • Arithmetic hyperbolic 3-manifolds
  • Abstract: The study of virtual properties of 3-manifolds groups has played a key role in major recent developments in 3-manifold topology. In this talk I will motivate and introduce the study of arithmetic hyperbolic manifolds and discuss some recent results on quantifying their virtual properties.
• • Thursday December 05 at 3:40-4:30pm in Ry 358
  • Tuomas Sahlsten, Manchester
  • Quantum chaos on random surfaces of large genus
  • Abstract: Quantum chaos and problems in the field (quantum unique ergodicity, random wave conjecture) provide a challenging frontier that have numerous connections to various fields around mathematics. Some of the central works in the field are in the context of arithmetic surfaces and also recently in large random graphs. Inspired by these works, we study quantum chaos on random surfaces when modelled under Weil-Petersson probability in the moduli space of genus g Riemann surfaces. We employ the recent advancements of Maryam Mirzakhani on computing expectations of geometric-topological observables on the moduli space to establish quantum ergodicity and new bounds for L^p norms of eigenfunctions of the Laplacian for large genus random surfaces. Based on joint works with Cliff Gilmore (Cork), Etienne Le Masson (Cergy), and Joe Thomas (Manchester).

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
• Nate Harman, nharman at math.uchicago.edu
• Kasia Jankiewicz, kasia at math.uchicago.edu
• Kevin Schreve, kschreve at math.uchicago.edu