Geometry/Topology Seminar

Fall 2015

Thursdays (and sometimes Tuesdays) 3-4pm, in Eckhart 308

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• 4/16/15
• 4/30/15
• 5/4/15
• 5/5/15
• 5/7/15
• 5/8/15
• 5/19/15
• 5/26/15
• 6/2/15
• 10/13/15
• 10/22/15

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• contacts

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• previous quarters
• 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 April 16 at 3-4pm in Eck 308
  • Steven Boyer, UQAM
  • Foliations, left-orders, and L-spaces
  • Abstract: Much work has been devoted in recent years to examining relationships between the existence of a co-oriented taut foliation on a closed, connected, prime 3-manifold W, the left-orderability of the fundamental group of W, and the property that W not be a Heegaard-Floer L-space. Classic work shows that each of these conditions holds when W has a positive first Betti number and it has been conjectured that they coincide when the first Betti number of W âis zero. In this talk I will discuss the known connections between these conditions and survey the current status of the conjectures.
• • Thursday April 30 at 3-4pm in Eck 308
  • Stefan Witzel, Bielefeld
  • Higher finiteness properties of arithmetic groups in positive characteristic
  • Abstract: Topological finiteness properties of groups generalize the properties of being finitely generated and of being finitely presented. Groups having some but not all of these properties naturally arise in the realm of arithmetic groups. For example, SL₃(F_q[t]) is finitely generated but not finitely presented. I will speak about how this €example generalizes. The talk is based on joint work with Kai-Uwe Bux and Ralf Kohl.
• • Monday May 4 at 1:30-2:30pm in Eck 202
  • Alex Kontorovich, Rutgers
  • The Unreasonable Effectiveness of Thin Groups
  • Abstract: Thin groups are certain arithmetic subgroups of Lie groups whose quotient manifolds have infinite volume. We will describe a number of problems in pseudorandom numbers, numerical integration, and homogeneous dynamics which, it turns out, are all the same problem when viewed through the lens of thin groups.
• • Tuesday May 5 at 3-4pm in Eck 308
  • Dave Witte Morris, Lethbridge
  • Arithmetic subgroups whose representations all map into GL(n,Z)
  • Abstract: Suppose Gamma is an arithmetic subgroup of a semisimple Lie group G. For any homomorphism rho from Gamma to GL(n,R), a classical paper of J.Tits determines whether rho(Gamma) is conjugate to a subgroup of GL(n,Z). Combining this with an easy surjectivity result in Galois cohomology provides a short proof of the known fact that every G has an arithmetic subgroup Gamma, such that the containment is true for every homomorphism rho. We will not assume the audience is acquainted with arithmetic groups, Galois cohomology, or the theorem of Tits.
• • Thursday May 7 at 3-4pm in Eck 308
  • Jeff Danciger, Austin
  • Convex projective structures on non-hyperbolic three-manifolds
  • Abstract: Y. Benoist proved that if a closed three-manifold M admits a convex real projective structure, then M is topologically the union along tori of finitely many sub-manifolds each of which admits a complete finite volume hyperbolic structure on its interior. We describe some initial results in the direction of a potential converse to Benoist's theorem. Specifically, we show that a cusped hyperbolic three-manifold may (under assumptions) be deformed to convex projective structures with totally geodesic torus boundary. Such structures may be (convexly) glued together whenever the geometry at the boundary matches up. In particular, we prove that many doubles of cusped hyperbolic three-manifolds admit convex projective structures. Joint work with Sam Ballas and Gye Seon Lee.
• • Friday May 8 at 1:30-2:30pm in Eck 202
  • Alex Suciu, Northeastern University
  • Resonance, representations, and the Johnson filtration
  • Abstract: I will describe a relationship between the classical representation theory of a complex, semisimple Lie algebra \mathfrak{g} and the resonance varieties R(V,K)\subset V^* attached to irreducible \mathfrak{g}-modules V and submodules K\subset V\wedge V. In the process, I will give a precise roots-and-weights criterion insuring the vanishing of these varieties. In the case when \mathfrak{g}= \mathfrak{sl}_n(C) or \mathfrak{sp}_2g(C), our approach yields a unified proof of two vanishing results for the resonance varieties of the (outer) Torelli groups of finitely generated free groups and surface groups. In turn, these vanishing results reveal certain homological finiteness properties of the Johnson filtration. This is joint work with Stefan Papadima.
• • Tuesday May 19 at 4-5pm in Eck 308
  • Christoforos Neofytidis, Binghamton
  • Groups presentable by products and an ordering of Thurston geometries by maps of non-zero degree
  • Abstract: The existence of a map of non-zero degree defines a transitive relation on the homotopy types of closed oriented manifolds of the same dimension, called domination relation. Gromov suggested investigating the domination relation as defining an ordering of manifolds and formulated numerous conjectures regarding classes that might (not) be comparable under this relation. A particular case of the domination question, which specifies Steenrod's classical problem on the realization of homology classes by manifolds, and is motivated by Milnor-Thurston and Gromov's theory of functorial semi-norms on homology (such as the simplicial volume), is when the domain is a non-trivial direct product. In this talk, I will discuss the (non-)existence of maps of non-zero degree from direct products to aspherical manifolds with fundamental groups with non-trivial center. As an application, I will describe an ordering of all non-hyperbolic aspherical 4-manifolds possessing a Thurston geometry.
• • Tuesday May 26 at 3-4pm in Eck 308
  • Ailsa Keating, Columbia University
  • Higher-dimensional Dehn twists and symplectic mapping class groups
  • Abstract: Given a Lagrangian sphere S in a symplectic manifold M of any dimension, one can associate to it a symplectomorphism of M, the Dehn twist about S. This generalises the classical two-dimensional notion. These higher-dimensional Dehn twists naturally give elements of the symplectic mapping class group of M, i.e. \pi₀ (Symp (M)). The goal of the talk is to present parallels between properties of Dehn twists in dimension 2 and in higher dimensions, with an emphasis on relations in the mapping class group.
• • Tuesday June 2 at 3-4pm in Eck 308
  • Michael Hull, UIC
  • Acylindrically hyperbolic groups
  • Abstract: Hyperbolic and relatively hyperbolic groups have played an important role in the development of geometric group theory. However, there are many other groups which admit interesting and useful actions on hyperbolic metric spaces, including mapping class groups, Out(F_n), right-angled Artin groups, and many 3-manifold groups. The class of acylindrically hyperbolic groups provides a framework for studying all of these groups (and many more) using many of the same techniques developed for hyperbolic and relatively hyperbolic groups. We will give a brief survey of examples and properties of acylindrically hyperbolic groups and show how the study of this class has yielded new results in a number of particular cases.
• • Tuesday October 13 at 3-4pm in Eck 308
  • Igor Belegradek, Georgia Tech
  • TBA
  • Abstract: TBA
• • Thursday October 22 at 3-4pm in Eck 308
  • Andrew Saunders, UIC
  • TBA
  • Abstract: TBA

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
• Dominic Dotterrer, d.dotterrer at math.uchicago.edu
• Rohit Nagpal, nagpal@math.uchicago.edu