General information

Some of the talks I have given, both seminar and conference talks and general expository ones.

Seminar and Conference Talks

Title: Finitary failure of co-Hopficity: Self-embeddings of groups.

?Feb 2017?: CUNY, New York Group Theory seminar.

Abstract: TBD

Title: Coarse geometry of expanders from homogeneous spaces.

?Feb 2017?: UT Austin, Groups and Dynamics Seminar.

?Feb 2017?: University of Notre Dame, Klein Seminar.

?Feb 2017?: Courant Institute, NYU -- Dynamical Systems Seminar

Nov 2017: University of Chicago, Geometry and Topology Seminar

Nov 2017: `Hyperbolic 3-manifolds and Beyond' -- Virginia Topology Conference.

Oct 2017: Lightning talk at `No Boundaries' (Farb Conference) -- University of Chicago.

Abstract: Subgroups of compact Lie groups give rise to expander graphs via a warped cone construction. We study the dependence of the coarse geometry of such expander graphs on the original subgroup and show they must exhibit a wide range of geometric behavior: The coarse geometry of the warped cone determines the subgroup up to conjugacy. As an application, we produce uncountably many non-quasi-isometric expanders. This is joint work with David Fisher and Thang Nguyen.

Title: Towers of regular self-covers and linear endomorphisms of tori.

July 2017: Seoul National University, South Korea -- Geometry/Topology/Dynamics Seminar.

July 2017: KAIST, Daejeon, South Korea -- Geometry and Topology Seminar.

May 2017: Workshop `Approximation, Deformation, Quasification', Isaac Newton Institute, Cambridge.

Apr 2017: Graduate Student Geometry and Topology Conference, Early career speaker.

Oct 2016: Penn State Dynamical Systems Seminar.

May 2016: Midwest Topology Seminar.

Abstract: Let M be a closed manifold that admits a nontrivial cover diffeomorphic to itself. Which manifolds have such a self-similar structure? Examples are provided by tori, in which case the covering is homotopic to a linear endomorphism. Under the assumption that all iterates of the covering of M are regular, we show that any self-cover is induced by a linear endomorphism of a torus on a quotient of the fundamental group. Under further hypotheses we show that a finite cover of M is a principal torus bundle. We use this to give an application to holomorphic self-covers of Kaehler manifolds.

Title: Rigidity of convex divisible domains in flag manifolds.

April 2017: Bloomington Geometry Workshop.

3 Dec 2015: Ohio State University, Geometric Group Theory Seminar

6 November 2015: University of Michigan -- Ann Arbor, Geometry Seminar.

Abstract: A projective structure on a manifold is a local modeling of the geometry on the geometry of projective space. Projective structures are notoriously flexibile: E.g. any hyperbolic manifold is canonically projective, but oftentimes the structure can be deformed. There are also projective structures on other manifolds altogether. A natural generalization of these structures is obtained by modeling the local geometry on other Grassmannians. In contrast to the plethora of examples of projective structures, we establish rigidity in this new context: We prove that in the Grassmannian of p-planes in \bbR^{2p}, p>1, every bounded convex domain with a compact quotient is a symmetric space. This is joint work with Andrew Zimmer.

Title: Higher rank rigidity in projective geometry.

30 March 2015: Rice University, Topology Seminar.

Abstract: Locally symmetric spaces (e.g. hyperbolic manifolds) form a class of manifolds exhibiting amazing rigidity from both algebraic and geometric points of view, especially when they have so-called "higher rank". They arise as quotients of bounded convex domains in either a projective space or a Grassmannian by a group of projective transformations. In the setting of projective geometry, Kac-Vinberg have shown there is no rigidity in a "projective rank one" setting, namely they construct nonsymmetric domains in projective spaces that nevertheless admit compact quotients. On the other hand we establish rigidity in the case of "highest possible projective rank": we prove that in the Grassmannian of p-planes in \bbR^{2p} every bounded convex domain with a compact quotient is symmetric. This is joint work with Andrew Zimmer.

Title: Symmetry gaps in Riemannian geometry and minimal orbifolds.

August 2016: `Reflections on Global Riemannian Geometry' (Grove Conference), Townsend, TN. Video.

March 2016: University of Chicago, Geometry and Topology Seminar.

15 March 2015: AMS Sectional Meeting (East Lansing, MI), Geometry of manifolds, singular spaces, and groups.

14 Nov 2014: Michigan State University, Geometry Seminar

30 Oct 2014: Indiana University - Bloomington, Geometry Seminar

19 Sep 2014: University of Michigan - Ann Arbor, Geometry Seminar

15 Sep 2014: Purdue University, Geometry Seminar

Abstract: In 1893 Hurwitz showed that a hyperbolic surface of genus at least 2 has isometry group of order at most 84(g-1). Do such bounds on the order of isometry groups exist more generally? It was conjectured by Farb-Weinberger that this is the case for certain aspherical manifolds. In this spirit we prove that the size of the isometry group of an arbitrary closed manifold is bounded in terms of certain geometric quantities (such as curvature and volume), unless the manifold admits an action by a compact connected Lie group. We give two applications of this result: First we characterize locally symmetric spaces among all Riemannian manifolds, and secondly, we generalize results of Kazhdan-Margulis and Gromov on the existence of minimal quotients of locally symmetric spaces and negatively curved manifolds.

Title: Riemannian manifolds with local symmetry.

22 Oct 2013: Ohio State University, Topology Seminar

23 Sep 2013: University of Maryland, Geometry and Topology Seminar

13 Jun 2013: Workshop in Geometric Topology (Grand Rapids, MI)

20 Oct 2012: AMS Sectional Meeting (Akron, OH), Interactions between Geometry and Topology

Abstract: In this talk I will discuss the problem of classifying all closed Riemannian manifolds whose universal cover has nondiscrete isometry group. Farb and Weinberger solved this under the assumption that M is aspherical: roughly, they proved that any such M is a fiber bundle with locally homogeneous fibers. However, if M is not aspherical, many new examples and phenomena appear. I will exhibit some of these, and discuss progress towards a classification.

Expository Talks

4 December 2014: FFSS Student Geometry Seminar (University of Chicago)

Title: Critical points of distance functions

21 October 2014: Topology reading group on Diffeomorphism groups

Title: The Bott Vanishing Theorem

25 and 28 August 2014: Geometry reading group on Entropy Rigidity and Bounded Cohomology

Title: The Milnor-Wood inequality and Goldman's Theorem

7 August 2014: Geometry reading group on Entropy Rigidity and Bounded Cohomology

Title: Volume rigidity after Bucher-Burger-Iozzi

29 July 2014: Geometry reading group on Entropy Rigidity and Bounded Cohomology

Title: Positivity of Simplicial Volume of Locally Symmetric Spaces (after Lafont-Schmidt)

3 and 8 July 2014: Geometry reading group on Entropy Rigidity and Bounded Cohomology

Title: Introduction, and Gromov's proof of Mostow Rigidity

29 May 2014: FFSS Student Geometry Seminar (University of Chicago)

Title: Symmetry gaps and minimal orbifolds

8 May 2014: FFSS Student Geometry Seminar (University of Chicago)

Title: The saga of strong approximation for thin groups

6 Mar 2014: FFSS Student Geometry Seminar (University of Chicago)

Title: The wild world of aspherical manifolds and the Davis trick

5, 7, and 10 Feb 2014: Morse Theory (course taught by Benson Farb at University of Chicago))

Title: Morse theory and closed geodesics

31 Oct 2013: FFSS Student Geometry Seminar (University of Chicago)

Title:The Auslander Conjecture

15 Oct 2013: Dynamics Reading Group on Complex Dynamics

Title: Basic Properties of Julia Sets

6 Jun 2013: FFSS Student Geometry Seminar (University of Chicago)

Title: Arithmetic Quantum Unique Ergodicity

4 Jun 2013: Dynamics Reading Group on Diagonalizable Flows on Homogeneous Spaces

Title: The High Entropy Method of Einsiedler-Katok

31 Oct 2012: Graduate Student Pizza Seminar (University of Chicago)

Title: Monstrous Moonshine

18 Oct 2012: FFSS Student Geometry Seminar (University of Chicago)

Title: Riemannian Manifolds with Nontrivial Local Symmetry

20 Sep 2012: First-Year Graduate Warmup Program (University of Chicago)

Title: Differential Topology

26 Jun 2012: Dynamics Reading Group | Ratner's Theorems

Title: (End of) Proof of Ratner's Theorems for $\bbR^2\rtimes SL(2,\bbR)$

14 Jun 2012: Dynamics Reading Group | Ratner's Theorems

Title: (End of) Proof of Ratner's Theorems for $SL(2,\bbR)$

19 Apr 2012: FFSS Student Geometry Seminar | University of Chicago

Title: Gromov's Polyonomial Growth Theorem

18 Apr 2012: Graduate Student Pizza Seminar | University of Chicago

Title: The geometry of nilpotent groups: A theorem of Milnor and Wolf

14 Jun 2012: Dynamics Reading Group | Counting in lie Groups after Eskin-McMullen

Title: Representations of G and integral points on homogeneous varieties

26 Jan 2012: FFSS Student Geometry Seminar | University of Chicago

Title: Margulis’ Normal Subgroups Theorem

10 Jan 2012: Topological Topics (course by Shmuel Weinberger) | University of Chicago

Title: Expander Graphs through Expansion, Random Walks, and Spectral Analysis

30 Nov 2011: Graduate Student Pizza Seminar | University of Chicago

Title: Hyperbolic Groups

20 Oct 2011: FFSS Student Geometry Seminar | University of Chicago

Title: Actions of higher rank lattices on low-dimensional manifolds

19 Sep 2011: First-Year Graduate Warmup Program (University of Chicago)

Title: Differential Topology