anuary 12: No talk, go to the special Colloquium at 4pm.
January 19: No talk, go to the special Colloquium at 4pm.
January 26: Colin Davalo (University of Heidelberg)
Title: Finite-sided Dirichlet domains for Anosov representations
Abstract: Dirichlet domains provide polyhedral fundamental domains for discrete subgroups of the isometries of hyperbolic space on the hyperbolic space. Selberg introduced a similar construction of apolyhedral fundamental domain for the action of discrete subgroups of the higher rank Lie group SL(n,R) on the projective model of the associated symmetric space. His motivation was to study uniform lattices, for which these domains are finite-sided. We will address the following question asked by Kapovich: for which Anosov subgroups are these domains finite-sided? Anosov subgroups are hyperbolic discrete subgroups satisfying strong dynamical properties that have infinite covolume in higher rank. We will first consider an example of an Anosov subgroup for which this fundamental domain can have infinitely many sides. We then provide a sufficient condition on a subgroup to ensure that the domain is finitely sided in a strong sense. This is joint work with Max Riestenberg.
February 2: Kejia Zhu (University of California Riverside)
Title: Relatively geometric actions of complex hyperbolic lattices on CAT(0) cube complexes
Abstract: We prove that for n>1, a non-uniform lattice in PU(n,1) does not admit a relatively geometric action on a CAT(0) cube complex. As a consequence, we prove that if a nonuniform lattice in a non-compact semisimple Lie group G admits a relatively geometric action on a CAT(0) cube complex, then G is isomorphic to SO(n,1). We also prove that given a relatively hyperbolic group with residually finite parabolic subgroups, if it is Kähler and acts relatively geometrically on a CAT(0) cube complex, then it is virtually a surface group. This work is joint with Corey Bregman and Daniel Groves.
February 9: Boris Tsvelikhovsky (University of California Riverside)
Title: Quandles: Knot Invariants and Structures on Topological Spaces.
Abstract: Initially developed for constructing knot invariants, quandles are defined as sets equipped with a binary operation. The three axioms of a quandle serve as algebraic counterparts to the three Reidemeister moves in classical knot theory.
In this talk, I will provide an introduction to quandles, elucidate their connection to knot theory, and explain how to produce nonisomorphic quandle structures on topological spaces, the main result of https://arxiv.org/pdf/1811.00886.pdf.
February 16: Amir Moradifam (Unviersity of California Riverside)
Title: Rigidity of Hawking Mass for Stable Constant Mean Curvature Surfaces
Abstract: I will talk about a recent work proving that all solutions of the mean field equation αΔu+2eu-2=0 on S2 are axially symmetric for 1/3≤ α ≤1. Our proof leverages the Sphere Covering Inequality and utilizes topological arguments on S2, enabling us to establish a symmetry result for α≥1/3, a significant improvement over our earlier results, which were applicable only for α≥1/2. We use this result to establish the rigidity of the Hawking mass for stable constant mean curvature (CMC) spheres, confirming a conjecture posed by Robert Bartnik in 2002. Indeed, we show that any complete Riemannian three-manifold (M,g) with non-negative scalar curvature, R(g)≥0, whose boundary Σ=∂M is a stable Constant Mean Curvature sphere with zero Hawking mass, is isometric to a Euclidean ball in R3. This is a joint work with Changfeng Gui.
February 23: Michael McNulty (Michigan State)
Title: Singularity formation for the higher-dimensional Skyrme model in the strong-field limit
Abstract: This talk concerns the formation of singularities in the classical (5+1)-dimensional, co-rotational Skyrme model. While it is well established that blowup is excluded in (3+1)-dimensions, nothing appears to be known in the higher dimensional case. We prove that the model, in the so-called strong field limit, admits an explicit self-similar solution which is asymptotically stable within backwards light cones. From a technical point of view, the main obstacle to this result is the presence of derivative nonlinearities in the corresponding evolution equation. These introduce first order terms in the linearized flow which render standard techniques useless. We demonstrate how this problem can be bypassed by using structural properties of the Skyrme model. This is joint work with Po-Ning Chen and Birgit Schörkhuber.
March 1: Ye-Kai Wang (National Yang Ming Chiao Tung University)
Title: Minkowski inequality on static manifolds (on Zoom)
Abstract: The classical Minkowski inequality provides a lower bound for the integral of mean curvature of a closed convex surface in Euclidean space. We describe how geometric flow method (Guan-Li, Brendle-Hung-Wang, etc) allows generalizations of the Minkowski inequality to non-convex surfaces in more general ambient spaces. The talk is based on joint work with Brian Harvie.
March 8: Morgan Opie (UCLA)
Title: Enumerating stably trivial topological vector bundles with higher real K-theories
Abstract: The zeroeth complex topological K-theory of a space encodes complex vector bundles up to stabilization. Since complex topological K-theory is highly computable, this is a great place to start when asking questions about topological vector bundles. But, in general, there are many non-equivalent vector bundles with the same K-theory class. Bridging the gap between K-theory and actual bundle theory is challenging, even for the simplest CW complexes. Building on work of Hu, we use Weiss calculus and a little chromatic homotopy theory to translate vector bundle enumeration questions to tractable stable homotopy theory computations. We compute lower bounds for the number of stably trivial rank complex rank r topological vector bundles on complex projective n-space, for infinitely many n and r. This is joint work with Hood Chatham and Yang Hu.
March 15: NO TALK
2024 Spring quarter
April 5: Panagiotis Dimakis (Université du Québec à Montréal)
Title: TBA
Abstract: TBA
April 12: Ye-Kai Wang (National Yang Ming Chiao Tung University)
Title: Minkowski inequality on static manifolds (on Zoom) TAKE 2
Abstract: The classical Minkowski inequality provides a lower bound for the integral of mean curvature of a closed convex surface in Euclidean space. We describe how geometric flow method (Guan-Li, Brendle-Hung-Wang, etc) allows generalizations of the Minkowski inequality to non-convex surfaces in more general ambient spaces. The talk is based on joint work with Brian Harvie.
April 19: TBA
Title: TBA
Abstract: TBA
April 26: Boris Chorny (University of Haifa)
Title: TBA
Abstract: TBA
May 3: No talk go to Geometric group theory workshop
May 10: Wenyuan Li (USC)
Title: TBA
Abstract: TBA
May 17: Hiro Lee Tanaka (Texas State University)
Title: TBA
Abstract: TBA
May 24: David Clausen (UCR)
Title: TBA
Abstract: TBA
May 31: TBA
Title: TBA
Abstract: TBA
June 7: TBA
Title: TBA
Abstract: TBA