UC San Diego Group Actions Seminar

In-person meetings are held in AP&M 7321 and Zoom meetings are held here.

If you would like to be included or removed from our email announcements, please email Brandon Seward.

If you would like to give a talk, please send the title, abstract and related papers (if available) of your proposed talk to one of the organizers by email.

Organizers: Amir Mohammadi, Anthony Sanchez, Brandon Seward

Current Quarter      Past Quarters

---

• April 13: Zhongkai Tao (UC Berkeley)
  
  Location: AP&M 7218 and Zoom
  
  Title: Fractal uncertainty principle via Dolgopyat's method in higher dimensions
  
  Abstract: The fractal uncertainty principle (FUP) was introduced by Dyatlov and Zahl which states that a function cannot be localized near a fractal set in both position and frequency spaces. It has rich applications in spectral gaps and quantum chaos on hyperbolic manifolds and has recently been an active area of research in harmonic analysis. I will talk about the history of the fractal uncertainty principle and explain its applications to spectral gaps. Then I will talk about our recent work, joint with Aidan Backus and James Leng, which proves a general fractal uncertainty principle for small fractal sets, improving the volume bound in higher dimensions. This generalizes the work of Dyatlov--Jin using Dolgopyat's method. As an application, we give effective essential spectral gaps for convex cocompact hyperbolic manifolds in higher dimensions with Zariski dense fundamental groups.
  
  
• April 27: Srivatsav Kunnawalkam Elayavalli (Institute for Pure and Applied Mathematics, UCLA)
  
  Location: AP&M 7218 and Zoom
  
  Title: Conjugacy for almost homomorphisms of sofic groups
  
  Abstract: I will discuss recent joint work with Hayes wherein we show that any sofic group G that is initially sub-amenable (a limit of amenable groups in Grigorchuk's space of marked groups) admits two embeddings into the universal sofic group S that are not conjugate by any automorphism of S. Time permintting, I will also characterise precisely when two almost homomorphisms of an amenable group G are conjugate, in terms of certain IRS's associated to the two actions of G. One of the applications of this is to recover the result of Becker-Lubotzky-Thom around permutation stability for amenable groups. The main novelty of our work is the usage of von Neumann algebraic techniques in a crucial way to obtain group theoretic consequences.
  
  
• May 4: Pratyush Sarkar (UCSD)
  
  Location: AP&M 7218 and Zoom
  
  Title: Exponential mixing of frame flows for geometrically finite hyperbolic manifolds
  
  Abstract: Let $\Gamma < G = \operatorname{SO}(n, 1)^\circ$ be a Zariski dense torsion-free discrete subgroup for $n \geq 2$. Then the frame bundle of the hyperbolic manifold $X = \Gamma \backslash \mathbb{H}^n$ is the homogeneous space $\Gamma \backslash G$ and the frame flow is given by the right translation action by a one-parameter diagonalizable subgroup of $G$. Suppose $X$ is geometrically finite, i.e., it need not be compact but has at most finitely many ends consisting of cusps and funnels. Endow $\Gamma \backslash G$ with the unique probability measure of maximal entropy called the Bowen-Margulis-Sullivan measure. In a joint work with Jialun Li and Wenyu Pan, we prove that the frame flow is exponentially mixing. The proof uses a countably infinite coding of the flow and the latest version of Dolgopyat's method. To overcome the difficulty in applying Dolgopyat's method due to the cusps of non-maximal rank, we prove a large deviation property for symbolic recurrence to certain large subsets of the limit set of $\Gamma$.
  
  video
  
  
• May 11 Sam Mellick (McGill University)
  
  Location: AP&M 7218 and Zoom
  
  Title: Vanishing of rank gradient for lattices in higher rank Lie groups via cost
  
  Abstract: In 2016 Abért, Gelander, and Nikolov made what they called a provocative conjecture: for lattices in higher rank simple Lie groups, the minimum size of a generating set (rank) is sublinear in the volume. I will discuss our solution to this conjecture. It is a corollary of our main result, where we establish "fixed price one" for a more general class of "higher rank" groups. No familiarity of fixed price or cost is required for the talk. Joint work with Mikołaj Frączyk and Amanda Wilkens.
  
  
• May 18: Cancelled
  
  
• May 25: Timothée Bénard (Centre for Mathematical Sciences, Cambridge UK)
  
  Title: Random walks with bounded first moment on finite volume spaces
  
  Abstract: We consider a finite volume homogeneous space endowed with a random walk whose driving measure is Zariski-dense. In the case where jumps have finite exponential moment, Eskin-Margulis and Benoist-Quint established recurrence properties for such a walk. I will explain how their results can be extended to walks with finite first moment. The key is to make sense of the following claim: "the walk in a cusp goes down faster that some iid Markov chain on R with negative mean". Joint work with N. de Saxcé.
  
  video
  
  
• June 1: Etienne BONNAFOUX (École Polytechnique)
  
  Title: Counting of pairs of saddle connections for a typical flat surface of a Sl(2,R)-invariant measure
  
  Abstract: Asymptotic counting of geometric objects has a long history. In the case of flat surfaces, various works by Masur and Veech showed the quadratic asymptotic growth of the number of saddle connections of bounded length. In this spirit, Athreya, Fairchild and Masur showed that, for almost any flat surface, the number of pairs of saddle connection of bounded length and bounded virtual area increases quadratically with the constraint on length. In this case the << almost all >> is with respect to the so-called Masur-Veech measure.
  
  To demonstrate this, they use tools of ergodic theory (hence the result is true almost everywhere). This result can be extended in several ways, giving an error term or extending it to almost any SL(2,R)-invariant measure. We will present several useful tools for tackling these questions.
  
  video
  
  
• June 8: Anthony Sanchez (UCSD)
  
  Location: AP&M 7218 and Zoom
  
  Title: Effective equidistribution of large dimensional measures on affine invariant submanifolds
  
  Abstract: The unstable foliation, that changes the horizontal components of period coordinates, plays an important role in study of translation surfaces, including their deformation theory and in the understanding of horocycle invariant measures.
  
  In this talk we show that measures of large dimension equidistribute in affine invariant manifolds and give an effective rate. An analogous result in the setting of homogeneous dynamics is crucially used in the effective equidistribution results of Lindenstrauss-Mohammadi and Lindenstrauss--Mohammadi--Wang. Background knowledge on translation surfaces and homogenous dynamics will be explained.