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

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• April 24: Pratyush Sarkar (UCSD)
  
  Location: AP&M 7321
  
  Title: Effective equidistribution of translates of tori in arithmetic homogeneous spaces and applications
  
  Abstract: A celebrated theorem of Eskin—Mozes—Shah gives an asymptotic counting formula for the number of integral (n x n)-matrices with a prescribed irreducible (over the integers/rationals) integral characteristic polynomial. We obtain a power saving error term for the counting problem for (3 x 3)-matrices. We do this by using the connection to homogeneous dynamics and proving effective equidistribution of translates of tori in SL_3(R)/SL_3(Z). A key tool is that the limiting Lie algebra corresponding to the translates of tori is a certain nilpotent Lie algebra. This allows us to use the recent breakthrough work of Lindenstrauss—Mohammadi—Wang—Yang on effective versions of Shah's/Ratner's theorems. We actually study the phenomenon more generally for any semisimple Lie group which we may discuss if time permits.
  
  
• May 1: Gaurav Aggarwal (Tata Institute for Fundamental Research, Mumbai)
  
  Location: Zoom
  
  Title: Lévy-Khintchine Theorems: effective results and central limit theorems
  
  Abstract: The Lévy-Khintchine theorem is a classical result in Diophantine approximation that describes the growth rate of denominators of convergents in the continued fraction expansion of a typical real number. We make this theorem effective by establishing a quantitative rate of convergence. More recently, Cheung and Chevallier (Annales scientifiques de l'ENS, 2024) established a higher-dimensional analogue of the Lévy-Khintchine theorem in the setting of simultaneous Diophantine approximation, providing a limiting distribution for the denominators of best approximations. We also make their result effective by proving a convergence rate, and in addition, we establish a central limit theorem in this context. Our approach is entirely different and relies on techniques from homogeneous dynamics.
  
  video
  
  
• May 8: Benjamin Dozier (Cornell University)
  
  Location: AP&M 7321
  
  Title: The boundary of a totally geodesic subvariety of moduli space
  
  Abstract: The moduli space of genus g Riemann surfaces equipped with the Teichmuller metric exhibits rich geometric, analytic, and dynamical properties. A major challenge is to understand the totally geodesic submanifolds -- these share many properties with the moduli space itself. For many decades, research focused on the one (complex) dimensional case, i.e. the fascinating Teichmuller cuves. The discovery of interesting higher-dimensional examples in recent years has led to new questions. In this talk, I will discuss joint work with Benirschke and Rached in which we study the boundary of a totally geodesic subvariety in the Deligne-Mumford compactification, showing that the boundary is itself totally geodesic.
  
  
• May 15: Omri Solan (Hebrew University of Jerusalem)
  
  Location: Zoom
  
  Title: Critical exponent gap in hyperbolic geometry
  
  Abstract: We will discuss the following result. For every geometrically finite Kleinian group $\Gamma < SL_2(\mathbb C)$ there is $\epsilon_\Gamma$ such that for every $g \in SL_2(\mathbb C)$ the intersection $g \Gamma g^{-1} \cap SL_2(\mathbb R)$ is either a lattice or has critical exponent $\delta(g \Gamma g^{-1} \cap SL_2(\mathbb R)) \leq 1 - \epsilon_\Gamma$. This result extends Margulis-Mohammadi and Bader-Fisher-Milier-Strover. We will discuss some ideas of the proof. We will focus on the applications of a new ergodic component: the preservation of entropy in a direction.
  
  video
  
  
• May 22: Sunrose Shrestha (Carleton College)
  
  Location: AP&M 7321
  
  Title: Two combinatorial models for random square-tiled surfaces
  
  Abstract: A square-tiled surface (STS) is a (finite, possibly branched) cover of the standard square-torus with possible branching over exactly 1 point. Alternately, STSs can be viewed as finitely many axis-parallel squares with sides glued in parallel pairs. This description allows us to encode an STS combinatorially by a pair of permutations -- one of which encodes the gluing of the vertical edges and the other the gluing of the horizontal edges. In this talk I will use the combinatorial description of STSs to consider two models for random STSs. The first model will encompass all square-tiled surfaces while the second will encompass a horizontally restricted class of them. I will discuss topological and geometric properties of a random STS from each of these models.
  
  
• May 29: Joshua Bowman (Pepperdine University)
  
  Location: AP&M 7321
  
  Title: Cycles in digraphs
  
  Abstract: Directed graphs, or digraphs, are useful in many areas of theoretical and applied mathematics, including for describing other combinatorial objects. We will review a method for counting closed walks in a digraph using a transfer matrix. Then we will use a group action to count singular cycles (closed walks for which the initial vertex has been forgotten). Finally we will apply these results to count structures in circulant graphs, up to rotational equivalence.