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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• Oct. 10: Rodolfo Gutiérrez-Romo (Universidad de Chile)
  
  Location: Zoom
  
  Title: The diagonal flow detects the topology of strata of quadratic differentials
  
  Abstract: A half-translation surface is a collection of polygons on the plane with side identifications by translations or half-turns in such a way that the resulting topological surface is closed and orientable. We also assume that the total Euclidean area of the polygons is finite. Two half-translations are equivalent if a sequence of cut-and-paste operations takes one to the other. From the view of complex geometry, an equivalent definition is a Riemann surface endowed with a meromorphic quadratic differential with poles of order at most one.
  
  A stratum of half-translation surfaces consists of those with prescribed cone angles at the vertices of the polygons. Strata are, in general, not connected. A natural flow, the diagonal or Teichmüller flow, acts on stratum components.
  
  In this talk, we investigate some topological properties of stratum components. We show that the (orbifold) fundamental group of such a component is "detected" by the diagonal flow in that every loop is homotopic to a concatenation of closed geodesics (coned to a base-point). Using this result, we show that the Lyapunov spectrum of the homological action of the diagonal flow is simple, thus establishing the Kontsevich–Zorich conjecture for quadratic differentials.
  
  This is a joint work with Mark Bell, Vincent Delecroix, Vaibhav Gadre, and Saul Schleimer.
  
  
• Oct. 17: Sidhanth Raman (UC Irvine)
  
  Location: AP&M 7321
  
  Title: Uniform Central Limit Theorems on Lie Groups
  
  Abstract: Random walks on groups have been utilized to study a wide array of mathematics, e.g. number theory, the spectral theory of Schrodinger operators, and homogeneous dynamics. Under sufficiently nice dynamical assumptions, these random walks obey central limit theorems. We will discuss some joint work with Omar Hurtado in which we introduce a natural family of topologies on spaces of probability measures, and study continuity and stability of statistical properties of random walks on linear groups over local fields. We are able to extend large deviation results known in the Archimedean case to non-Archimedean local fields and also demonstrate certain large deviation estimates for heavy tailed distributions unknown even in the Archimedean case. Time permitting, we will discuss applications to Schodinger operators (an Anderson localization result) and hyperbolic geometry (a stable geodesic counting result).
  
  
• Oct. 24: Ben Johnsrude (UCLA)
  
  Location: AP&M 7321
  
  Title: Exceptional set estimates for projection theorems over non-Archimedean local fields
  
  Abstract: How do linear projections affect the dimensions of subsets of Cartesian space? Marstrand's result from 1954 demonstrates that each Borel set behaves generically under projections onto almost every linear subspace. Recent developments in Fourier analysis have permitted these results to be expanded significantly to apply to much more restricted families of projections, and even effectively bound the dimension of the set of exceptional projections.
  
  We discuss the special case of projecting subsets of three dimensions onto lines, working over non-Archimedean local fields of characteristic not equal to 2. We will briefly discuss the relevancy to polynomial effective equidistribution in homogeneous dynamics. The main technical input will be a refined decoupling theorem for non-Archimedean local fields. This work mirrors the work in the real setting by the authors Gan, Guo, Guth, Harris, Iosevich, Maldague, Ou, and Wang, and builds on previous work by the speaker and Zuo Lin.
  
  
• Nov. 7: Shreyasi Datta (University of York)
  
  Location: Zoom
  
  Title: Fourier Asymptotics and Effective Equidistribution
  
  Abstract: We talk about effective equidistribution of the expanding horocycles on the unit cotangent bundle of the modular surface with respect to various classes of Borel probability measures on the reals, depending on their Fourier asymptotics. This is a joint work with Subhajit Jana.
  
  video
  
  
• Nov. 14: Tattwamasi Amrutam (Institute of Mathematics, Polish Academy of Sciences)
  
  Location: AP&M 7321
  
  Title: A Continuous Version of the Intermediate Factor Theorem
  
  Abstract: Let G be a discrete group. A G-space X is called a G-boundary if the action of G on X is minimal and strongly proximal. In this talk, we shall prove a continuous version of the well-studied Intermediate Factor Theorem in the context of measurable dynamics. When a product group G = Γ_1 × Γ_2 acts (by a product action) on the product of corresponding Γ_i-boundaries ∂Γ_i, we show that every intermediate factor
  X × (∂Γ_1 × ∂Γ_2) → Y → X
  is a product (under some additional assumptions on X). We shall also compare it to its measurable analog proved by Bader-Shalom. This is a recent joint work with Yongle Jiang.
  
  
• Nov. 21: Camilo Arosemena Serrato (Rice University)
  
  Location: Zoom
  
  Title: Rigidity of Codimension One Higher Rank Actions
  
  Abstract: We discuss work in progress regarding the following assertion: Let G be a simple higher rank Lie group, then any closed manifold M, admitting a smooth locally free action of G, with codimension one orbits, is finitely and equivariantly covered by G/Gamma x S^1, for some cocompact lattice Gamma of G, where G acts by left translations on the first factor, and trivially on S^1. This result is in the spirit of the Zimmer program. We will focus on the case G = SL(3,R) for the talk.
  
  video
  
  
• Dec. 5: Nicolas Monod (École Polytechnique Fédérale de Lausanne)
  
  Location: AP&M 7321
  
  Title: The Furstenberg boundary of Gelfand pairs
  
  Abstract: Many classical locally compact groups G admit a very large compact subgroup K, where "very large" has been formalized by Gelfand in 1950. Examples include G=SL_n(R) with K=SO(n), or G=SL_n(Q_p) with K=SL_n(Z_p). More generally, all semi-simple algebraic groups and some tree automorphism groups.
  
  In these explicit examples, there is also an "Iwasawa decomposition" which formalizes the fact that G has a homogeneous Frustenberg boundary, even homogeneous under K. This is a very strong restriction for general groups.
  
  Using no structure theory whatsoever, we prove that this homogeneity (and Iwasawa decompostion) holds for all Gelfand Pairs. This implies, in some geometric cases, a classification of Gelfand pairs. (This is related to a small part of my 2021 zoom colloquium at UCSD).
  
  
• Dec. 12: Andreas Wieser (UCSD / Institute for Advanced Study)
  
  Location: AP&M 7321
  
  Title: Local-global principles and effective rates of equidistribution for semisimple orbits
  
  Abstract: We prove an effective equidistribution theorem for semisimple closed orbits on compact adelic quotients. The obtained error depends polynomially on the minimal complexity of intermediate orbits and the complexity of the ambient space. As an application, we establish a local-global principle for representations of quadratic forms, improving the codimension assumptions and providing effective bounds in a theorem of Ellenberg and Venkatesh. We will discuss these theorems not assuming any prior knowledge of any of the above concepts. This is based on joint work with Manfred Einsiedler, Elon Lindenstrauss, and Amir Mohammadi.