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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• June 1: Osama Khalil (University of Utah)

  
  Title: On the Mozes-Shah phenomenon for horocycle flows on moduli spaces

  Abstract: The Mozes-Shah phenomenon on homogeneous spaces of Lie groups asserts that the space of ergodic measures under the action by subgroups generated by unipotents is closed. A key input to their work is Ratner's fundamental rigidity theorems. Beyond its intrinsic interest, this result has many applications to counting problems in number theory. The problem of counting saddle connections on flat surfaces has motivated the search for analogous phenomena for horocycle flows on moduli spaces of flat structures. In this setting, Eskin, Mirzakhani, and Mohammadi showed that this property is enjoyed by the space of ergodic measures under the action of (the full upper triangular subgroup of) SL(2,ℝ). We will discuss joint work with Jon Chaika and John Smillie showing that this phenomenon fails to hold for the horocycle flow on the stratum of genus two flat surfaces with one cone point. As a corollary, we show that a dense set of horocycle flow orbits are not generic for any measure; in contrast with Ratner's genericity theorem.
  
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• May 25: Joshua Frisch (Caltech)

  
  Title: Which Linear Groups have bounded harmonic functions?

  Abstract: The Poisson boundary of a group is a measure space which serves a dual purpose. From the perspective of random walks it represents the range of distinct asymptotic possibilities that a random walk on the group might take. From the perspective of harmonic analysis it classifies the space of bounded harmonic functions on that group. Understanding the Poisson boundary of a group is intimately related to the algebraic and geometric properties of that group. The most basic question one can ask about the Poisson boundary is whether it is trivial (equivalently whether there are any non-constant bounded harmonic functions on that group). In this talk I will survey some core ideas around the Poisson boundary and then focus on the case of linear groups. In particular I will give a complete characterization of when linear groups over positive characteristic fields admit any non-constant bounded harmonic functions. This is joint work with Anna Erschler.
  
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• May 18: Philipp Kunde (Anatole Katok Center for Dynamical Systems and Geometry)

  
  Title: Anti-classification results for the Kakutani equivalence relation

  Abstract: Dating back to the foundational paper by John von Neumann, a fundamental theme in ergodic theory is the \emph{isomorphism problem} to classify invertible measure-preserving transformations (MPT's) up to isomorphism. In a series of papers, Matthew Foreman, Daniel Rudolph and Benjamin Weiss have shown in a rigorous way that such a classification is impossible. Besides isomorphism, Kakutani equivalence is the best known and most natural equivalence relation on ergodic MPT's for which the classification problem can be considered. In joint work with Marlies Gerber we prove that the Kakutani equivalence relation of ergodic MPT's is not a Borel set. This shows in a precise way that the problem of classifying such transformations up to Kakutani equivalence is also intractable.
  
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• May 11: Cagri Sert (UZH)

  
  Title: Expanding measures: Random walks and rigidity on homogeneous spaces

  Abstract: We will start by reviewing recent developments in random walks on homogeneous spaces. In a second part, we will discuss the notion of a H-expanding probability measure on a connected semisimple Lie group H. As we shall see, for a H-expanding μ with H < G, on the one hand, one can obtain a description of μ-stationary probability measures on the homogeneous space G/Λ (G Lie group, Λ lattice) using the measure classification results of Eskin-Lindenstrauss, and on the other hand, the recurrence techniques of Benoist-Quint and Eskin-Mirzakhani-Mohammadi can be adapted to this setting. With some further work, these allow us to deduce equidistribution and orbit closure description results simultaneously for a class of subgroups which contains Zariski-dense subgroups and further epimorphic subgroups of H. If time allows, we will see how, utilizing an idea of Simmons-Weiss, these also allow us to deduce Birkhoff genericity of a class of fractal measures with respect to certain diagonal flows, which, in turn, has applications in diophantine approximation problems. Joint work with Roland Prohaska and Ronggang Shi.
  
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• May 4, 2021: Lingfu Zhang (Princeton University)

  
  Title: Factor of IID for the free Ising model on the d-regular tree

  Abstract: It is known that there are factors of IID for the free Ising model on the d-regular tree when it has a unique Gibbs measure and not when reconstruction holds (when it is not extremal). We construct a factor of IID for the free Ising model on the d-regular tree in (part of) its intermediate regime, where there is non-uniqueness but still extremality. The construction is via the limit of a system of stochastic differential equations. This is a joint work with Danny Nam and Allan Sly.
  
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• April 27: Prasuna Bandi (Tata Institute)

  
  Title: Density at integer points of an inhomogeneous quadratic form and linear form

  Abstract: In 1987, Margulis solved an old conjecture of Oppenheim which states that for a nondegenerate, indefinite and irrational quadratic form Q in n≥3 variables, Q(ℤⁿ) is dense in ℝ. Following this, Dani and Margulis proved the simultaneous density at integer points for a pair consisting of quadratic and linear form in 3 variables when certain conditions are satisfied. We prove an analogue of this for the case of an inhomogeneous quadratic form and a linear form. This is based on joint work with Anish Ghosh.
  
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• April 20: Rachel Greenfeld (UC Los Angeles)

  
  Title: Translational tilings in lattices

  Abstract: Let F be a finite subset of ℤ^d. We say that F is a translational tile of ℤ^d if it is possible to cover ℤ^d by translates of F without any overlaps. The periodic tiling conjecture, which is perhaps the most well-known conjecture in the area, suggests that any translational tile admits at least one periodic tiling. In the talk, we will motivate and discuss the study of this conjecture. We will also present some new results, joint with Terence Tao, on the structure of translational tilings in lattices and introduce some applications.

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• April 6: Jenna Zomback (University of Illinois Urbana-Champaign)

  
  Title: A backward ergodic theorem and its forward implications

  Abstract: In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation T, one takes averages of a given integrable function over the intervals {x, T(x), T²(x),..., Tⁿ(x)} in front of the point x. We prove a "backward" ergodic theorem for a countable-to-one pmp T, where the averages are taken over subtrees of the graph of T that are rooted at x and lie behind x (in the direction of T^-1). Surprisingly, this theorem yields forward ergodic theorems for countable groups, in particular, one for pmp actions of free groups of finite rank, where the averages are taken along subtrees of the standard Cayley graph rooted at the identity. This strengthens Bufetov's theorem from 2000, which was the most general result in this vein. This is joint work with Anush Tserunyan.

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