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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• Jan. 9: Luke Jeffreys (University of Bristol)
  
  Location: Zoom
  
  Title: Local dimension in the Lagrange and Markov spectra
  
  Abstract: Initially studied by Markov around 1880, the Lagrange spectrum, L, and the Markov spectrum, M, are complicated subsets of the real line that play a crucial role in the study of Diophantine approximation and binary quadratic forms. Perron's 1920s description of the spectra in terms of continued fractions allowed powerful dynamical machinery to come to bear on many problems. In this talk, I will discuss recent work with Harold Erazo and Carlos Gustavo Moreira investigating the function d_loc(t) that determines the local Hausdorff dimension at a point t in L'.
  
  video, slides
  
  
• Feb. 13 at 4:00 PM: Siyuan Tang (Beijing International Center for Mathematical Research)
  
  Location: Zoom
  
  Title: Effective density of surfaces near Teichmüller curves
  
  Abstract: The study of orbit dynamics for the upper triangular subgroup P in SL(2, R) holds fundamental significance in homogeneous and Teichmüller dynamics. In this talk, we shall discuss the quantitative density properties of P-orbits for translation surfaces near Teichmüller curves. In particular, we discuss the Teichmüller space H(2) of genus two Riemann surfaces with a single zero of order two, and its corresponding absolute period coordinates, and examine the asymptotic dynamics of P-orbits in these spaces.
  
  video
  
  
• Feb. 20: Paul Apisa (University of Wisconsin)
  
  Location: Zoom
  
  Title: SL(2, R)-invariant measures on the moduli space of twisted holomorphic 1-forms and dilation surfaces
  
  Abstract: A dilation surface is roughly a surface made up of polygons in the complex plane with parallel sides glued together by complex linear maps. The action of SL(2, R) on the plane induces an action of SL(2, R) on the collection of all dilation surfaces, aka the moduli space, which possesses a natural manifold structure. As with translation surfaces, which can be identified with holomorphic 1-forms on Riemann surfaces, dilation surfaces can be thought of as “twisted” holomorphic 1-forms.
  
  The first result that I will present, joint with Nick Salter, produces an SL(2, R)-invariant measure on the moduli space of dilation surfaces that is mutually absolutely continuous with respect to Lebesgue measure. The construction fundamentally uses the group cohomology of the mapping class group with coefficients in the homology of the surface. It also relies on joint work with Matt Bainbridge and Jane Wang, showing that the moduli space of dilation surfaces is a K(pi,1) where pi is the framed mapping class group.
  
  The second result that I will present, joint with Bainbridge and Wang, is that an open and dense set of dilation surfaces have Morse-Smale dynamics, i.e. the horizontal straight lines spiral towards a finite set of simple closed curves in their forward and backward direction. A consequence is that any fully supported SL(2, R) invariant measure on the moduli space of dilation surfaces cannot be a finite measure.
  
  video
  
  
• Feb. 27: Ilya Gekhtman (Technion)
  
  Location: AP&M 7321
  
  Title: Linearly growing injectivity radius in negatively curved manifolds with small critical exponent
  
  Abstract: Let X be a proper geodesic Gromov hyperbolic space whose isometry group contains a uniform lattice \Gamma. For instance, X could be a negatively curved contractible manifold or a Cayley graph of a hyperbolic group. Let H be a discrete subgroup of isometries of X with critical exponent (exponential growth rate) strictly less than half of the growth rate of \Gamma. We show that the injectivity radius of X/H grows linearly along almost every geodesic in X (with respect to the Patterson-Sullivan measure on the Gromov boundary of X). The proof will involve an elementary analysis of a novel concept called the "sublinearly horosherical limit set" of H which is a generalization of the classical concept of "horospherical limit set" for Kleinian groups. This talk is based on joint work with Inhyeok Choi and Keivan Mallahi-Kerai.
  
  
• March 6: Waltraud Lederle (UCLouvain)
  
  Location: AP&M 7321
  
  Title: Boomerang subgroups
  
  Abstract: Given a locally compact group, its set of closed subgroups can be endowed with a compact, Hausdorff topology. With this topology, it is called the Chabauty space of the group. Every group acts on its Chabauty space via conjugation. This action has connections to rigidity theory, Margulis' normal subgroup theorem and measure preserving actions of the group via invariant random subgroups (IRS).
  
  I will give a gentle introduction into Chabauty spaces and IRS and state a few classical results. I will define boomerang subgroups and explain how special cases of the classical results can be proven via them.
  
  Based on joint work with Yair Glasner.
  
  
• March 13: Filippo Calderoni (Rutgers University)
  
  Title: TBA
  
  Abstract: TBA