UC San Diego Group Actions Seminar
Thursdays 10:00 - 10:50 AM
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
Winter 2021
- March 9: Pengyu Yang (ETH Zurich)
Title: Equidistribution of expanding translates of lines in SL3(ℝ)/SL3(ℤ)Abstract: Let X=SL3(ℝ)/SL3(ℤ) and a(t)=diag(t2,t-1,t-1). The expanding horospherical group U+ is isomorphic to ℝ2. A result of Shah tells us that the a(t)-translates of a non-degenerate real-analytic curve in a (U+)-orbit get equidistributed in X. It remains to study degenerate curves, i.e. planar lines y=ax+b. In this talk, we give a Diophantine condition on the parameter (a,b) which serves as a necessary and sufficient condition for equidistribution. Joint work with Kleinbock, Saxcé and Shah. If time permits, I will also talk about generalisations to SLn(ℝ)/SLn(ℤ). Joint work with Shah.
- March 2: Sam Mellick (UMPA ens de Lyon)
Title: Point processes on groups, their cost, and fixed price for G x ZAbstract: Invariant point processes on groups are a rich class of probability measure preserving (pmp) actions. In fact, every essentially free pmp action of a nondiscrete locally compact second countable group is isomorphic to a point process. The cost of a point process is a numerical invariant that, informally speaking, measures how hard it is to "connect up" the point process. This notion has been very profitably studied for discrete groups, but little is known for nondiscrete groups. This talk will not assume any sophisticated knowledge of probability theory. I will define point processes, their cost, and discuss why every point process on groups of the form G x Z has cost one. Joint work with Miklós Abért.
- Febuary 23: Gianluca Basso (Université Claude Bernard Lyon 1)
Title: Topological dynamics beyond Polish groupsAbstract: When G is a Polish group, one way of knowing that it has nice dynamics is to show that M(G), the universal minimal flow of G, is metrizable. For non-Polish groups, this is not the relevant dividing line: the universal minimal flow of the symmetric group of a set of cardinality κ is the space of linear orders on κ-not a metrizable space, but still nice, for example. In this talk, we present a set of equivalent properties of topological groups which characterize having nice dynamics. We show that the class of groups satisfying such properties is closed under some topological operations and use this to compute the universal minimal flows of some concrete groups, like Homeo(ω1). This is joint work with Andy Zucker.
- Febuary 16: Nishant Chandgotia (TIFR Bangalore)
Title: About Borel and almost Borel embeddings for ZD actionsAbstract: Krieger's generator theorem says that all free ergodic measure preserving actions (under natural entropy constraints) can be modelled by a full shift. Recently, in a sequence of two papers Mike Hochman noticed that this theorem can be strengthened: He showed that all free homeomorphisms of a Polish space (under entropy constraints) can be Borel embedded into the full shift. In this talk we will discuss some results along this line from a recent paper with Tom Meyerovitch and ongoing work with Spencer Unger.
- Febuary 9: Tamara Kucherenko (City College of New York)
Title: Flexibility of the Pressure FunctionAbstract: Our settings are one-dimensional compact symbolic systems. We discuss the flexibility of the pressure function of a continuous potential (observable) with respect to a parameter regarded as the inverse temperature. The points of non-differentiability of this function are of particular interest in statistical physics since they correspond to qualitative changes of the characteristics of a dynamical system referred to as phase transitions. It is well known that the pressure function is convex, Lipschitz, and has an asymptote at infinity. We show that these are the only restrictions. We present a method to explicitly construct a continuous potential whose pressure function coincides with any prescribed convex Lipschitz asymptotically linear function starting at a given positive value of the parameter. This is based on joint work with Anthony Quas.
- Febuary 2: Matthieu Joseph (ENS de Lyon)
Title: Rigidity and flexibility phenomenons in isometric orbit equivalenceAbstract: TBAIn an ongoing work, we introduce the notion of isometric orbit equivalence for probability measure preserving actions of marked groups. This notion asks the Schreier graphings defined by the actions of the marked groups to be isomorphic. In the first part of the talk, we will prove that pmp actions of a marked group whose Cayley graph has a discrete automorphisms group are rigid up to isometric orbit equivalence. In a second time, we will explain how to construct pmp actions of the free group that are isometric orbit equivalent but not conjugate.
- January 19: Felix Weilacher (Carnegie Mellon University)
Title: Marked groups with isomorphic Cayley graphs but different Descriptive combinatorics.Abstract: We discuss the relationship between the Borel/Baire measurable/measurable combinatorics of the action of a finitely generated group on its Bernoulli shift and the discrete combinatorics of the multiplication action of that group on itself. Our focus is on various chromatic numbers of graphs generated by these actions. We show that marked groups with isomorphic Cayley graphs can have Borel/Baire measurable/measurable chromatic numbers which differ by arbitrarily much. In the Borel two-ended, Baire measurable, and measurable hyperfinite settings, we show our constructions are nearly best possible (up to only a single additional color), and we discuss prospects for improving our constructions in the general Borel setting. Along the way, we will get tightness of some bounds of Conley and Miller on Baire measurable and measurable chromatic numbers of locally finite Borel graphs.
- January 12: Minju Lee (Yale University)
Title: Invariant measures for horospherical actions and Anosov groups.Abstract: Let Γ be an Anosov subgroup of a connected semisimple real linear Lie group G. For a maximal horospherical subgroup N of G, we show that the space of all non-trivial NM-invariant ergodic and A-quasi-invariant Radon measures on Γ \ G, up to proportionality, is homeomorphic to ℝrank G-1, where A is a maximal real split torus and M is a maximal compact subgroup which normalizes N. This is joint work with Hee Oh.
- January 5: François Le Maître (Institut de Mathématiques de Jussieu-PRG)
Title: A decomposition for measure-preserving near-actions of ergodic full groupsAbstract: Given a measure-preserving action of a countable group on a standard probability space, one associates to it a full group which by Dye's reconstruction theorem completely remembers the associated equivalence relation whose classes are the action's orbits. A natural question is then to understand how exactly this full group encodes the properties of the associated (measure-preserving) equivalence relation. In this talk, we will see that all non-free ergodic near-actions of the full group actually come from measure-preserving actions of the equivalence relation (or of its symmetric powers), paralleling a recent result of Matte-Bon concerning actions by homeomorphisms of topological full groups. If time permits, we will explain how this can be used to show that a measure-preserving ergodic equivalence relation has property (T) if and only if all the non-free ergodic near-actions of its full group are strongly ergodic. This talk is based on an ongoing joint work with Alessandro Carderi and Alice Giraud.