| Date: | October 5, 2022 |
| Time: | 3:00pm |
| Location: | BLOC 506a |
| Speaker: | Volodymyr Nekrashevych, Texas A&M University |
| Title: | Dimensions of self-similar groups |
| Abstract: | Every contracting self-similar group defines the associated limit
space with a natural class of metrics on it. If the group is the iterated
monodromy group of a locally expanding covering map of a compact metric space
X, then the limit space is canonically homeomorphic to X. For example, the
Julia set of a sub-hyperbolic rational function is the limit space of its
iterated monodromy group (where the usual metric on the Riemann sphere belongs
to the above mentioned natural class of metrics on the limit space). We will
discuss how the topological dimension of the limit space can be interpreted in
terms of the action of the group on the rooted tree. We will also discuss
possible applications of the Ahlfors-regular dimension of the limit space (for
the natural class of metrics). |
|
| Date: | October 19, 2022 |
| Time: | 3:00pm |
| Location: | BLOC 506a |
| Speaker: | Volodymyr Nekrashevych, Texas A&M University |
| Title: | Dimensions of self-similar groups (part 2) |
| Abstract: | Every contracting self-similar group defines the associated limit
space with a natural class of metrics on it. If the group is the iterated
monodromy group of a locally expanding covering map of a compact metric space
X, then the limit space is canonically homeomorphic to X. For example, the
Julia set of a sub-hyperbolic rational function is the limit space of its
iterated monodromy group (where the usual metric on the Riemann sphere belongs
to the above mentioned natural class of metrics on the limit space). We will
discuss how the topological dimension of the limit space can be interpreted in
terms of the action of the group on the rooted tree. We will also discuss
possible applications of the Ahlfors-regular dimension of the limit space (for
the natural class of metrics). |
|
| Date: | November 2, 2022 |
| Time: | 3:00pm |
| Location: | BLOC 506a |
| Speaker: | Yury Kudriashov, Texas A&M University |
| Title: | Bifurcations of vector fields on the two-sphere |
| Abstract: | A generic vector field on the two-sphere is structurally stable: any
other vector field that is sufficiently close to the original one is conjugate
to it by a homeomorphism of the sphere. In 1960-s, V. Arnold conjectured that
the same is true for a generaic finite parameter family of vector fields on the
sphere. Recently, Yu. Ilyashenko, I. Schurov and me disproved this conjecture:
we described a locally generic type of a 3-parameter bifurcation such that
topological classification of bifurcations of this type has at least one numeric
parameter. We also described a 6-parameter bifurcation that have functional
moduli of topological classification. I will talk about these examples and their
improvements constructed by N. Goncharuk, N. Solodvnikov and me. This talk is an
extended version of my talk at the postdoc talks series. |
|
| Date: | November 16, 2022 |
| Time: | 3:00pm |
| Location: | BLOC 506a |
| Speaker: | Chris Shriver, University of Texas, Austin |
| Title: | Non-equilibrium Gibbs states on a tree |
| Abstract: | We consider two notions of statistical equilibrium for a probability-measure-preserving shift system: an “equilibrium state” maximizes a functional called the pressure while a “Gibbs state” satisfies a local equilibrium condition. Classical results of Dobrushin, Lanford, and Ruelle show that these notions are equivalent for Z^d systems, under some assumptions on the interaction, and the equivalence has been extended to arbitrary amenable groups. Barbieri and Meyerovitch have recently shown that one direction still holds for sofic groups: equilibrium states are always Gibbs.
We will show that the converse fails in a nontrivial way using the example of the free boundary Ising state on an infinite regular tree (i.e. a free group): we show that for all temperatures below the uniqueness threshold this state is nonequilibrium over some sofic approximation, and below the reconstruction threshold it is nonequilibrium over every sofic approximation. |
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