| Date: | September 11, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Nguyen-Bac Dang, Stonybrook |
| Title: | Spectral gap in the dynamical degrees of tame automorphism preserving an affine quadric threefold |
| Abstract: | In this talk, I will present the tame automorphisms group preserving an
affine quadric threefold. The main focus of my talk is the
understanding of the degree sequences induced by the elements of this
group. Precisely, I will explain how one can apply some ideas from
geometric group theory in combination with valuative techniques to show
that the values of the dynamical degrees of these tame automorphisms
admit a spectral gap.
Finally I will apply these techniques to study random walks on this particular group. |
|
| Date: | September 18, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Misha Lyubich, Stonybrook |
| Title: | Quasisymmetries of Julia sets |
| Abstract: | We will describe the group of quasisymmetric self-homeomorphisms of some
Julia sets J. The answer depends substantially on the Julia set in
question. For instance, for a ``Sierpinski carpet" J, this group turns out
to be finite, for the ``basilica" J it is an uncountably infinite group
containing the circle Thompson group, while for an ``Apollonian gasket" J,
it is countably infinite.
Based on joint results with M. Bonk, R. Lodge, S. Merenkov,
and S. Mukherjee. |
|
| Date: | September 25, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Frank Lin, UT Austin |
| Title: | A topological dynamical system with two different positive sofic entropies |
| Abstract: | Dynamical entropy is an important tool in classifying measure-preserving or topological dynamical systems up to measure or topological conjugacy. Classical dynamical entropy theory, of an action of a single transformation, has been studied since the 50s and 60s. Recently Lewis Bowen and Kerr-Li have developed entropy theory for actions of sofic groups. Although a conjugacy invariant, sofic entropy in general appears to be less well-behaved than classical entropy. In particular, sofic entropy may depend on the choice of sofic approximation, although only degenerate examples have been known until now.
We present an example, inspired by hypergraph 2-colorings from statistical physics literature, of a topological dynamical system with two different positive topological sofic entropies corresponding to different sofic approximations. The measure-theoretic case remains open. |
|
| Date: | October 16, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Vadim Kaimanovich, University of Ottawa |
| Title: | Freedom and boundaries |
| Abstract: | I will outline recent results (joint with Anna Erschler) on the
stabilizers of the group action on its Poisson boundary: existence of a
free boundary action for any group with infinite conjugacy classes, a
complete description of the possible kernels of such actions, and an
example of a totally non-free boundary action. |
|
| Date: | October 23, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Nóra Gabriella Szőke |
| Title: | A Tits alternative for topological full groups |
| Abstract: | I will present a Tits alternative for topological full groups of minimal actions of finitely generated groups. On the one hand, the topological full group of a minimal action of a virtually cyclic group is amenable. This is a generalization of the result of Juschenko and Monod for Z-actions. On the other hand, when a finitely generated group G is not virtually cyclic, then we can construct a minimal free action of G on a Cantor space such that the topological full group contains a non-abelian free group. |
|
| Date: | November 13, 2019 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Tatiana Nagnibeda, University of Geneva |
| Title: | Various types of spectra and spectral measures on Schreier and Cayley graphs. |
| Abstract: | We will be interested in the Laplacian on graphs associated with finitely generated groups: Cayley graphs and more generally Schreier graphs corresponding to some natural group actions. The spectrum of such an operator is a compact subset of the closed interval [-1,1], but not much more can be said about it in general.
We will discuss various techniques that allow to construct examples with different types of spectra: connected, union of two intervals, totally disconnected…, and how this depends on the choice of the generating set in the group. Types of spectral measures that can arise in these examples will also be discussed. |
|
| Date: | November 13, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Paul Schupp, UIUC |
| Title: | Closures of Turing Degrees |
| Abstract: | This talk is on aspect of my general project with Carl Jockusch on “the coarsification of computability theory”, that is, bringing the asymptotic-generic point of view of geometric group theory into the theory of computability. Classically, computability theory studies Turing degrees, that is, equivalence classes of subsets of N which are computationally equivalent. Coarse computability studies how closely arbitrary subsets of N can be approximated by computable sets. The idea of coarse computabilty leads to a natural definition of the closure of a Turing degree in the space S of coarse similarity classes of subsets of N with the Besicovich metric. It
turns out that S is an interesting space. We will discuss interactions of the topology of S and properties of Turing degrees. |
|
| Date: | November 13, 2019 |
| Time: | 4:00pm |
| Location: | BLOC 220 |
| Speaker: | Tullio Ceccherini-Silberstein |
| Title: | Hecke algebras of multiplicity-free induced representations |
| Abstract: | Given a finite group G and a subgroup K, one says that (G,K) is a Gelfand pair
provided the associated permutation representation (\lambda, L(G/K)) is multiplicity-free
(that is, decomposes into pairwise non-equivalent irreducible subrepresentations). This condition
is equivalent to the algebra End_G(L(G/K)) of interwining operators being commutative.
Observe that \lambda is nothing but the induced representation Ind_K^G \iota_K of the trival representation
\iota_K of K. In [CS-S-T] we consider triples (G,K,\theta), where \theta is, more generally, an irreducible K-representation
and introduce a Hecke-type algebra H(G,K,\theta) - analogous to End_G(L(G/K)) - and show that
that Ind_K^G\theta is multiplicity-free if and only if H (G,K,\theta) is commutative.
We apply our results in the context of the representation theory of GL_2(q), the general linear group
of a field with q elements.
[CS-S-T] Harmonic analysis and spherical functions for multiplicity-free induced representations of finite groups.
Springer (to appear) arXiv: 1811.09526. |
|
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