| Résumé: |
There exists a notion of a Fourier Transform on $L^1(G)$ for any (possibly non-commutative) locally compact group $G$. The definition extends the classical definition of the Fourier Transform on $L^1(\mathbb{R})$, $L^1(\mathbb{T})$, $\ell^1(\mathbb{Z})$ etc. Abstract harmonic analysis then deals with, to a large extent, understanding how much the properties of the Fourier Transform on an arbitrary locally compact group resemble the properties of the Fourier Transform in the classical cases. It is a classical problem, dating back to the 1930’s, to determine which locally compact groups are Wiener, which is a question about their Fourier Transform. The problem is well understood for abelian groups, compact groups and connected groups, however, it is not well understood which totally disconnected groups are Wiener.
The Wiener property can also be formulated more generally for any Banach $*$-algebra. In this talk, I will discuss recent progress on the Wiener property for weighted $L^p$-algebras on locally elliptic groups, along with some related representation theoretic properties. Locally elliptic groups are an interesting class of totally disconnected groups that include non-archimedean local fields and unipotent linear algebraic groups over these fields. My results make progress on understanding which totally disconnected groups are Wiener and have connections with ongoing work on the representation theory of these groups. |
| Résumé: | TBA |
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| Résumé: | Dyer groups are a family encompassing both Coxeter groups and right-angled Artin groups. Among many common properties, these two families admit the same solution to the word problem. Each of these two classes of groups also have natural piecewise Euclidean CAT(0) spaces associated to them. In this talk I will introduce Dyer groups, give some of their properties. I will not assume the audience to be familiar with Coxeter groups or right-angled Artin groups. |
| Résumé: |
A group has Property (T) if every action by affine isometry on a Hilbert space admits a fixed point. This definition emphasizes the idea that group actions having Property (T) are rigid.
In his work on the Baum-Connes conjecture, Vincent Lafforgue defined in 2007 a strengthening of Property (T) that implies a fixed point result for affine actions on Hilbert spaces that are no longer isometric, but for which the operator norm growth is sub-exponential. Lafforgue also showed that any action by isometry on a uniformly locally finite Gromov-hyperbolic space of a group having the strong Property (T) admits bounded orbits.
I will present a work where I show that relatively hyperbolic groups do not have the strong Property (T). The idea of the proof, similar to Lafforgue's proof for hyperbolic groups, is to use a natural action on a hyperbolic graph to construct a representation of our group into a Hilbert space that has sub-exponential growth and no fixed point. |
| Résumé: | We show that, up to a natural equivalence relation, the only non-trivial, non-identity holomorphic maps $\mathrm{Conf}_n\mathbb{C}\to\mathrm{Conf}_m\mathbb{C}$ between unordered configuration spaces, where $m\in\{3,4\}$, are the resolving quartic map $R\colon\mathrm{Conf}_4\mathbb{C}\to\mathrm{Conf}_3\mathbb{C}$, a map $\Psi_3\colon\mathrm{Conf}_3\mathbb{C}\to\mathrm{Conf}_4\mathbb{C}$ constructed from the inflection points of elliptic curves in a family, and $\Psi_3\circ R$. This completes the classification of holomorphic maps $\mathrm{Conf}_n\mathbb{C}\to\mathrm{Conf}_m\mathbb{C}$ for $m\leq n$, extending results of Lin, Chen and Salter, and partially resolves a conjecture of Farb. We also classify the holomorphic families of elliptic curves over $\mathrm{Conf}_n\mathbb{C}$. To do this we classify homomorphisms between braid groups with few strands and $\mathrm{PSL}_2\mathbb{Z}$, then apply powerful results from complex analysis and Teichmüller theory. Furthermore, we prove a conjecture of Castel about the equivalence classes of endomorphisms of the braid group with three strands. Joint with Peter Huxford. |