| Date: | January 24, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Mikael de la Salle, Université Lyon I and IAS |
| Title: | Higher-rank lattices and uniformly convex Banach spaces (this is a joint seminar with the Groups and Dynamics seminar) |
| Abstract: | Lattices in higher rank simple Lie groups, such as SL(3,Z), are very rigid. It is in general expected, and often proved, that they can only act non-trivially if there is an arithmetic reason for it. In particular, it was conjectured by Bader, Furman, Gelander and Monod that any action by isometries of such a group on a uniformly convex Banach Space has a fixed point. I will present the solution to this conjecture, that I obtained recently with Tim de Laat, following a breakthrough by Izhar Oppenheim. As a consequence, the finite quotients of higher rank lattices are super-expanders in the sense of Mendel and Naor. |
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| Date: | February 1, 2024 |
| Time: | 10:00am |
| Location: | BLOC 302 |
| Speaker: | Hung Viet Chu, Texas A&M University |
| Title: | Higher Order Tsirelson Spaces and their Modified Version are Isomorphic |
| Abstract: | In this talk, we sketch the proof that the Tsirelson space $T_\xi$ is naturally isomorphic to its modified version $T^M_\xi$ for each countable ordinal $\xi$. We begin by recalling the definition of Schreier families $\mathcal{S}_\xi$, where $\xi$ is a countable ordinal, and their modified version $\mathcal{S}^M_\xi$. These families are defined by transfinite induction on $\xi$. In the case of a successor ordinal $\xi$, i.e., $\xi = \gamma+1$, the family $\mathcal{S}_\xi$ is the collection of unions of sets $E_1< E_2 < \cdots < E_d$ in $\mathcal{S}_\gamma$ with $\min E_1\geqslant d$, where $E_i < E_j$ means that $\max E_i < \min E_j$. On the other hand, the modified version $\mathcal{S}^M_\xi$ only requires the sets $E_i$ to be disjoint instead of being consecutive. From these definitions, we know that $\mathcal{S}_\xi \subset \mathcal{S}^M_\xi$. Our first result shows that $\mathcal{S}_\xi$ is actually equal to $\mathcal{S}^M_\xi$ for each countable ordinal $\xi$, thus answering a question by Argyros and Tolias. This result together with certain tree analysis of the norming sets of $T_\xi$ and $^M_\xi$ give us their isomorphism. As an application, we show that the algebra of linear bounded operators on $T_\xi$ has $2^{\mathfrak c}$ closed ideals. The speaker is thankful to Dr. Schlumprecht for his excellent guidance in this joint work. |
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| Date: | February 22, 2024 |
| Time: | 10:00am |
| Location: | BLOC 302 |
| Speaker: | Chris Gartland, University of California, San Diego |
| Title: | Stochastic Embeddings of Hyperbolic Metric Spaces |
| Abstract: | This talk is based on ongoing work of the speaker. We will discuss the stochastic embeddability of snowflakes of doubling metric spaces into ultrametric spaces and the induced stochastic embeddings of their hyperbolic fillings into trees. As an application, we obtain that finitely generated Gromov hyperbolic groups admit proper, uniformly Lipschitz affine actions on L1. |
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| Date: | March 21, 2024 |
| Time: | 10:00am |
| Location: | BLOC 302 |
| Speaker: | Harisson Gaebler, University of North Texas |
| Title: | Riemann integration and asymptotic structure of Banach spaces |
| Abstract: | Let X be a Banach space. A bounded and Lebesgue almost-everywhere continuous function f:[0,1]\to X is Riemann-integrable. However, the converse statement is false in general. This motivates the following definition: X is said to have the Lebesgue property if every Riemann-integrable function f:[0,1]\to X is Lebesgue almost-everywhere continuous. In this talk, I will discuss my work during the last several years on the relationship between the Lebesgue property and asymptotic structures. I will begin by giving an overview of the Lebesgue property that includes relevant examples, older results, and a brief mention of my first paper on this topic which ultimately led to two more recent joint works. I will then spend the majority of the talk discussing the these two more recent papers whose results include 1) the characterization of the Lebesgue property in terms of a new asymptotic structure that sits strictly between the notions of a unique \ell_{1} spreading model and a unique \ell_{1} asymptotic model (this is a joint work with Bunyamin Sari) and 2) the complete separation of the Lebesgue property from a (uniformly) unique \ell_{1} spreading model (this is a joint work with Pavlos Motakis and Bunyamin Sari). Lastly, I will mention two relevant open problems. |
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| Date: | April 4, 2024 |
| Time: | 10:00am |
| Location: | BLOC 302 |
| Speaker: | Rubén Medina, Universidad de Granada |
| Title: | On nonlinear approximation properties and a problem of Godefroy and Ozawa |
| Abstract: | In this talk we will focus on nonlinear analogues of classical
approximation properties in separable Banach spaces. More specifically,
we will present different properties that have been conjectured to hold
in every separable Banach space by N. Kalton (2012) as well as by G.
Godefroy and N. Ozawa (2014). Regarding the problem raised by Godefroy
and Ozawa, we will present certain advances produced in the last three
years in collaboration with Petr Hájek, some of which hint towards a
negative solution. We will also comment some positive results for
'large' families of spaces and connections with linear approximation
properties. |
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