C3

The stable uniqueness theorem for KK-theory asserts that a Cuntz-pair of *-homomorphisms between separable C*-algebras gives the zero element in KK if and only if the *-homomorphisms are stably homotopic through a unitary path, in a specific sense. This result, along with its group equivariant analogue, has been crucial in the classification theory of C*-algebras and C*-dynamics. In this talk, I will present a unitary tensor category analogue of the stable uniqueness theorem and explore its application to a duality in tensor category equivariant KK-theory. To make the talk approachable even for those unfamiliar with actions of unitary tensor categories or KK-theory, I will introduce the relevant definitions and concepts, drawing comparisons with the case of group actions. This is joint work with Kan Kitamura and Robert Neagu.

The notion of semi-uniform stability of a strongly continuous semi-group refers to the stability of classical solutions of a linear evolution equation, and this has analogues with the classical Katznelson-Tzafriri theorem. The co-generator of a strongly continuous semigroup is a bounded linear operator that comes from a particular discrete approximation to the semigroup. After reviewing some background on (quantified) stability theory for semigroups and the Katznelson-Tzafriri theorem, I will present some results relating the stability of a strongly continuous semigroup with that of its cogenerator. This talk is based on joint work with David Seifert.

Roe type algebras are operator algebras designed to catch the large-scale behaviour of metric spaces. This talk focuses on the following question: if two Roe type algebras associated to spaces (X,d_X) and (Y,d_Y) are isomorphic, how similar are X and Y? We provide positive results proved in the last 5 years, and, if time allows it, we show that sometimes answers to this question are subject to set theoretic considerations

In this talk, based on a joint work with Ilijas Farah, I will present an application of an old continuous selection theorem due to Michael to the study of II1 factors. More precisely, I'll show that if two strongly continuous paths (or loops) of projections (p_t), (q_t), for t in [0,1], in a II1 factor are such that every p_t is subequivalent to q_t, then the subequivalence can be realized by a strongly continuous path (or loop) of partial isometries. I will then use an extension of this result to solve affirmatively the so-called trace problem for factorial W*-bundles whose base space is 1-dimensional.

Let G be the points of a reductive group over a p-adic field. According to Bernstein, the category of smooth complex representations of G decomposes as a product of indecomposable subcategories (blocks), each determined by inertial supercuspidal support. Moreover, each of these blocks is equivalent to the category of modules over a Hecke algebra, which is understood in many (most) cases. However, when the coefficients of the representations are now allowed to be in a more general ring (in which p is invertible), much of this fails in general. I will survey some of what is known, and not known.

Consider definable sets over the family of finite fields $\mathbb{F}_q$. Ax proved a quantifier-elimination result for this theory, in a reasonable geometric language. Chatzidakis, Van den Dries and Macintyre showed that to a first-order approximation, the cardinality of a definable set $X$ is definable in a very mild expansion of Ax's theory.  Can such a statement be true of the next higher order approximation, i.e. can we write $|X(\mathbb{F}_q)| = aq^{d} + bq^{d-1/2} + o(q^{d-1/2})$, with $d,a,b$ varying definably with $X$ in a tame theory?    Here $b$ must be viewed as real-valued so continuous logic is needed. I will report on joint work in progress with Will Johnson.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

Solovay defined the inner model $L(\mathbb{R}, \mu)$ in the context of $\mathsf{AD}_{\mathbb{R}}$ by using it to define the supercompactness measure $\mu$ on $\mathcal{P}_{\omega_1}(\mathbb{R})$ naturally given by $\mathsf{AD}_{\mathbb{R}}$. Solovay speculated that stronger versions of this inner model should exist, corresponding to stronger versions of the measure $\mu$. Woodin, in his unpublished work, defined $\mu_{\infty}$ which is arguably the ultimate version of the supercompactness measure $\mu$ that Solovay had defined. I will talk about $\mu_{\infty}$ in the context of $\mathsf{AD}^+$ and the axiom $\mathsf{V} = \mathsf{Ultimate\ L}$.

https://woloszyn.org/

I will report on a joint work with Pablo Destic and Nuno Hultberg, about some applications of Globally Valued Fields (GVFs) and I will describe a density result that we needed, which turns out to be connected to Riemann-Zariski and Berkovich spaces.