Activities This Week

The conceptual completeness theorem answers a fundamental question: given the category of models of a theory, how much of the theory’s syntax can be recovered? After a gentle introduction to categorical logic and classifying categories, I will explain Makkai’s remarkable result showing that the semantic category, together with ultraproduct-preserving functors, determines the theory up to definitional equivalence.

Given a natural number $n \geq 1$, the odometer semigroup $O_n$, also known as the adding machine or the Baumslag–Solitar monoid with two generators, is a well-known object in group theory. This talk will examine the odometer semigroup in relation to representations of bounded linear operators. We will focus on noncommutative operators and show that contractive representations of $O_n$ always admit nicer representations. A complete description of representations of $O_n$ on the Fock space will be presented, along with connections to odometer lifting and subrepresentations. Along the way, we will also classify Nica–covariant representations of $O_n$.

The Kazhdan–Lusztig isomorphism, relating the affine Hecke algebra of a p-adic group to the equivariant K-theory of the Steinberg variety of its Langlands dual, played a key role in the proof of the Deligne–Langlands conjectures concerning the classification of tamely ramified irreducible representations. In this talk, I will recall the statement of the Kazhdan–Lusztig isomorphism. I will also introduce the relative Langlands duality and propose a conjectural relative version of the Kazhdan–Lusztig isomorphism. I will focus on specific examples which we can prove.

The theory of characters of infinite groups, initiated by Thoma, is a generalization of the representation theory of finite groups. More explicitly, a character of a group is an (extremal) conjugation invariant positive definite function. A group said to be character rigid if every character of the group is either supported on the center or comes from a finite dimensional representation. Connes conjecture that any irreducible lattice in a higher rank Lie group is character rigid. Surprisingly, this conjecture is a generalization of the celebrated Margulis normal subgroup theorem and of the Stuck-Zimmer conjecture on IRS rigidity. I will discuss a recent joint work with Alon Dogon, Yuval Gorfine, Liam Hanany, and Arie Levit showing that any nonuniform higher rank lattice is character rigid, proving the Stuck-Zimmer conjecture for such lattices.