UEA

Introduced by Lusztig in the eighties, character sheaves are the geometrical counterpart of irreducible representations of finite groups of Lie type. Defined over algebraic groups, they allow us to use geometrical tools to deduce information on the finite groups. In this talk, Marie Roth will give a definition of character sheaves before explaining how to compute their restriction to conjugacy classes (to some extent). This work was part of her PhD thesis under the supervision of Olivier Dudas and Gunter Malle.

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.

I will present Pila's Modular Zilber-Pink with Derivatives (MZPD) conjecture, which is a Zilber-Pink type statement for the j-function and its derivatives, and discuss some weak and functional/differential analogues. In particular, I will define special varieties in each setting and explain the relationship between them. I will then show how one can prove the aforementioned weak/functional/differential MZPD statements using the Ax-Schanuel theorem for the j-function and its derivatives and some basic complex analytic geometry. Note that I gave a similar talk in Oxford last year (where I discussed a differential MZPD conjecture and proved it assuming an Existential Closedness conjecture for j), but this talk is going to be significantly different from that one (the approach presented in this talk will be mostly complex analytic rather than differential algebraic, and the results will be unconditional).

We discuss the connections and differences between the ZFC set theory and univalent foundations and answer the above question in the negative.
 

I will explain the framework of quasiminimal structures and quasiminimal classes, and give some basic examples and open questions. Then I will explain some joint work with Martin Bays in which we have constructed variants of the pseudo-exponential fields (originally due to Boris Zilber) which are quasimininal and discuss progress towards the problem of showing that complex exponentiation is quasiminimal. I will also discuss some joint work with Adam Harris in which we try to build a pseudo-j-function.

In joint work with Todor Tsankov, we show that the automorphism groups of countable, omega-categorical structures have Kazhdan's property (T). The proof uses Tsankov's work on the unitary representations of these groups, together with a construction of a particular free subgroup of the automorphism group.

Little is known about C_exp, the complex field with the exponential function. Model theoretically it is difficult due to the definability of the integers (so its theory is not stable), and a lack of clear algebraic structure; for instance, it is not known whether or not pi+e is irrational. In order to study C_exp, Boris Zilber constructed a class of pseudo-exponential fields which satisfy all the properties we desire of C_exp. This class is categorical for every uncountable cardinal, and other more general classes have been defined. I shall define the three main classes of exponential fields that I study, one of which being Zilber's class, and show that they exhibit "stable-like" behaviour modulo the integers by defining a notion of independence for each class. I shall also explicitly apply one of these independence relations to show that in the class of exponential fields ECF, types that are orthogonal to the kernel are exactly the generically stable types.
 

I will outline some recent work with Jonathan Kirby regarding the first stage in the construction of the pseudo j-function. In particular, I will go through the construction of the analogue of the canonical countable pseudo exponential field as the "Fraisse limit" of a category of "partial j-fields". Although I will be talking about the j-function throughout the talk, it is not necessary to know anything about the j-function to get something from the talk. In particular, even if you don't know what the j-function is, you will still hopefully have an understanding of how to construct the countable pseudo-exp by the end of the talk.
 

In an o-minimal expansion of the real field, while few holomorphic functions are globally definable, many may be locally definable. Wilkie conjectured that a few basic operations suffice to obtain all of them from the basic functions in the language, and proved the conjecture at generic points. However, it is false in general. Using Ax's theorem, I will explain one counterexample. However, this is not the end of the story.
This is joint work with Jones and Servi.