We will follow a suggestion by Udi to construct a decidable field which has an undecidable finite extension.

# Past Advanced Logic Class

I will discuss my ongoing project towards a version of the Modular Andre-Oort Conjecture incorporating the derivatives of the j function. The work originates with Jonathan Pila, who formulated the first "Modular Andre-Oort with Derivatives" conjecture. The problem can be approached via o-minimality; I will discuss two categories of result. The first is a weakened version of Jonathan's conjecture. Under an algebraic independence conjecture (of my own, though it follows from standard conjectures), the result is equivalent to the statement that Jonathan's conjecture holds.

The second result is conditional on the same algebraic independence conjecture - it specifies more precisely how the special points in varieties can occur in this context.

If time permits, I will discuss my most recent work towards making the two results uniform in algebraic families.

I will talk about a result on meromorphic continuation of Euler products over primes p of definable p-adic or motivic integrals, and applications to zeta functions of groups. If time permitting, I'll state an analogue for counting rational points of bounded height in some adelic homogeneous spac

We discuss a recent preprint by Aschenbrenner, Khélif, Naziazeno and

Scanlon, giving a positive solution to the ring-analogue of Pop's

problem on elementary equivalence vs isomorphism.

This talk will discuss the so-called ``generic cohomology’’ of function fields over algebraically closed fields, from the point of view of motives and/or Zariski geometry. In particular, I will describe some interesting connections between cup products, algebraic dependence, and (geometric) valuation theory. As an application, I will mention a new result which reconstructs higher-dimensional function fields from their generic cohomology, endowed with some additional motivic data.

Everyone welcome!

Generalising previous definability results in global fields using

quaternion algebras, I will present a technique for first-order

definitions in finite extensions of Q(t). Applications include partial

answers to Pop's question on Isomorphism versus Elementary Equivalence,

and some results on Anscombe's and Fehm's notion of embedded residue.

Given a Shimura variety, I will show how to define a corresponding two-sorted structure. Based on work of Chris Daw and Adam Harris, we will study what is needed for the class of this structures to be categorical. Of course, an introduction to Shimura varieties will be given.

Given a Shimura variety, I will show how to define a corresponding two-sorted structure. Based on work of Chris Daw and Adam Harris, we will study what is needed for the class of this structures to be categorical. Of course, an introduction to Shimura varieties will be given.

In this talk, we will explain how the counting theorems of Pila and Wilkie lead to a conditional proof of the aforementioned conjecture. In particular, we will explain how to generalise the work of Habegger and Pila on a product of modular curves.

Habegger and Pila were able to prove that the Zilber-Pink conjecture holds in such a product if the so-called weak complex Ax and large Galois orbits conjectures are true. In fact, around the same time, Pila and Tsimerman proved a stronger statement than the weak complex Ax conjecture, namely, the Ax-Schanuel conjecture for the $j$-function. We will formulate Ax-Schanuel and large Galois orbits conjectures for general Shimura varieties and attempt to imitate the Habegger-Pila strategy. However, we will encounter an additional difficulty in bounding the height of a pre-special subvariety.

This is joint work with Jinbo Ren.