Having outlined the problems, this seminar will be devoted to describing the tools which have been developed to help us solve them. First and foremost of these is the Ax-Schanuel theorem, the functional equivalent of Schanuel's conjecture. We will give a proof, and then discuss how this is applied to prove weak Zilber-Pink, another highly useful result. Given time, we will also explain how it was used by Bays-Kirby to remove Schanuel's conjecture from the list of requirements necessary for Zilber's Quasi-minimality conjecture.
We shall present a general method of constructing uncountable structures that are homogeneous with respect to their finitely generated substructures. The method is based on the category-theoretic Fraisse limits, using the monoid of self-embeddings of a countable homogeneous structure. Joint work with Adam Bartoš.
In this talk we will present a 1989 paper of Chatzidakis which consists in the following. In a field of characteristic $p>0$, the Artin-Shreier map $x \mapsto x^p-x$ is an additive homomorphism. Chatzidakis considers and solves the question of the existence of a cross section of the Artin-Shreier map which has a decidable theory, by describing a model-companion. Then she constructs an explicit model within the algebraic closure $𝔽_p^{\textrm{alg}}$ of the prime field. The construction is quite involved and has a gap, and we will propose a fix for that gap, which is joint work with Martin Hils.
The notion of primitive permutation group for group actions resembles that of simple group for abstract groups. I will discuss work in progress with Katrin Tent (building on earlier work with Liebeck and Tent) on primitive pseudofinite permutation groups. Some methods are analogous to those used in John Wilson’s classification of simple pseudofinite groups.
The open core of a structure is the reduct generated by the open definable sets. Tame topological structures (e.g. o-minimal) are inter-definable with their open core. Structures such as M = (ℝ, <, +, ℚ) are wild in the sense that they define a dense co-dense set. Still, M is NIP and its open core is o-minimal. In this talk we push forward the thesis that the open core of an NTP2 (a generalization of NIP) topological structure is tame. Our main result is that, under suitable conditions, the open core has quantifier elimination, and its definable functions are generically continuous.
An element $x$ in a group $G$ is a left Engel element if for each $x ∈ G$ there exists a positive integer $n = n(x)$ such that $[[[g,x],x],··· ,x] = 1$ ($n$ times). If $n = n(x)$ can be chosen independently of $x$, then we say that $x$ is a left $n$-Engel element. There are some connections to groups of prime power exponent and for example, every element in a group of exponent 3 is a left 2-Engel element. Whereas it is easy to see that the normal closure of a left 2-Engel element is abelian, it is still an open question whether the normal closure of a left 3-Engel element is locally nilpotent. We will give some overview of this problem, focusing on advances in recent years.
A valued difference field is a valued field equipped with an automorphism which fixes the valuation ring setwise. I will discuss various properties of the existentially closed valued difference fields, both from the algebraic and the model-theoretic perspective, and I will highlight the role of tropical geometry in some of the proofs. This is joint work with Jan Dobrowolski and Rosario Mennuni.
A difference field is a field equipped with a given automorphism and a difference variety is the natural analogue of an algebraic varieties in this setting. Complex numbers with complex conjugation or finite fields with the Frobenius automorphism are natural examples of difference fields.
For finite fields and varieties over them, the celebrated Lang-Weil estimate gives a universal estimate of number of rational points of varieties over finite fields in terms of several notions of the complexities of the given variety. In this talk, we will discuss an analogue to Lang-Weil estimate for difference varieties in finite difference fields. The proof uses pseudofinite difference fields, where the automorphism is the nonstandard Frobenius. This is joint work with Martin Hils, Ehud Hrushovski and Tingxiang Zou.
Lie algebras have an interesting relationship with model companions: Whether a given theory has a model companions depends nontrivially on the chosen language and restrictions on the Lie algebras. We will discuss several of these results, and how they can be applied to answer a question of Mennuni.
