The Elekes-Szabó Theorem roughly says the following: Let R be an algebraic ternary relation in W1*W2*W3 defined in a field K of characteristic 0, such that any two coordinate is interalgebraic with the third one, for example the collinear relation for three points in a curve.
Suppose there are arbitrarily large finite subsets Xi of Wi each of size n and has bounded intersection with any proper subvariety of Wi, such that the intersection of R with X1*X2*X3 has size approximately n^2, then R must be essentially the graph of addition of some commutative algebraic group G. In this talk, I will give an overview of several results (joint work with Martin Bays and Jan Dobrowolski) in the effort of removing the assumption of Xi having bounded intersection with proper subvarieties of Wi. This assumption is closely related to Wi being 1-dimensional. Our motivation is to find a genuine higher-dimensional generalisation of the Elekes-Szabó Theorem.
Many model-theoretic dividing lines, such as NIP, stability, and distality, are naturally defined in terms of binary formulas. One can nonetheless study higher-arity formulas under an NIP/stability/distality assumption, but this often turns out to be too strong. This has led to the development of higher-arity NIP, stability, and distality (which are strict weakenings of the binary versions). Of these, higher-arity distality has received the least treatment, and much more work remains before the theory can be applied. In particular, the theory of strong honest definitions for higher-arity distality is relatively undeveloped. I will survey what is known in the literature and discuss my ongoing work.
The representation type of a finite-dimensional k-algebra is an algebraic measure of how hard it is to classify its finite-dimensional indecomposable modules.
Intuitively, a finite-dimensional k-algebra is of tame representation type if we can classify its finite-dimensional modules and wild representation type if its module category contains a copy of the category of finite-dimensional modules of all other finite-dimensional k-algebras. An archetypical (although not finite-dimensional) tame algebra is k[x]. The structure theorem for finitely generated modules over a PID describes its finite-dimensional modules. Drozd’s famous dichotomy theorem states that all finite-dimensional algebras are either wild or tame.
The tame/wild dividing line is not seen by standard model theoretic invariants or even the more specialised invariants coming from Model Theory of Modules. A long-standing conjecture of Mike Prest claims that a finite-dimensional algebra has decidable theory of modules if and only if it is of tame representation type. More recently, I conjectured that a finite-dimensional algebra has decidable theory of (pseudo)finite dimensional modules if and only if it is of tame representation type. This talk will focus on recent work providing evidence for the second conjecture.
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.
