C4

Here, a connection is a algebraic structure that is weaker than an algebra and stronger than a module. I will define this structure and give examples. I will then define the quantum product and explain how connections capture important properties of this product. I will finish by stating a new result which describes how this extends to equivariant Floer cohomology. No knowledge of symplectic topology will be assumed in this talk.
 

We will provide a general overview on some recent work on non-archimedean parametrizations and their applications. We will start by presenting our work with Cluckers and Comte on the existence of good Yomdin-Gromov parametrizations in the non-archimedean context and a $p$-adic Pila-Wilkie theorem.   We will then explain how this is used in our work with Chambert-Loir to prove bialgebraicity results in products of Mumford curves. 
 

 

Given a finite set X, is an easy exercise to show that a binary operation * from XxX to X which is injective in each variable separately, and which is also associative, makes (X,*) into a group. Hrushovski and others have asked what happens if * is only partially associative - do we still get something resembling a group? The answer is known to be yes (in a strong sense) if almost all triples satisfy the associative law. In joint work with Tim Gowers, we consider the so-called `1%' regime, in which we only have an epsilon fraction of triples satisfying the associative law. In this regime, the answer turns out to be rather more subtle, involving certain group-like structures which we call rough approximate groups. I will discuss these objects, and try to give a sense of how they arise, by describing a somewhat combinatorial interpretation of partial associativity.
 

A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs.  Given a hereditary graph property H, the speed of H is the function which sends an integer n to the number of distinct elements in H with underlying set {1,...,n}.  Not just any function can occur as the speed of hereditary graph property.  Specifically, there are discrete ``jumps" in the possible speeds.  Study of these jumps began with work of Scheinerman and Zito in the 90's, and culminated in a series of papers from the 2000's by Balogh, Bollob\'{a}s, and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized.  In contrast to this, many aspects of this problem in the hypergraph setting remained unknown.  In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds.  The jumps in the factorial range turned out to have surprising connections to the model theoretic notion of mutual algebricity, which we also discuss.  This is joint work with Chris Laskowski.

We take a look at difference fields with several commuting automorphisms. The theory of difference fields with one distinguished automorphism has a model companion known as ACFA, which Zoe Chatzidakis and Ehud Hrushovski have studied in depth. However, Hrushovski has proved that if you look at fields with two or more commuting automorphisms, then the existentially closed models of the theory do not form a first order model class. We introduce a non-elementary framework for studying them. We then discuss how to generalise a result of Kowalski and Pillay that every definable group (in ACFA) virtually embeds into an algebraic group. This is joint work in progress with Zoe Chatzidakis and Nick Ramsey.

In the talk we will define higher K-groups, and explain some of their relations to number theory

 This is a longstanding problem. We will present candidate axioms that we believe form a complete axiom system satisfied by $F_p^{alg}((t))$.  A currently "unfinished" proof will be discussed.   This work is joint with Françoise Delon (Paris) and Arno Fehm (Dresden).

 In Classification Theory, Shelah defined several cardinal invariants of a complete theory which detect the presence of certain trees among the definable sets, which in turn quantify the complexity of forking.  In later model-theoretic developments, local versions of these invariants were recognized as marking important dividing lines - e.g. simplicity and NTP2.  Around these dividing lines, a dichotomy theorem of Shelah states that a theory has the tree property if and only if it is witnessed in one of two extremal forms--the tree property of the first or second kind--and it was asked if there is a 'quantitative' analogue of this dichotomy in the form of a certain equation among these invariants.  We will describe these model-theoretic invariants and explain why the quantitative version of the dichotomy fails, via a construction that relies upon some unexpected tools from combinatorial set theory. 

 

I will be talking of recent results by Ayse Berkman and myself, as well as about a more general program of research in this area.

I will present joint work with A. Pillay on the theory of NIP formulas in arbitrary groups, which exhibit a local formulation of the notion of finitely satisfiable generics (as defined by Hrushovski, Peterzil, and Pillay). This setting generalizes ``local stable group theory" (i.e., the study of stable formulas in groups) and also the case of arbitrary NIP formulas in pseudofinite groups. Time permitting, I will mention an application of these results in additive combinatorics.