L6

We consider the differentiability of definable functions in tame expansions
of algebraically closed valued fields.
As the Frobenius inverse shows such a function may be nowhere
differentiable.
We prove differentiability almost everywhere in valued fields of
characteristic 0
that are C-minimal, definably complete and such that, in the valuation
group,
definable functions are strongly eventually linear.
This is joint work with Pablo Cubides-Kovacsics.

For mapping class groups of surfaces it is well-understood that their homology stability is closely related to the fact that they give rise to an infinite loop space. Indeed, they define an operad whose algebras group complete to infinite loop spaces.

In recent work with Basterra, Bobkova, Ponto and Yaekel we define operads with homology stability (OHS) more generally and prove that they are infinite loop space operads in the above sense. The strong homology stability results of Galatius and Randal-Williams for moduli spaces of manifolds can be used to construct examples of OHSs. As a consequence the map to K-theory defined by the action of the diffeomorphisms on the middle dimensional homology can be shown to be a map of infinite loop spaces.

Given two $k$-graphs $H$ and $F$, a perfect $F$-packing in $H$ is a collection of vertex-disjoint copies of $F$ in $H$ which together cover all the vertices in $H$. In the case when $F$ is a single edge, a perfect $F$-packing is simply a perfect matching. For a given fixed $F$, it is generally the case that the decision problem whether an $n$-vertex $k$-graph $H$ contains a perfect $F$-packing is NP-complete.

In this talk we describe a general tool which can be used to determine classes of (hyper)graphs for which the corresponding decision problem for perfect $F$-packings is polynomial time solvable. We then give applications of this tool. For example, we give a minimum $\ell$-degree condition for which it is polynomial time solvable to determine whether a $k$-graph satisfying this condition has a perfect matching (partially resolving a conjecture of Keevash, Knox and Mycroft). We also answer a question of Yuster concerning perfect $F$-packings in graphs.

This is joint work with Jie Han (Sao Paulo).
 

Groups which are "attached" to theories of fields, appearing in models of the theory  
as the automorphism groups of intermediate fields fixing an elementary submodel are called geometrically represented. 
We will discuss the concept ``geometric representation" in the case of pseudo finite fields.  Then will show that any group which is geometrically represented in a complete theory of a pseudo-finite field must be abelian. 
This result also generalizes to bounded PAC fields. This is joint work with Zoe Chatzidakis.
 

We prove tight upper and lower bounds on an observable of the antiferromagnetic Potts model. From this we deduce the case d=3 of a conjecture of Galvin and Tetali on maximising the number of proper colourings in d-regular graphs.

Using some new results in probabilistic recognition of black box groups
(Joint work with Sukru Yalcinkaya), I will discuss some emerging
structures that beg for a model-theoretic analysis.

In this talk on joint work with REBECCA BELLOVIN we discuss the “local ε-isomorphism” conjecture of Fukaya and Kato for (crystalline) families of G_{Q_p}-representations. This can be regarded as a local analogue of the global Iwasawa main conjecture for families, extending earlier work of Kato for rank one modules, of Benois and Berger for crystalline representations with respect to the cyclotomic extension as well as of Loeffler, Venjakob and Zerbes for crystalline representations with respect to abelian p-adic Lie extensions of Q_p. Nakamura has shown Kato’s - conjecture for (ϕ,\Gamma)-modules over the Robba ring, which means in particular only after inverting p, for rank one and trianguline families. The main ingredient of (the integrality part of) the proof consists of the construction of families of Wach modules generalizing work of Wach and Berger and following Kisin’s approach via a corresponding moduli space.
 

We will consider the following deceptively simple question, formulated recently by Po Shen Loh who connected it to an open problem in Ramsey Theory. Define the '2-less than' relation on the set of triples of integers by saying that a triple x is 2-less than a triple y if x is less than y in at least two coordinates. What is the maximal length of a sequence of triples taking values in {1,...,n} which is totally ordered by the '2-less than' relation?

In his paper, Loh uses the triangle removal lemma to improve slightly on the trivial upper bound of n^2, and conjectures that the truth should be of order n^(3/2). The gap between these bounds has proved to be surprisingly resistant. We shall discuss joint work with Tim Gowers, giving some developments towards this conjecture and a wide array of natural extensions of the problem. Many of these extensions remain open.
 

In a recent breakthrough, Peter Keevash proved the Existence conjecture for combinatorial designs, which has its roots in the 19th century. In joint work with Daniela Kühn, Allan Lo and Deryk Osthus, we gave a new proof of this result, based on the method of iterative absorption. In fact, `regularity boosting’ allows us to extend our main decomposition result beyond the quasirandom setting and thus to generalise the results of Keevash. In particular, we obtain a resilience version and a minimum degree version. In this talk, we will present our new results within a brief outline of the history of the Existence conjecture and provide an overview of the proof.