L6

A fundamental theorem of Wilson states that, for every graph $F$, every sufficiently large $F$-divisible clique has an $F$-decomposition. Here $G$ has an $F$-decomposition if the edges of $G$ can be covered by edge-disjoint copies of $F$ (and $F$-divisibility is a trivial necessary condition for this). We extend Wilson's theorem to graphs which are allowed to be far from complete (joint work with B. Barber, D. Kuhn, A. Lo).

I will also discuss some results and open problems on decompositions of dense graphs and hypergraphs into Hamilton cycles and perfect matchings.

In 1937 Quine introduced an interesting, rather unusual, set theory called New Foundations - NF for short.  Since then the consistency of NF has been a problem that remains open today.  But there has been considerable progress in our understanding of the problem. In particular NF was shown, by Specker in 1962, to be equiconsistent with a certain theory, TST^+ of simple types. Moreover Randall Holmes, who has been a long-term investigator of the problem, claims to have  solved the problem by showing that TST^+ is indeed consistent.  But the working manuscripts available on his web page that describe his possible proofs are not easy to understand - at least not by me.

I will explain the framework of quasiminimal structures and quasiminimal classes, and give some basic examples and open questions. Then I will explain some joint work with Martin Bays in which we have constructed variants of the pseudo-exponential fields (originally due to Boris Zilber) which are quasimininal and discuss progress towards the problem of showing that complex exponentiation is quasiminimal. I will also discuss some joint work with Adam Harris in which we try to build a pseudo-j-function.

Standard commutative algebra is based on the notions of commutative monoid, Abelian group and commutative ring. In recent years, motivations from category theory, algebraic geometry, and mathematical logic led to the development of an area that may be called commutative 2-algebra, in which the notions used in commutative algebra are replaced by their category-theoretic counterparts (e.g. commutative monoids are replaced by  symmetric monoidal categories). The aim of this talk is to explain the analogy between standard commutative algebra and commutative 2-algebra, and to outline how this suggests counterparts of basic aspects of algebraic geometry. In particular, I will describe some joint work with Andre’ Joyal on operads and analytic functors in this context.

There is a tight fit between type theories à la Martin-Löf and constructive set theories such as Constructive Zermelo-Fraenkel set theory, CZF, and its extension as well as classical Kripke-Platek set theory and extensions thereof. The technology for determining their (exact) proof-theoretic strength was developed in the 1990s. The situation is rather different when it comes to type theories (with universes) having the impredicative type of propositions Prop from the Calculus of Constructions that features in some powerful proof assistants. Aczel's sets-as-types interpretation into these type theories gives rise to  rather unusual set-theoretic axioms: negative power set and negative separation. But it is not known how to determine the proof-theoretic strengths of intuitionistic set theories with such axioms via familiar classical set theories (though it is not difficult to see that ZFC plus infinitely many inaccessibles provides an upper bound). The first part of the talk will be a survey of known results from this area. The second part will be concerned with the rather special computational and proof-theoretic behavior of such theories.

Unordered configuration spaces on (connected) manifolds are basic objects
that appear in connection with many different areas of topology. When the
manifold M is non-compact, a theorem of McDuff and Segal states that these
spaces satisfy a phenomenon known as homological stability: fixing q, the
homology groups H_q(C_k(M)) are eventually independent of k. Here, C_k(M)
denotes the space of k-point configurations and homology is taken with
coefficients in Z. However, this statement is in general false for closed
manifolds M, although some conditional results in this direction are known.

I will explain some recent joint work with Federico Cantero, in which we
extend all the previously known results in this situation. One key idea is
to introduce so-called "replication maps" between configuration spaces,
which in a sense replace the "stabilisation maps" that exist only in the
case of non-compact manifolds. One corollary of our results is to recover a
"homological periodicity" theorem of Nagpal -- taking homology with field
coefficients and fixing q, the sequence of homology groups H_q(C_k(M)) is
eventually periodic in k -- and we obtain a much simpler estimate for the
period. Another result is that homological stability holds with Z[1/2]
coefficients whenever M is odd-dimensional, and in fact we improve this to
stability with Z coefficients for 3- and 7-dimensional manifolds.

I will discuss a puzzling theorem about smooth, periodic, real-valued functions on the real line. After introducing the classical Hardy-Littlewood maximal function (which just takes averages over intervals centered at a point), we will prove that if a function has the property that the computation of the maximal function is simple (in the sense that it's enough to check two intervals), then the function is already sin(x) (up to symmetries). I do not know what maximal local averages have to do with the trigonometric function. Differentiation does not help either: the statement equivalently says that a delay differential equation with a solution space of size comparable to C^1(0,1) has only the trigonometric function as periodic solutions.

I will present some recent results concerning the higher gradient integrability of

σ-harmonic functions u with discontinuous coefficients σ, i.e. weak solutions of

div(σ∇u) = 0. When σ is assumed to be symmetric, then the optimal integrability

exponent of the gradient field is known thanks to the work of Astala and Leonetti

& Nesi. I will discuss the case when only the ellipticity is fixed and σ is otherwise

unconstrained and show that the optimal exponent is attained on the class of

two-phase conductivities σ: Ω⊂R27→ {σ1,σ2} ⊂M2×2. The optimal exponent

is established, in the strongest possible way of the existence of so-called

exact solutions, via the exhibition of optimal microgeometries.

(Joint work with V. Nesi and M. Ponsiglione.)