Oxford

The talk will focus on results of two related strands of research undertaken by the speaker. The first is a model of quantum mechanics based on the idea of 'structural approximation'. The earlier paper 'The semantics of the canonical commutation relations' established a method of calculation, essentially integration, for quantum mechanics with quadratic Hamiltonians. Currently, we worked out a (model-theoretic) formalism for the method, which allows us to
perform more subtle calculations, in particular, we prove that our path integral calculation produce correct formula for quadratic Hamiltonians avoiding non-conventional limits used by physicists. Then we focus on the model-theoretic analysis of the notion of structural approximation and show that it can be seen as a positive model theory version of the theory of measurable structures, compact domination and integration (p-adic and adelic).

A generic surface homeomorphism (up to isotopy) is what we call it pseudo-Anosov. These maps come equipped with an algebraic integer that measures
how much the map stretches/shrinks in different direction, called the stretch factor. Given a surface homeomorsphism, one can ask if it is the lift (by a branched or unbranched cover) of another homeomorphism on a simpler surface possibly of small genus. Farb conjectured that if the algebraic degree of the stretch factor is bounded above, then the map can be obtained by lifting another homeomorphism on a surface of bounded genus.
This was known to be true for quadratic algebraic integers by a Theorem of Franks-Rykken. We construct counterexamples to Farb's conjecture.

The smooth homotopy category is a simultaneous enlargement of the usual homotopy category and of the category of smooth manifolds. Its structure can be described very simply and explicitly by a version of van Est's theorem.  It provides us with an  interpolation between topology and geometry (and with a toy model of derived algebraic geometry and motivic homotopy theory, though I shall not pursue those directions).  My talk will list some situations which the category seems to illuminate: one will be Kapranov's beautiful description of the Lie algebra of the 'group' of based loops in a manifold.
 

We show that a closed almost Kähler 4-manifold of globally constant holomorphic sectional curvature k<=0 with respect to the canonical Hermitian connection is automatically Kähler. The same result holds for k < 0 if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observation that such manifolds are self-dual, so that Chern–Weil theory implies useful integral formulas, which are then combined with results from Seiberg–Witten theory.

We will follow a suggestion by Udi to construct a decidable field which has an undecidable finite extension.

I will talk about a result on meromorphic continuation of Euler products over primes p of definable p-adic or motivic integrals, and applications to zeta functions of groups. If time permitting, I'll state an analogue for counting rational points of bounded height in some adelic homogeneous spac

Transport properties of liquids and gases in the regime of weak coupling (or effective weak coupling) are determined by the solutions of relevant kinetic equations for particles or quasiparticles, with transport coefficients being proportional to the minimal eigenvalue of the linearized kinetic operator. At strong coupling, the same physical quantities can sometimes be determined from dual gravity, where quasinormal spectra enter as the eigenvalues of the linearized Einstein's equations. We discuss the problem of interpolating between the two regimes using results from higher derivative gravity.

Over the last few decades, a number of authors have discussed the analogy between linking numbers in three manifold topology and symbols in arithmetic. This talk will outline some results that make this precise in terms of natural complexes associated to arithmetic duality theorems. In particular, we will describe a ‘finite path integral’ formula for power residue symbols.

We discuss a recent preprint by Aschenbrenner, Khélif, Naziazeno and
Scanlon, giving a positive solution to the ring-analogue of Pop's
problem on elementary equivalence vs isomorphism. 

.. showing that a field K is isomorphic to Q if it has the same absolute Galois group and if it satisfies a very small additional condition (very similar to my talk 2 years ago).