Leeds

I will discuss joint work with Gismatullin, Halupczok, and Simonetta on the following problem: given a henselian valued field of characteristic 0, possibly equipped with analytic structure (in the sense stemming originally from Denef and van den Dries), describe the possibilities for a definable group G in the valued field sort which is definably almost simple, that is, has no proper infinite definable normal subgroups. We also have results for an algebraically closed valued field K in characteristic p, but assuming also that the group is a definable subgroup of GL(n, K).

I will review some classical results on geometric scattering
theory for linear hyperbolic evolution equations
on globally hyperbolic spacetimes and its relation to particle and charge
creation in QFT. I will then show that some index formulae for the
scattering matrix can be interpreted as a special case of the  Lorentzian
analog of the Atyiah-Patodi-Singer index theorem. I will also discuss a
local version of this theorem and its relation to anomalies in QFT.
(Joint work with C. Baer)

Transseries arise naturally when solving differential equations around essential singularities. Just like most Taylor series are not convergent, most transseries do not converge to real functions, even when using advanced summation techniques.

On the other hand, we can show that all classical transseries induce analytic functions on the surreal line. In fact, this holds for an even larger (proper) class of series which we call "omega-series".

Omega-series can be composed and differentiated, like LE-series, and they form a differential subfield of surreal numbers equipped with the simplest derivation. This raises once again the question whether all surreal numbers can be also interpreted as functions. Unfortunately, it turns out that the simplest derivation is in fact incompatible with this goal.

This is joint work with A. Berarducci.

 Anyone who has worked in $\beta $N will not be surprised to learn that some of the algebraically defined subsets of $\beta N$ are not topologically simple, even though their algebraic definition may be very simple.  I shall show that the following subsets of $\beta N$ are not Borel: $N^*+N^*$; the smallest ideal of $\beta N$; the set of idempotents in $\beta N$; any semiprincipal right ideal in $\beta N$; the set of idempotents in any left ideal in $\beta N$.

Abstract:  Anyone who has worked in $\beta $N will not be surprised to learn that some of the algebraically defined subsets of $\beta N$ are not topologically simple, even though their algebraic definition may be very simple.  I shall show that the following subsets of $\beta N$ are not Borel: $N^*+N^*$; the smallest ideal of $\beta N$; the set of idempotents in $\beta N$; any semiprincipal right ideal in $\beta N$; the set of idempotents in any left ideal in $\beta N$.

I will describe joint work with Katrin Tent, in which we consider a profinite group equipped with a uniformly definable family of open subgroups. We show that if the family is `full’ (i.e. includes all open subgroups) then the group has NIP theory if and only if it has NTP_2 theory, if and only if it has an (open) normal subgroup of finite index which is a direct product of finitely many compact p-adic analytic groups (for distinct primes p). Without the `fullness’ assumption, if the group has NIP theory then it  has a prosoluble open normal subgroup of finite index.

The concept of pseudofinite dimension for ultraproducts of finite structures was introduced by Hrushovski and Wagner. In this talk, I will present joint work with D. Macpherson and C. Steinhorn in which we explored conditions on the (fine) pseudofinite dimension that guarantee simplicity or supersimplicity of the underlying theory of an ultraproduct of finite structures, as well as a characterization of forking in terms of droping of the pseudofinite dimension. Also, under a suitable assumption, it can be shown that a measure-theoretic condition is equivalent to loc

The prime spectrum of a quantum algebra has a finite stratification in terms
of a set of distinguished primes called H-primes, and we can study these
strata by passing to certain nice localizations of the algebra.  H-primes
are now starting to show up in some surprising new areas, including
combinatorics (totally nonnegative matrices) and physics, and we can borrow
techniques from these areas to answer questions about quantum algebras and
their localizations.    In particular, we can use Grassmann necklaces -- a
purely combinatorial construction -- to study the topological structure of
the prime spectrum of quantum matrices.