Week 10.06.2024 – 16.06.2024

see
https://disorderedsystemskcl.weebly.com/disordered-systems-days.html

The Nielsen-Thurston classification theorem states that there are three kinds of surface homeomorphisms up to homotopy: periodic, reducible, and pseudo-Anosov.

In the introductory part of the talk, we will investigate the differences between these three categories, focusing on the wide array of geometric, topological, and dynamical properties that set pseudo-Anosov mapping classes apart from the rest. To this end, we will also introduce the curve graph, a combinatorial object associated with a surface, and describe how the dynamics of the action of a mapping class on the curve graph can be used to detect pseudo-Anosovness.

In the second part of the talk, we will see how to turn this characterisation into an algorithm for deciding whether a given mapping class is pseudo-Anosov. The key tools will be the theory of train tracks and their connection to the curve graph developed by Masur and Minsky.

This is part of the London number theory study group on uniform Mordell. This talk will give an introduction to theory of abelian varieties and their moduli, especially over the complex numbers. The notion of mixed Shimura varieties may be touched on. The notion of Kawamata or Ueno locus may also be touched on.

The website is here:
https://sites.google.com/site/netandogra/seminars/uniform-mordell

A good deal of the arithmetic of a field can be expressed by sentences in the first-order language of rings. The theories
of the characteristic zero local fields have been axiomatized and are decidable: in the case of $Q_p$ and its finite extensions,
Ax, Kochen, and (independently) Ershov, gave complete axiomatizations that are centred on a formalization of Hensel’s
Lemma. In fact the theory of any field of characteristic zero which is complete with respect to a non-archimedean
valuation can be likewise axiomatized.

I will explain recent joint work with Jahnke, and also with Dittmann and Jahnke, in which we extend the classical
work on these theories to include the case of imperfect residue fields. In particular we show that “Hilbert’s Tenth
Problem” (H10) in these fields (i.e. the problem of effectively determining whether a given Diophantine equation has
solutions) is solvable if and only if the analogous problem is solvable on a structure we define on the residue field. This
follows a pattern of such “transfer” results for H10 — established for valued fields of positive characteristic in earlier
work with Fehm — although in the current case we really need the extra structure.

I will describe these results, focusing on the extent to which they depend (or not) on the residue field. If there is
time I will discuss the aforementioned H10 transfer for complete valued fields in positive characteristic, including more
recent uniform aspects.

I will not assume a background in logic.

We consider, for $h,E>0$, the semiclassical Schrödinger operator $-h^2\Delta + V - E$ in dimension two and higher. The potential $V$, and its radial derivative $\partial_{r}V$ are bounded away from the origin, have long-range decay and $V$ is bounded by $r^{-\delta}$ near the origin while $\partial_{r}V$ is bounded by $r^{-1-\delta}$, where $0\leq\delta < 4(\sqrt{2}-1)$. In this setting, we show that the resolvent bound is exponential in $h^{-1}$, while the exterior resolvent bound is linear in $h^{-1}$.

One of the oldest problems in low-dimensional topology is the unknot recognition problem, posed by Max Dehn in 1910: Is there an algorithm to decide if a given knot can be untangled? You know that this is a challenging problem if you owned a pair of earphones that are tangled! The unknot recognition problem was highlighted by Alan Turing in his last article in 1954, and the first solution was given by Wolfgang Haken in 1961. However, it remains widely open whether there exists a polynomial time algorithm to detect the unknot. The current state-of-the-art is Lackenby’s announcement for a quasi-polynomial time algorithm, which puts it in similar standing to the graph isomorphism problem. I will discuss what is known about the unknot recognition, how it is related to the theory of foliations on three-dimensional manifolds, as well as recent developments on related algorithmic problems.