Past Seminars

We prove an analogue of the Tate isogeny conjecture and the
semi-simplicity conjecture for overconvergent crystalline Dieudonne modules
of abelian varieties defined over global function fields of characteristic
p, combining methods of de Jong and Faltings. As a corollary we deduce that
the monodromy groups of such overconvergent crystalline Dieudonne modules
are reductive, and after base change to the field of complex numbers they
are the same as the monodromy groups of Galois representations on the
corresponding l-adic Tate modules, for l different from p.

A Landau-Ginzburg B-model is a smooth scheme $X$, equipped with a global function $W$. From $(X,W)$ we can construct a category $D(X,W)$, which is called by various names, including ‘the category of B-branes’. In the case $W=0$ it is exactly the derived category $D(X)$, and in the case that $X$ is affine it is the category of matrix factorizations of $W$. There has been a lot of foundational work on this category in recent years, I’ll describe the most modern and flexible approach to its construction. I’ll then interpret Nick Addington’s thesis in this language. We’ll consider the case that $W$ is a quadratic form on a vector bundle, and the corresponding global version of Knorrer periodicity. We’ll see that interesting gerbe structures arise, related to the bundle of isotropic Grassmannians.

We consider minimisers of integral functionals $F$ over a ‘constrained’ class of $W^{1,p}$-mappings from a bounded domain into a compact Riemannian manifold $M$, i.e. minimisers of $F$ subject to holonomic constraints. Integrands of the form $f(Du)$ and the general $f(x,u,Du)$ are considered under natural strict $p$-quasiconvexity and $p$-growth assumptions for any exponent $1 < p <+\infty$. Unlike the harmonic and $p$-harmonic map case, the quasiconvexity condition requires one to linearise the map at the level of the gradient. In a bid to give a direct proof of partial $C^{1,α}-regularity for such minimisers, we developing an appropriate notion of a tangential harmonic approximation together with a discussion on the difficulties in establishing Caccioppoli-type inequalities. The need in the latter problem to construct suitable competitors to the minimiser via the so-called Luckhaus Lemma presents difficulties quite separate to that of the unconstrained case. We will give a proof of this lemma together with a discussion on the implications for higher integrability.

In my talk I shall give a small survey on some algorithmic properties of amalgamated products of finite rank 
free groups. In particular, I'm going to concentrate on Membership Problem for this groups. Apart from being algorithmically interesting, amalgams of free groups admit a lot of interpretations. I shall show how to 
characterize this construction from the point of view of geometry and linguistic.