L2

The spectral presheaf of a nonabelian von Neumann algebra or C*-algebra was introduced as a generalised phase space for a quantum system in the so-called topos approach to quantum theory. Here, it will be shown that the spectral presheaf has many features of a spectrum of a noncommutative operator algebra (and that it can be defined for other classes of algebras as well). The main idea is that the spectrum of a nonabelian algebra may not be a set, but a presheaf or sheaf over the base category of abelian subalgebras. In general, the spectral presheaf has no points, i.e., no global sections. I will show that there is a contravariant functor from unital C*-algebras to their spectral presheaves, and that a C*-algebra is determined up to Jordan *-isomorphisms by its spectral presheaf in many cases. Moreover, time evolution of a quantum system can be described in terms of flows on the spectral presheaf, and commutators show up in a natural way. I will indicate how combining the Jordan and Lie algebra structures may lead to a full reconstruction of nonabelian C*- or von Neumann algebra from its spectral presheaf.

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.

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.

I will explain some recent joint work with Georgios Dimitroglou Rizell in which we use moduli spaces of holomorphic discs with boundary on a monotone Lagrangian torus in ${\mathbb C}^n$ to prove that all such tori are smoothly isotopic when $n$ is odd and at least 5

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.  

The plan: start with an introduction to several random matrix ensembles and discuss asymptotic properties of the eigenvalues of the matrices, the last one being the so-called "Normal Matrix Model", and the connection described in the title will be explained. If all goes well I will end with an explanation of asymptotic computations for a new normal matrix model example, which demonstrates a form of universality.

(NOTE CHANGE OF VENUE TO L2)

"We introduce some type of generalized Poisson formula which is equivalent 
to Langlands' automorphic transfer from an arbitrary reductive group over a 
global field to a general linear group."

The quantification and management of risk in financial markets
is at the center of modern financial mathematics. But until recently, risk
assessment models did not consider the effects of inter-connectedness of
financial agents and the way risk diversification impacts the stability of
markets. I will give an introduction to these problems and discuss the
implications of some mathematical models for dealing with them. 

I will discuss similarities and differences between the geometry of
relatively hyperbolic groups and that of mapping class groups.
I will then discuss results about random walks on such groups that can
be proven using their common geometric features, namely the facts that
generic elements of (non-trivial) relatively hyperbolic groups are
hyperbolic, generic elements in mapping class groups are pseudo-Anosovs
and random paths of length $n$ stay $O(\log(n))$-close to geodesics in
(non-trivial) relatively hyperbolic groups and
$O(\sqrt{n}\log(n))$-close to geodesics in mapping class groups.