The Keller-Segel equations model chemotaxis of bio-organisms. In a reduced form, considered in this talk, they are related to Vlasov equation for self-gravitating systems and are used in social sciences in description of crime patterns.  

It is relatively easy to show that in the critical dimension 2 and for mass of the initial condition greater than 8 \pi, the solutions 'blowup' (or 'collapse') in finite time. This blowup is supposed to describe the chemotactic aggregation of the organisms and understanding its mechanism, especially its universal features, would allow to compare theoretical results with experimental observations. Understanding this mechanism turned out to be a very subtle problem defying solution for a long time. 

In this talk I discuss recent results on dynamics of solutions of the (reduced) Keller-Segel equations in the critical dimension 2 which include a formal derivation and partial rigorous results on the blowup dynamics of solutions. The talk is based on the joint work with S. I. Dejak, D. Egli and  P.M. Lushnikov.

There are many examples in mathematics, both pure and applied, in which
problems with symmetric formulations have non-symmetric solutions.
Sometimes this symmetry breaking is total, as in the example of
turbulence, but often the symmetry breaking is only partial. One technique
that can sometimes be used to constrain the symmetry breaking is
reflection positivity. It is a simple and useful concept that will be
explained in the talk, together with some examples. One of these concerns
the minimum eigenvalues of the Laplace operator on a distorted hexagonal
lattice. Another example that we will discuss is a functional inequality
due to Onofri.
The talk is based on joint work with E. Lieb.

I will discuss the properties of discrete random Schrödinger operators in which the random part of the potential is supported on a sublattice. For the standard Anderson model, no results concerning localization/delocalization transition are rigorously established. For trimmed Anderson model described above, one can trace out the onset of the localization breakup, in the strong disorder regime (for some examples). This is a joint work with Sasha Sodin.

The classical Lieb-Robinson bounds provide control over the speed of
propagation in quantum spin systems. In analogy to relativistic systems,
they establish a ``light cone'' $x \leq vt$ outside of which commutators
of initially localized observables are exponentially small. We consider an
XY spin chain in a quasiperiodic magnetic field and prove a new anomalous
Lieb-Robinson bound which features the modified light cone $x \leq
vt^\alpha$ for some $0<\alpha<1$. In fact, we can characterize $\alpha$
exactly as the upper transport exponent of a one-body Schr\"odinger
operator. This may be interpreted as a rigorous proof of anomalous quantum
many-body transport. Joint work with David Damanik, Milivoje Lukic and
William Yessen.