Given a polynomial map between two vector spaces over a field,
how many values can it miss? The lecture will present a number of
new results on this question. They were inspired by the work of
Daqing Wan, and obtained jointly with Michiel Kosters (Leiden).

We discuss self-organized dynamics of agent-based models
with focus on a prototype model driven by non-symmetric self-alignment
introduced in [1].
Unconditional consensus and flocking emerge when the self-alignment is
driven by global interactions with a sufficiently slow decay rate.  In
more realistic models, however, the interaction of self-alignment is
compactly supported, and open questions arise regarding the emergence
of clusters/flocks/consensus, which are related to the propagation of
connectivity of the underlying graph.
In particular, we discuss heterophilious self-alignment: here, the
pairwise interaction between agents increases with the diversity of
their positions and we assert that this diversity enhances
flocking/consensus. The methodology carries over from agent-based to
kinetic and hydrodynamic descriptions.

[1] A new model for self-organized dynamics and its flocking behavior, J.
Stat. Physics 144(5) (2011) 923-947.
 

Given some class of "geometric spaces", we can make a ring as follows.

1. (additive structure)  When U is an open subset of such a space X, [X] = [U] + [(X \ U)];
2. (multiplicative structure)  [X x Y] = [X] [Y].

In the algebraic setting, this ring (the "Grothendieck ring of varieties") contains surprising structure, connecting geometry to arithmetic and topology.  I will discuss some remarkable statements about this ring (both known and conjectural), and present new statements (again, both known and conjectural).  A motivating example will be the case of "points on a line" --- polynomials in one variable.  (This talk is intended for a broad audience.)  This is joint work with Melanie Matchett
Wood.

There is a well known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible topological spherical buildings of rank at least three.

We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type M associated with any polar action of cohomogeneity at least two on any simply connected closed positively curved manifold. Although this chamber system is typically not a Tits geometry of type M, we prove that in all cases but one that its universal Tits cover indeed is a building. We construct a topology on this universal cover making it into a topological building in the sense of Burns and Spatzier. Using this structure we classify all polar actions on (simply connected) positively curved manifolds of cohomegeneity at least two.
(Joint work with K.Grove and G. Thorbergsson)

 
Thermoacoustic (TAT) and Photoacoustic Tomography (PAT) are examples of multiwave imaging methods allowing to combine the high imaging contrast of one wave (an electromagnetic or a photoacoustic one) with the high resolution of ultrasound. We present recent results obtained in collaboration with Gunther Uhlmann, Jianliang Qian and Hongkai Zhao on the mathematical theory behind TAT, PAT and other multiwave methods. We allows the acoustic speed to be variable, and consider the partial data case as well. We will also discuss the case of a discontinuous speed modeling brain imaging.  Numerical reconstructions will be shown as well.
Most of the progress is due to the use of microlocal methods. One of the goals of the talk is to show the usefulness of microlocal methods to solving real life problems.