The theory of polynomial functors allows one to make sense of the stable polynomial representation theory of the general linear group over a field of characteristic 0. It also has a good notion of specialization, so that calculations done in the "infinite limit" can be used to get information in the usual finite-dimensional siutation. (At the combinatorial level this is understood as the theory of symmetric functions vs. symmetric polynomials.) Such a theory is less well-understood in the other classical groups, but the analogous category has been introduced by Olshanskii and Penkov-Strykas. However, the specialization from the infinite case to the finite case (which is needed for applications) was not previously understood. This talk will explain joint work with Andrew Snowden and Jerzy Weyman where such a specialization is constructed and its properties analyzed. Time permitting, I will explain how this is intricately connected with the invariant theory and commutative algebra surrounding determinantal varieties. I will not assume prior knowledge of polynomial functors and representation theory of classical groups.

In this talk, I will report on a joint project with Yichao Tian. Let p be a prime unramified in a totally real field F. The Goren-Oort stratification is defined by the vanishing locus of the partial Hasse-invariants; it is an analog of the stratification of modular curve mod p by the ordinary locus and the supersingular locus. We give explicit global description of the Goren-Oort stratification of the special fiber of the Hilbert modular variety for F.

We give an introduction to the principle ISP and its relatives as well as their connections to supercompact cardinals and the proper forcing axiom.

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.
 

I will present a new PDE approach to obtain large time behavior
of Hamilton-Jacobi equations. This applies to usual Hamilton-Jacobi
equations, as well as the degenerate viscous cases, and weakly coupled
systems. The degenerate viscous case was an open problem in last 15 years.
This is the joint work with Cagnetti, Gomes, and Mitake.