we present a technique for proving convergence of h and hp
adaptive finite element methods through comparison with certain
reference refinement schemes based on interpolation error. we then
construct a testing environment where properties of different
adaptive approaches can be evaluated and improved.

i will discuss recent results concerning the kuramoto-sivashinky equation in two space dimensions with periodic boundary conditions. in particular, i will present a global existence result in the wiener algebra, when growing modes are absent, and bounds on the analyticity radius when the data is only $l^2$. this is joint work with david ambrose (drexel university).

we will discuss rigidity, compactness, and finite quantization of entropy for the space of self-shrinking lagrangian surfaces. this is based on joint work with john ma

let $p$ be a uniform random permutation matrix of size $n$ and let $\chi_n(z)= \det(zi - p)$ denote its characteristic polynomial. we prove a law of large numbers for the maximum modulus of $\chi_n$ on the unit circle, specifically,
\[
\sup_{|z|=1}|\chi_n(z)|= n^{x_c + o(1)}
\]
with probability tending to one as $n\to \infty$, for a numerical constant $x_c\approx 0.677$. the main idea of the proof is to uncover an approximate branching structure in the distribution of (the logarithm of) $\chi_n$, viewed as a random field on the circle, and to adapt a well-known second moment argument for the maximum of the branching random walk. unlike the well-studied \emph{cue field} in which $p_n$ is replaced with a haar unitary, the distribution of $\chi_n(z)$ is sensitive to diophantine properties of the argument of $z$. to deal with this we borrow tools from the hardy--littlewood circle method in analytic number theory. based on joint work with ofer zeitouni.

how do the choices made by individual pedestrians influence the large-scale crowd dynamics?
what are the factors that slow them down and motivate them to seek detours?
what happens when multiple crowds pursuing different targets interact with each other?
we will consider how answers to these questions shape a class of popular pde-based models, in which a conservation law models the evolution of pedestrian density while a hamilton-jacobi-bellman pde is used to determine the directions of pedestrian flux. this presentation will emphasize the role of anisotropy in pedestrian interactions, the geometric intuition behind our choice of optimal directions, and connections to the non-zero-sum game theory. (joint work with elliot cartee.)

let g be a simple and simply connected algebraic group over a field. we can attach to a it the sheaf of conformal blocks: a vector bundle on $m_g$ whose fibres are identified with global sections of a certain line bundle on the stack of g-torsors. we generalize the construction of conformal blocks to the case in which g is replaced by a ``twisted group'' defined over curves in terms of covering data. in this case the associated conformal blocks define a sheaf on a hurwitz stack and have properties analogous to the classical case.