Given a Lagrangian L, I will discuss the existence of a
neighborhood W of L with the following property: for any
Hamiltonian diffeomorphism f, if f(L) is contained inside W, then
f(L) intersects L. On the one hand, for any symplectic manifold
of...
Central predictions of arithmetic quantum chaos such as the
Quantum Unique Ergodicity conjecture and the sup-norm problem ask
about the mass distribution of automorphic forms, most classically
in terms of their weight or Laplace eigenvalue (for...
I will show that any Schubert or Richardson variety R in a flag
manifold G/P is equivariantly rigid and convex. Equivariantly rigid
means that R is uniquely determined by its equivariant cohomology
class, and convex means that R contains any torus...
We use the min-max construction to find closed hypersurfaces
which are stationary with respect to anisotropic elliptic
integrands in any closed n-dimensional manifold . These surfaces
are regular outside a closed set of zero n-3 dimension. The...
A result of Jan Nekovář says that the Galois action on p-adic
intersection cohomology of Hilbert modular varieties with
coefficients in automorphic local systems is semisimple. We will
explain a new proof of this result for the non-CM part of
the...
It is well-known that the geodesic flow on ellipsoids of
revolution is integrable. In joint work with Ferreira and Vicente,
we used this fact to obtain a symplectomorphism between the unit
disk bundle of such an ellipsoid without fiber and a toric...
In part 1, I will survey the history of total positivity,
beginning in the 1930's with the introduction of totally positive
matrices, which turn out to have surprising linear-algebraic and
combinatorial properties. I will discuss some modern...
Consider a closed surface M of genus greater than or equal to 2.
For negatively curved metrics on M and their corresponding geodesic
flow, we can study the topological entropy, the Liouville entropy,
and the mean root curvature. In 2004, Manning...
The notion of singular support has its origin in the theory of
partial differential equations, and was introduced to the world of
constructible sheaves by M. Kashiwara and P. Schapira in the 1970s.
It is a basic invariant (like the support), and...
In 1962, Yudovich established the well-posedness of the
two-dimensional incompressible Euler equations for solutions with
bounded vorticity. However, uniqueness within the broader class of
solutions with L^p vorticity remains a key unresolved...
