The subject of invisibility has fascinated people for thousands
of years. There has recently been considerable theoretical and practical
progress in understanding how to cloak objects. We will discuss some of
the recent work on the subject of invisibility which involves using
singular electromagnetic parameters, or singular Riemannian metrics.

Spaces of sections of tensor powers of the theta line bundle on moduli spaces of semistable arbitrary rank bundles on a compact Riemann surface are subject to a level-rank duality: each space of sections is geometrically isomorphic to the dual of the space of sections obtained by interchanging the tensor power (level) of the theta bundle on the moduli space and the rank of the bundles that make up the moduli space.
This corresponds in representation theory to an isomorphism of conformal blocks of representations of affine Lie algebras, when the rank of the algebra and the level of the representation are interchanged.
Dr. Marian will sketch a proof of the geometric statement, which is the result of joint work with Dragos Oprea, and draws inspiration from work by Prakash Belkale who established the isomorphism for a generic Riemann surface.

Dr. Krieger will discuss a recent result, joint with W. Schlag and D. Tataru, which establishes reguarity breakdown for wave maps with suitable initial data and target S^{2} in the energy critical dimension. The breakdown occurs via the bubbling off of a ground state harmonic map.

This is joint work with Xin Zhou.

The speaker will consider, in particular, different aspects of the
solution of the Cauchy problem for the perturbed defocusing NLS equation,
(1) iq_t + q_xx - 2(|q|^2)q -(\epsilon)W(|q|^2)q = 0
q(x,0)=q_0(x)--> 0 as |x|--> \infty.

Here (\epsilon)>0, W(s) is non-negative and W(s) behaves like s^k as s --> 0 for some (sufficiently large) exponent l.

For fixed k>7/4, and \epsilon sufficiently small, the authors
(i) describe the long-time behavior of solutions of (1)
(ii) show that on an invariant, open, connected set in phase space, equation (1) is completely integrable in the sense of Liouville
(iii)show that the solution of (1) is universal in the following sense: one uses W to set the macroscopic scales for the solution, but once the scale is set, the solution of (1) looks the same independent of W.

The main technical tool in proving (i)(ii)(iii) is to use the Zakaharov-Shabat scattering map for NLS to transform the problem to normal form in the manner of Kaup and Newell, and then to analyze the normal form using Riemann-Hilbert/steepest-descent-type methods.