The interaction of flowing fluids with free bodies -- sometimes
compliant, sometimes active, sometimes multiple -- constitutes a class
of beautiful dynamic boundary problems that are central to biology and
engineering. Examples range from how organisms locomote in fluids
(which depends strongly on scale) to how non-Newtonian stresses
develop in complex liquids (strongly dependent on the nature of
fluidic microstructure). I will discuss several interesting examples,
emphasizing how they are formulated mathematically so as to yield
models tractable for analysis or simulation, and show how this work
has interacted with experimental studies.

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.