What does it mean for a collection to be finite? On the one hand, we have our preschool notion that a collection is finite when can be counted with natural numbers in a way that terminates. On the other hand, there is a definition due to Dedekind that a set is finite if and only if it cannot be put in one-to-one correspondence with a proper subset. Intuitively these two notions should be equivalent, but can we prove it? I will argue that to avoid a circular argument, one direction requires more care than one would initially think. Further, the other direction is true only by virtue of the Axiom of Choice. To outline the proof of this fact, we will examine formal notions of definiabilty and the set-theoretic technique of forcing.

In 1963 V. I. Yudovich proved the existence and uniqueness of weak solutions of the incompressible 2D Euler equations assuming that the vorticity, which is the curl of velocity, is bounded and integrable in the full plane. A few extensions of this result have been established, most notably by Yudovich himself and, also, by M. Vishik, always assuming some decay of vorticity at infinity. Paradoxically, if the vorticity is doubly-periodic then there is no difficulty in establishing well-posedness of weak solutions, as long as the vorticity is also bounded. In this talk I will report on work in progress aimed at extending, for 2D flows in a domain exterior to an island, well-posedness of weak solutions to include all vorticities which are bounded and are the curl of a bounded velocity field. This work is related to recent results by Taniuchi, Tashiro and Yoneda and it builds on previous, albeit incomplete, work due to Ph. Serfati, where flow in the full plane was considered.

This is joint work with J. P. Kelliher (UCR) and M. C. Lopes Filho (UNICAMP).

In this lecture, we will talk about a recent joint
work ofGordon Heier and myself about curvature characterizations
ofuniruledness and rational connectivity of projective manifolds. A
result on projective manifolds with zero total scalar curvature will
also be discusse.

Homomorphic encryption allows you to delegate the processing of your data or your query to a "worker" (e.g., the "cloud") without sacrificing your privacy. The worker can process your private data or private query even though it is encrypted, and send back to you the (encrypted) response that you were seeking, without learning anything significant.

This talk will partly be a tutorial designed to give a taste of how homomorphic encryption schemes work, and why they have been so inefficient. Next, the talk will sketch some very recent results that dramatically improve efficiency and give us hope that homomorphic computation may one day be truly practical.