2008 Clay Research Conference, May 12-13, Cambridge, Massachusetts
On May 12-13, at MIT, the Clay Mathematics Institute held its 2008 Clay Research Conference. The program consisted of a two-day series of lectures on recent research developments, together with presentation of the Clay Research Awards. The conference was hosted by the MIT Mathematics Department.
The conference was held at:
MIT Media Lab - Bartos Theatre
Wiesner Building, E15
Titles and abstracts
I'll discuss an approach to renormalization of quantum field theories based on Wilson's effective action picture and the Batalin-Vilkovisky formalism. The general theory will be applied to give a conceptual proof of renormalizability of pure Yang-Mills theory on R4.
Question: Let ƒ(z1,…,zn) be a holomorphic function on an open set U ⊂ Cn. For which s ∈ R is |ƒ|-s locally integrable?
It is not hard to see that there is a largest value s0 (depending on ƒ and p) such that |ƒ|-s is integrable in a neighborhood of p for s < s0 but not integrable for s > s0. Our aim is to study this "critical value" s0. Subtle properties of these critical values are connected with Mori's program (especially the termination of flips), with the existence of Kähler-Einstein metrics in the positive curvature case and many other topics.
This talk will survey Cliff Taubes' recent proof of the Weinstein conjecture in dimension three and related topics. Taubes shows how to construct periodic orbits of Reeb vector fields on contact three manifolds from special cycles in the Seiberg-Witten Monopole Floer homology. The proof follows ideas from Taubes' work relating the Seiberg-Witten and Gromov invariants of four-manifolds but with a new twist. It hinges on new results describing the asymptotic behavior of spectral flow for Dirac type operators.
Many problems of an asymptotic and quantitative nature in geometry have recently been solved using a variety of probabilistic tools. Apart from the classical use of the probabilistic method to prove existence results, it turns out thinking "that probabilistic", or interpreting certain geometric invariants in a probabilistic way, is a powerful way to bound a variety of geometric quantities. This talk is devoted to surveying the ways in which probabilistic reasoning plays a sometimes unexpected role in topics such as bi-Lipschitz and uniform embedding theory, extension problems for Lipschitz maps, metric Ramsey problems, harmonic analysis, and theoretical computer science. We will show how random partitions of metric spaces can be used to embed them in normed spaces, find large Euclidean subsets, extend Lipschitz functions, and bound the weak (1,1) norm of maximal functions. We will also discuss the role of random projections, and describe the connections between the behavior of Markov chains in metric spaces and Lipschitz extension, lower bounds for bi-Lipschitz embeddings, and the computation of compression exponents for discrete groups.
Let X be a projective 3-fold. We construct a moduli space of stable pairs in the derived category of X with a well-defined enumerative geometry. The enumerative invariants are conjectured to be equivalent to the Gromov-Witten theory of X. The geometry is a very natural place to study recent derived category wall-crossing formulae. Fibrations of K3 surfaces provide computable examples. Connections to work of Kawai-Yoshioka and the Yau-Zaslow formula for enumerating rational curves on K3 surfaces are made. Joint work with R. Thomas.
Many "quantum gravity" models in mathematical physics can be interpreted as probability measures on the space of metrics on a Riemannian manifold. We describe several recently derived connections between these random metrics and certain random fractal curves called "Schramm-Loewner evolutions" (SLE).
We show that there exist, starting from complex dimension 4, compact Kaehler manifolds not having the cohomology algebra of a projective complex manifold. In particular their complex structure does not deform to that of a projective manifold, while Kodaira proved that compact Kaehler surfaces deform to projective ones. The argument uses the notion of Hodge structure on a cohomology algebra, and exhibits algebraic obstructions for the existence of such Hodge structure admitting a rational polarization. We will also explain further applications of this notion, for examples the fact that cohomology algebras of compact Kaehler manifolds are strongly restricted amongst cohomology algebras of compact symplectic manifolds satisfying the hard Lefschetz property.