One of the fundamental problems in the classification of complex surfaces is to find a new family of simply connected surfaces with p_g = 0 and K^2 > 0. In this
talk, I will sketch how to construct a new family of simply connected symplectic 4- manifolds using a rational blow-down surgery and how to show that such 4-manifolds
admit a complex structure using a Q-Gorenstein smoothing theory. In particular, I will show explicitly how to construct a simply connected minimal surface of general
type with p_g = 0 and K^2 = 3.
If time allows, I will also sketch how to construct a simply
connected, minimal, symplectic 4-manifold with b_+2 = 1 (equivalently, p_g = 0) and K^2 = 4 using a rational blow-down surgery.
Consider a family of smooth compact connected $n$ dimensional Riemannian manifolds. What can one say about the spectral geometry of a limit of these?
This question has interested many spectral geometers; my talk focuses on conical metric degeneration in which the family converges "asymptotically conically'' to an open manifold with conical singularity. I will present spectral convergence results and discuss techniques including microlocal analysis on manifolds with corners and geometric blowup constructions. I will also summarize spectral convergence results for other geometric contexts and discuss applications and open questions.
Renyi Mathematics Institute, Budapest and Columbia
Time:
Tuesday, November 6, 2007 - 4:00pm
Location:
MSTB 254
After discussing the basics of contact surgery in dimension 3, and introducing contact Ozsvath-Szabo invarinats, we show that a Seifert fibered 3-manifold does admit a positive tight contact structure unless it is orientation preserving diffeomorphic to the result of (2n-1)-surgery along the T(2,2n+1) torus knot (for some positive integer n).
We proved that the integration of the Chern-Weil forms on CY moduli are always rational numbers. This result follows from a more general one: the integration of the Chern-Weil forms of the Hodge bundles on any coarse moduli spaces are rationa numbers. When the dimension of the moduli space is one, this was a result of Zucker and Peters. For the fisrt Chern class, this was proved by Kollar.
We will also discuss the applications in string theory. This is joint with M. Douglas.
In 1954 Hirzebruch asked which linear combinations of Chern numbers are topological invariants of smooth complex projective varieties. We give a complete answer to this question in small dimensions, and also prove partial results without restrictions on the dimension.
Many Hodge integral identities, including the ELSV formula of Hurwitz numbers and the lambda_g conjecture, are various limits of the formula of one-partition Hodge integrals conjectured by Marino and Vafa. Local Gromov-Witten invariants in all degrees and all genera of any toric surfaces in a Calabi-Yau threefold are determined by the formula of two-partition Hodge integrals. I will describe proofs of the formulae of one-partition and two-partition Hodge integrals based on joint works with Kefeng Liu and Jian Zhou.
I shall report some recent progress on the quasi-local mass associated to any space-like surface in space-time. The relation to isometric embedding problems shall also be discussed.
In this talk, I will report a recent joint work with Mu-Tao Wang. We construct examples of shrinkers and expanders for Lagrangian mean curvature flows. These examples are Hamiltonian stationary and asymptotic to the union of two Hamiltonian stationary cones found by Schoen and Wolfson. The Schoen-Wolfson cones are obstructions to the existence problems of special Lagragians or Lagrangian minimal surfaces in the variational approach. It is known that these cone singularities cannot be resolved by any smooth Lagrangian submanifolds. The shrinkers and expanders that we found can be glued together to yield solutions of the Brakke motion-a weak formulation of the mean curvature flow, and thus provide a canonical way to resolve the union of two such cone singularities. Our theorem is analogus to the Feldman-Ilmanen-Knopf gluing construction for the K\"ahler-Ricci flows.