Tag - Differential geometry

Steve Zelditch: Local and global analysis of nodal sets

30th April, 2017

Nodal sets are zero sets of eigenfunctions of the Laplacian on a Riemannian manifold. Local analysis studies nodal sets in small balls, ignoring the global geometry. Global analysis exploits the dynamics of the geodesic flow to obtain information on nodal sets. First, I will describe the recent proof by Alexander Logunov of Yau's lower bound conjecture for hypersurface volumes of nodal sets. It is a local proof based mainly on the combinatorics of the Donnelly-Fefferman doubling exponent bounds. Second, I will describe recent results on numbers of nodal domains on surfaces of non-positive curvature. These results are based on the ergodicity of the geodesic flow.

Bong Lian: The Riemann-Hilbert problem for period integrals: recent applications

30th April, 2017

I will discuss two recent applications of the Riemann-Hilbert problem for periods of CY manifolds. One
of them deals with the hyperplane conjecture for toric hypersurfaces that goes back to the mid 1990s.
The second application provide descriptions for zeros of derivatives of generalized hypergeometric
functions.

Jean-Pierre Demailly: L2 Extension theorem for sections defined on non-reduced analytic subvarieties

30th April, 2017

The goal of the talk will be to discuss L2 extension properties for holomorphic sections of vector bundles satisfying weak semi-positivity properties. Using techniques borrowed from recent proofs of the Ohsawa-Takegoshi extension theorem, we obtain several surjectivity results for the restriction morphism to a non necessarily reduced subvariety, provided the latter is defined as the zero variety of a multiplier ideal sheaf. These extension results are derived from L2 approximation techniques, and they hold under rather general geometric assumptions.

Mu-Tao Wang: Linear stability of Schwarzschild black hole: the Cauchy problem of metric coefficients

30th April, 2017

The Schwarzschild solution of the vacuum Einstein equation in general relativity is the unique static solution that represents an isolated gravitating system of a single black hole. Studies, both theoretically and experimentally, of such a system are modeled on the Schwarzschild solution and its perturbation. The stability of the Schwarzschild solution is thus of utmost importance. I will address the linear stability of the Schwarzschild solution, which has a long history and rich literature involving the works of both physicists and mathematicians, and culminating in the recent breakthrough of Dafermos-Holzegel-Rodnianski. In joint work with Pei-Ken Hung and Jordan Keller, we provide a different and simpler proof that reveals the underlying geometric structure of the vacuum Einstein equation at a more elementary level.

Alena Pirutka: Irrationality problems

30th April, 2017

Let X be a projective algebraic variety, the set of solutions of a system of homogeneous polynomial equations. Several classical notions describe how "unconstrained" the solutions are, i.e., how close X is to projective space: there are notions of rational, unirational and stably rational varieties. Over the field of complex numbers, these notions coincide in dimensions 1 and 2, but diverge in higher dimensions. In this talk I will discuss classical and recent advances in this area, examples and deformation properties.

Yujiro Kawamata: Birational geometry and derived categories

30th April, 2017

I will talk about the recent progress on the DK conjecture connecting birational geometry and the
derived categories, and related conjectures such as DL conjecture, etc. I will also discuss two kinds of
factorizations of birational maps; those into flips, flops and divisorial contractions according to the
minimal model programme, and more traditional factorizations into blow-ups and blow-downs with
smooth centres.

Si Li: Vertex algebras, quantum master equation and mirror symmetry

30th April, 2017

We develop the effective Batalin-Vilkovisky quantization theory for chiral deformation of 2-dimensional conformal field theories. We establish an exact correspondence between renormalized quantum master equations for effective functionals and Maurer-Cartan equations for chiral vertex operators. As an application, we explain a universal approach to KdV type integrable hierarchies via B-twisted topological string field theory. This leads to an exact solution of quantum B-model (BCOV theory) in complex one dimension that solves the higher genus mirror symmetry conjecture on elliptic curves.

Jun Li: New recursion relation for GW of quintic CY via Mixed-Spin-P fields

30th April, 2017

We introduce the notion of Mixed-Spin-P fields as an algebro-geometric model interpolating all genus GW and FJRW invariants of the quintic CY threefolds, realizing the vision of Witten’s transformation relating these two theories. This theory of mixed-P-fields produces recursive relations among GW and FJRW invariants. These relations hopefully will provide the mean to solve all genus GW invariants of quintic CY threefolds.

Ciprian Manolescu: Homology cobordism and triangulations

29th April, 2017

In the 1970s, Galewski-Stern and Matumoto studied the existence and the classification of triangulations on topological manifolds of dimension at least 5. They reduced these problems to questions about the three-dimensional homology cobordism group, ΘH3, and the Rokhlin homomorphism from this group to ℤ/2. The structure of the homology cobordism group is still unknown, but some information can be obtained using tools from gauge theory and symplectic geometry, such as the Seiberg-Witten Floer spectrum and involutive Heegaard Floer homology. I will describe the proof of the existence of non-triangulable high-dimensional manifolds (using gauge theory), and some open problems.

Fernando Coda Marques: The space of cycles, a Weyl’s law for minimal hypersurfaces and Morse index estimates

29th April, 2017

The space of cycles in a compact Riemannian manifold has very rich topological structure. The space of hypercycles, for instance, taken with coefficients modulo 2, is weakly homotopically equivalent to the infinite dimensional real projective space. This reveals the existence of non-trivial k-parameter sweepouts for every k. We will discuss a proof of a Weyl's law conjectured by Gromov (joint work with Liokumovich and Neves) in which the eigenvalues of the Laplacian are replaced by the areas of minimal hypersurfaces constructed by minimax methods. We will also discuss current work with Neves about Morse index bounds in the min-max theory of minimal surfaces and the problem of multiplicity.