Conference Abstracts for 35th Annual Geometry Festival

35th Annual Geometry Festival

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Friday, April 23
4:30pm Joel Spruck, Johns Hopkins University
A Personal Tribute to Louis Nirenberg

I first met Louis Nirenberg in person in 1972 when I became a Courant Instructor. He was already a celebrated mathematician and a suave sophisticated New Yorker, even though he was born in Hamilton, Canada and grew up in Montreal. In this talk I will describe some of his famous papers, some of our joint work and other work he inspired. For reasons of exposition, I will not follow a strict chronological order and I will concentrate on some of Louis' work inspired by geometric problems beginning around 1974, especially the method of moving planes and implicit fully nonlinear elliptic equations. During the twenty year period 1953-1973 he produced an incredible body of work in many fields of pde including pseudo differential operators and local solvability. Of course I cannot begin to talk about this work. Louis loved to collaborate and I apologize for omitting many other important results of the last twenty ve years, a majority of which were in collaboration with his brilliant and devoted student Yanyan Li.

Saturday, April 24

Akito Futaki, Yau Center, Tsinghua University
Deformation Quantization, and Obstructions to the Existence of Closed Star Products

A star product is a non-commutative product on the set of formal functions, i.e. formal power series with coefficients in smooth functions. Giving a star product is called deformation quantization. The trace of a star product is an algebra character from the non-commutative algebra of formal functions into the abelian algebra of formal constants. The trace is expressed as an $L^2$ product with a function called the trace density. A star product is said to be closed if the trace density is constant, i.e. the trace is given by the integration. In this talk, we discuss on obstructions to the existence of closed star product as in the similar spirit of Kähler geometry.


Jean-Pierre Demailly, Institut Fourier, Université Grenoble Alpes
Holomorphic Morse inequalities, old and new

Holomorphic Morse inequalities provide asymptotic estimates for the dimension of cohomology groups of large tensor powers of holomorphic line bundles on compact complex manifolds. They are analytic in nature, but an interesting purely algebraic formulation has been recently proposed by BenoƮt Cadorel. We will discuss these results as well as several geometric applications, for example to existence theorems for jet differentials, and related unsolved problems.


Tristan Collins, MIT
SYZ Mirror Symmetry for del Pezzo Surfaces

If X is a del Pezzo surface and D is a smooth anti-canonical divisor, we can regard the complement X\D as a non-compact Calabi-Yau surface. I will discuss a proof of a strong form of the Strominger-Yau-Zaslow mirror symmetry conjecture for these non-compact surfaces. It turns out the mirror Calabi-Yau is a rational elliptic surface (in particular, it has an elliptic fibration onto $P^1$) with a singular fiber which is a chain of nodal spheres. I will discuss how we can construct special Lagrangian fibrations on these manifolds, as well as moduli of complex and symplectic structures and how hyper-Kähler rotation allows us to construct an identification of these moduli spaces. This is joint work with A. Jacob and Y.-S. Lin.


Jim Isenberg, University of Oregon
Some Recent Results on Ricci Flow

We discuss the results of two recent collaborative works on Ricci Flow. The first of these results, done with Eric Bahuaud and Chris Guenther, shows that "convergence stability" holds for Ricci Flow solutions converging to the flat metric on the torus as well as for Ricci Flow solutions converging to the hyperbolic metric. Convergence stability tells us that if the Ricci flow starting at a metric h converges to a metric g, then it follows that the Ricci Flow starting at metrics sufficiently close to h (relative to a specified topology) must also converge to g. Convergence stability is a consequence of stability at g combined with long-time continuous dependence for the class of geometries including h and g. Our verification that convergence stability holds for the hyperbolic metric depends on geometric analysis results for asymptotically hyperbolic metrics contained in a specified weighted Holder space. The second of our results, done with Tim Carson, Dan Knopf and Natasa Sessum, involves the study of singularity formation at spatial infinity for Ricci Flow of certain multi-warped complete geometries on non-compact manifolds. We use this analysis to show that there is unexpected behavior of blowup sequences for Ricci Flows developing Type I singularities at spatial infinity. In particular, we find that some blowup sequences form gradient solitons, while others form ancient solutions which are not solitons.


Chiu-Chu Melissa Liu, Columbia University
Topological Recursion and Crepant Transformation Conjecture

The Crepant Transformation Conjecture (CTC), first proposed by Yongbin Ruan and later refined/generalized by others, relates Gromov-Witten (GW) invariants of K-equivalent smooth varieties/orbifolds. The Remodeling Conjecture (proposed by Bouchard-Klemm-Marino-Pasquetti and proved in full generality by Fang, Zong and the speaker) relates open and closed GW invariants of a symplectic toric Calabi-Yau 3-orbifold to invariants of its mirror curve defined by Chekhov-Eynard-Orantin Topological Recursion. We will explain how to use the Remodeling Conjecture to derive all genus open and closed CTC for symplectic toric Calabi-Yau 3-orbifolds. This is based on joint work with Bohan Fang, Song Yu, and Zhengyu Zong.

Sunday, April 25

Bing Wang, University of Science and Technology of China
Local entropy along the Ricci flow

Inspired by the Li-Yau estimates, we localize the entropy functionals of G. Perelman, and generalize his no-local-collapsing theorem and pseudo-locality theorems. The improved no-local-collapsing theorem can be used to study the general Kähler-Ricci flow. The improved pseudo-locality theorem can be used to show the continuous dependence, with respect to the initial metric in the Gromov-Hausdorff topology, of the Ricci flow on manifolds satisfying a lower Ricci-curvature bound; and to prove the compactness for the moduli of Kähler manifolds with bounded scalar curvature that satisfy a rough locally-almost-Euclidean condition.


Simon Donaldson, SCGP, Stony Brook University, and Imperial College, London
Some boundary value and mapping problems for differential forms

Hitchin formulated the equations for $G_{2}$ holonomy in seven dimensions and Calabi-Yau structures in six dimensions in terms of variational problems for closed 3-forms. We will discuss versions of these ideas for manifolds with boundary. In the second case this leads to a mapping problem for maps from a five dimensional manifold to ${\bf C}^{3}$ which is related to CR geometry, contact geometry and four-dimensional Riemannian geometry and has a dimension reduction to the classical Minkowski problem for convex surfaces in 3-space.

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