When you arrive on Friday, please go to the fourth floor, room 4E17 for refreshments and to meet the rest of us. Friday talks will be in DRL A4 on the ground floor. Saturday and Sunday talks will be in DRL A1, also on the ground floor.

The address for the math building, DRL is 209 South 33rd St, Philadelphia. 

FRIDAY APRIL 19

12 noon   Arrivals in DRL 4E17 with snacks and drinks

3 - 4 Talk #1 in DRL A4 Xiuxiong Chen

      "Mathematics Inspired by E. Calabi: A narrative from his former student"

4 - 4:30 Refreshments in DRL 4E17

4:30 - 5:30 Talk #2 in DRL A4 Robert Bryant

        "The affine Bonnet problem and integrability"

SATURDAY APRIL 20

8:30 AM Breakfast in the lobby outside DRL A1

9:30 - 10:30 Talk #3 in DRL A1 Shing-Tung Yau via Zoom

        "Complete Calabi-Yau metrics"

10:30 - 11 Remainder of breakfast in the lobby

11 - 12 Talk #4 in DRL A1 Blaine Lawson 

      "Generalized Potential Theories and Nonlinear PDEs"

12 - 2 PM LUNCH in nearby restaurants

2 - 3  Talk #5 in DRL A1 Simon Donaldson via Zoom

      "Deformation theory for complex Calabi-Yau threefolds with boundary"

3 - 3:30 Refreshments in the lobby

3:30 - 4:30 Talk #6 in DRL A1 Yang Li 

      "Metric SYZ conjecture"

4:30 - 5 Refreshments in the lobby

5 - 6 Talk # 7 in DRL A1 Claude LeBrun 

      "Einstein Manifolds, Extremal Kähler Metrics, and Gravitational Instantons"

6 - 7:30 Reception in the physics grad lounge 3E9:  Wine, hors d'oeuvres, brief speeches

8  Dinner at Sang Kee Restaurant, 36th and Chestnut Streets

               For speakers and those who have registered and paid for the banquet. 

Banquet registration is now closed. If you would like to join, talk to one of the organizers to be added to the waitlist in case someone cancels.

SUNDAY APRIL 21

8:30 AM Breakfast in the lobby outside DRL A1

9:30 - 10:30 Talk #8 in DRL A1 Natasa Sesum via Zoom

      "Generalized cylinder limits of Ricci flow singularities"

10:30 - 11 Remainder of breakfast in the lobby

11 - 12 Talk #9 in DRL A1 Bing Wang 

      "On Kähler Ricci shrinkers"

CONFERENCE ENDS AT 12 NOON

TITLES AND ABSTRACTS

Robert Bryant

The affine Bonnet problem and integrability

The Bonnet problem in Euclidean surface theory is well-known:  Given a metric g on an oriented surface M^2 and a function H, when (and in how many ways) can (M,g) be isometrically immersed in R^3 with mean curvature H?  For generic data (g,H), such an isometric immersion is impossible and, in the case that it does exist, the immersion is unique.  Bonnet showed that, aside from the famous case of surfaces of constant mean curvature, there is a finite dimensional moduli space of (g,H) for which the space of such Bonnet immersions has positive dimension.

The corresponding problem in affine theory (a favorite topic of Eugenio Calabi) is still not completely solved.  After reviewing the Euclidean results on this problem by O.Bonnet, J.Radon, E.Cartan, and A.Bobenko, I will give a report on affine analogs of these results.  In particular, I will consider the classification of the data (g,H) for which the space of \emph{affine Bonnet immersions} has positive dimension, showing a surprising connection with integrable systems in the case of data with the highest possible dimension of solutions.

Xiuxiong Chen 

Mathematics Inspired by E. Calabi: A narrative from his former student

Calabi's influence on theoretical physics and mathematics has stretched far and wide. A large portion of his publication remains a major part of active research literature. His impact on geometric analysis alone would be already too great to recount here. So I will focus on Kähler geometry, an area in which I have privilege to be involved and where my own research has been greatly shaped by Calabi, starting with my thesis, up to the latest work with my young collaborators on extremal Kaehler metrics. As a former student, my reflections will inevitably be infused with strong personal flavor.

Simon Donaldson 

Deformation theory for complex Calabi-Yau threefolds with boundary 

The talk is based on joint work with Fabian Lehmann. A complex Calabi-Yau threefold is a 3-dimensional complex manifold with a nowhere-vanishing holomorphic 3-form. In the case of closed manifolds, the deformation theory is well-understood: the local deformations are parametrised by the de Rham cohomology class of the holomorphic form. Our main result is an extension of this to manifolds with boundary. We define a finite-dimensional space K and show that if K=0 the local deformations are parametrised by cohomological data and the restriction of the real part of the 3-form to the boundary.  The proof uses dbar-Neumann and Nash-Moser theory , combined with some ideas specific to the situation.  In the last part of the talk we will discuss examples and some invariants, local and global, of the boundary data.

Blaine Lawson

Generalized Potential Theories and Nonlinear PDE’s

Many years Reese Harvey and I wondered whether the spaces which hold large calibrated geometries, like K ̈ahler manifolds, also supported analogues of holomorphic or pluriharmonic functions. The answer is essentially no –however analogues of the plurisubharmonic functions always exist. They give rise to a useful generalized potential theory on these spaces, with much

of the structure of classical potential and pluripotential theories. Gradually we realized that this point of view could be greatly generalized. It led to new results in nonlinear PDE’s. In particular, it gave some answers to the following questions:

(1) How does one deal with a partial differential equation when there is no natural operator?

(2) Given a differential operator, are there other operators with the same solutions but different useful properties?

(3) In fact, can one radically change the operator to something tractable, in a way that enables solving the original equation? 

Yang Li

Metric SYZ conjecture

The Strominger-Yau-Zaslow conjecture asks for the existence of special Lagrangian torus fibrations on Calabi-Yau manifolds near the large complex structure limit. I will discuss some recent progress on this, especially the case of the Fermat family of hypersurfaces.

Claude LeBrun

Einstein Manifolds, Extremal Kähler Metrics, and Gravitational Instantons

This lecture will further explore a constellation of topics pioneered by Eugenio Calabi, albeit interacting in ways that Gene might never have expected. We will focus on an aspect of Einstein manifolds that is unique to real dimension four. Indeed, any 4-dimensional Einstein metric that is Hermitian (with respect to some integrable complex structure) must actually be conformal to an extremal K ̈ahler metric.Conversely, the world of conformally Kähler, Einstein metrics is rich and fertile in real dimension 4, although attempts to imitate these ideas in higher dimensions just lead to an empty desert. I will begin by describing the classification of such metrics on compact complex surfaces, and then outline more recent results concerning an analogous class of complete, non-compact, Ricci-flat 4-manifolds. Finally, I will describe rigidity results that assert that another Einstein metric that is sufficiently close to one of these special Einstein metrics must actually be a metric of the same special type.

Natasa Sesum

Generalized cylinder limits of Ricci flow singularities

We study multiply warped product geometries and show that for an open set of initial data within multiply warped product geometries the Ricci flow starting at any of those develops generalized cylinder as singularity model. More precisely, for any p and q we construct an open set of initial data within multiply warped product geometris whose Ricci flows develop S^p\times R^q as a singularity model. 

Bing Wang

On Kahler Ricci shrinkers

Gradient shrinking Ricci solitons are also called Ricci shrinkers. They are fundamental models for Ricci flow singularity analysis. 

We obtain a scalar curvature bound on each Kähler Ricci shrinker surface. Combining this estimate with earlier work by many authors, including the volume estimate of Cao-Zhou, the curvature estimate of Munteanu-Wang, the construction and classification of Bamler, Conlon-Cifarelli-Deruelle and Sun,  we provide a complete classification of all Kähler Ricci shrinker surfaces. This is joint work with Yu Li.

Shing-Tung Yau

Complete Calabi-Yau metrics

Abstract: After proving Calabi's conjecture in 1976, I started programs on constructing of Ricci-flat Kahler metrics with singularities and Ricci-flat metrics on noncompact manifolds. In this talk, I discuss several methods on these topics, particularly, including free boundary Monge-Ampere equation joint with Collins and Tong.