Columbia University in the City of New York


Deformations of Geometric Structures in Current Mathematics:

“A celebration of the works of Masatake Kuranishi



The Deformations of Geometric Structures in Current Mathematics conference will take place on Tuesday, May 3 through Friday, May 6, 2022. Columbia University, Math department will be hosting this event in memory of Professor Masatake Kuranishi.

Invited speakers: Simon Brendle (Columbia University) · Robert Bryant (Duke University) · Tristan Collins (Massachusetts Institute of Technology) · Jean-Pierre Demailly (Institut Fourier, Grenoble, France) · Simon K. Donaldson (Simons Center, Stony Brook ) · Dusa McDuff (Columbia University) · Hélène Esnault (Freie Universität, Berlin, Germany) · Charles Fefferman (Princeton University) · Teng Fei (Rutgers University, Newark) · Robert Friedman (Columbia University) · Kenji Fukaya (Simons Center, Stony Brook ) · Akito Futaki (Tsinghua University, PR China) · Phillip Griffiths (Institute for Advanced Study) · Victor Guillemin (Massachusetts Institute of Technology) · Nigel Hitchin (Oxford University, UK) · Dominic Joyce (Oxford University, UK) · Yujiro Kawamata (University of Tokyo) · Joseph J. Kohn (Princeton University) · H. Blaine Lawson (Stony Brook University) · Melissa C.C. Liu (Columbia University) · John Morgan (Columbia University) · Shigefumi Mori (RIMS, Kyoto, Japan) · Junjiro Noguchi (University of Tokyo, Japan) · Takeo Ohsawa (Nagoya University, Japan) · Duong H. Phong (Columbia University) · Paul Seidel (Massachusetts Institute of Technology) · Yum-Tong Siu (Harvard University) · Gabor Szekelyhidi (University of Notre Dame) · Clifford Taubes (Harvard University) · Claire Voisin (Institut de Mathematiques, Jussieu, France) · Shing-Tung Yau (Harvard University)














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Meeting ID: 930 2005 3589

Passcode: 572093




Tuesday, May 3, 2022







9:00 – 10:00am

Simon Donaldson (connecting via Zoom)


Closed 3-forms in Five Dimensions and Embedding Problems

The talk is on joint work with Fabian Lehmann (Simons Centre, Stony Brook). The first part will be an elementary discussion of the structure of closed 3-forms on 5-dimensional manifolds. We will define an open set of “pseudoconvex” 3-forms and consider the question of realising these by embeddings in C^3, or other complex Calabi-Yau threefolds. The main result is a deformation theorem, which is proved using Nash-Moser theory and an analysis of the linearised problem. The material is related to embedding questions for CR structures, to hyperkahler geometry and orthogonal symplectic pairs in four dimensions and to Hitchin’s approach to special structures in dimensions 6 and 7.

10:15 - 11:15am

Simon Brendle


The Isoperimetric Inequality on a Minimal Surface

We prove a Sobolev inequality which holds on submanifolds in Euclidean space of arbitrary dimension and codimension. This inequality is sharp if the codimension is at most 2. As a special case, we obtain a sharp isoperimetric inequality for minimal submanifolds in Euclidean space of codimension at most 2. Moreover, we indicate how the techniques employed in the proof can be applied to various other problems. 

11:30 - 12:30pm

Dusa McDuff


Kuranishi Atlases and Symplectic Curve Counting

I will explain some questions in symplectic embeddings (some solved, some still open) where it is important to be able to count curves.  Then I will discuss the relevance of Kuranishi atlases (in their various forms) to this problem. I hope also to explain my recent work with Wehrheim on constructing finite dimensional Kuranishi type models of polyfold Fredholm sections.






2:30 - 3:30pm

Cliff Taubes


Topological Aspects of Z/2 eigenfunctions of the Laplacian on the Round 2-Sphere

I will describe some very counter-intuitive properties of the ‘spherical harmonics’ that are associated to a configuration of an even number of points on the 2-sphere. (These eigenfunctions arise when studying non-convergent sequences in moduli spaces of solutions to generalizations of the Seiberg-Witten equations in dimension 3.)

4:00 - 5:00pm

Tristan Collins


Complete Calabi-Yau Metrics on the Complement of Two Divisors

In 1990 Tian-Yau proved that if Y is a Fano manifold and D is a smooth anti-canonical divisor, the complement X=Y\D admits a complete Calabi-Yau metric. A long standing problem has been to understand the existence of Calabi-Yau metrics when D is singular.  I will discuss the resolution of this problem when D=D_1+D_2 has two components and simple normal crossings.  I will also explain a general picture which suggests the case of general SNC divisors should be inductive on the number of components. This is joint work with Y. Li.

5:00 – 6:00pm









Wednesday, May 4, 2022






9:00 – 10:00am

Junjiro Noguchi (connecting via Zoom from Japan)

Analytic Ax-Schanuel for Semi-Abelian Varieties and Nevanlinna Theory

There is a so-called Schanuel conjecture on the transcendentacy of complex numbers. In 1972 Ax proved an formal analytic version of the conjecture by means of Kolchin's differential algebra (Ax-Schanuel).

Here we consider an analtyic version of the conjecture for a semi-abelian

variety $A$; we first give a Nevanlinna theoretic proof of the analytic

Ax-Schanuel for semi-abelian varieties. We then prove a 2nd Main Theorem for the associated entire curve into the product of a semi-abelian variety A and Lie (A).

10:15 - 11:15am

Dominic Joyce (connecting via Zoom from Great Britain)

What is a Kuranishi Space?

Kuranishi spaces first appeared in the work of Fukaya-Ono and Fukaya-Oh-Ohta-Ono, as the geometric structure on moduli spaces of J-holomorphic curves. I will explain a new definition of Kuranishi spaces which is better behaved than the original FOOO definition. It uses ideas from Derived Algebraic Geometry, and is based on the idea that Kuranishi spaces are really "derived smooth orbifolds".

Presentation Slides

11:30 - 12:30pm

Kenji Fukaya

Atiyah-Floer 2 Functor

This is a report of a joint work (a part is in progress) with A. Daemi and partially with M. Lipyanskiy. Atiyah-Floer conjecture concerns an isomorphism between two Floer homologies, one in Gauge theory and the other in Lagrangian Floer theory.

In this talk I will explain how we can `2-categorify' it.  We will define two 2 categories one by gauge theory and the other by Lagrangian Floer theory. We then construct a 2-functor which induces the above mentioned isomorphism in a certain case.





2:30 - 3:30pm

Robert Bryant

On the Cartan-Kuranishi Prolongation Theorem

An analytic system of PDE can be shown to have solutions provided that it is involutive, a property that generalizes and includes both the Cauchy-Kowalevskaya case and the Froebenius case.  This was the main tool used by Élie Cartan in his study of transformation groups and latter works in differential geometry.  For systems that are not involutive, Cartan devised a process of prolongation whose goal was to either uncover obstructions to existence of solutions of a given system or to replace it with an; 'equivalent; system that was involutive. Cartan tried to prove that prolongation did indeed achieve its goal, but was not able to do so.  In one of his first major works, M. Kuranishi gave a sufficient condition for the prolongation process to succeed. 


In this talk, I will review the historical background of Kuranishi's fundamental result and discuss some interesting geometric problems (some solved, some still open) where prolongation has turned out to be useful, even essential, in solving (or at least making progress in understanding) the problems described.

4:00 – 5:00pm

Charles Fefferman (connecting via Zoom)

Edge states in graphene

When graphene is truncated by a straight line cut, electrons sometimes propagate along the edge without diffracting into the bulk. Whether this happens depends sensitively on the orientation of the cut. The talk explains what's known and what's still mysterious about the existence of such edge states (Joint work with Sonia Fliss and Michael Weinstein).





Thursday, May 5, 2022






9:00 – 10:00am

Yujiro Kawamata

Deformations over non-commutative base.

Kuranishi considered deformation theory over commutative base space. If one allows the base to be non-commutative, then there are more deformations. The deformations over commutative base can sometimes be regarded as the 'first order' approximation of more general 'higher order' deformations. Though the formal theories of deformations are parallel and the extension to the non-commutative case is simple, some new phenomena and invariants appear. I will explain these by some examples.

10:15 - 11:15am

Hélène Esnault

Arithmetic Properties of Complex Rigid Local Systems

Rigid local systems are those for which the ‘Kuranishi’ space is reduced to a (possibly fat) point. Said differently it is a $0$-dimensional component of the Betti moduli space. One expects for them arithmetic properties which would follow from a positive answer to Simpson’s conjecture, now embedded by Petrov in the relative Fontaine-Mazur conjecture. We prove some of them (Joint work with Michael Groechenig).

11:30 - 12:30pm

Zhiqin Lu

The Spectrum of the Laplacian on Forms Over Open Manifolds

In this talk, we present the proof of the following theorem: let M be a complete non-compact Riemannian manifold whose curvature goes to zero at infinity, then its essential spectrum of the Laplacian on differential forms is a connected set. In particular, we study the case of form spectrum when the manifold is collapsing at infinity. This is joint with Nelia Charalambous.





                                      2:30 - 3:30pm

Takeo Ohsawa

$L^2$ estimates for $\dbar$ in Deformation, Approximation and Bundle-convexity

The method of $L^2$ estimates for the $\dbar$ operator will be recalled in connection to the local triviality questions for analytic families of complex manifolds. Emphasis is put on the isotrivial families of $\mathbb{C}$, which recently turned out to be locally trivial if the total space admit complete K\"ahler metrics. The $L^2$ method employed in the proof has been generalized to study some function-theoretic questions. It will be extended to prove an approximation theorem and a bundle-convexity theorem on complex manifolds which are close to weakly 1-complete.

                                      4:00 – 5:00pm

John Morgan

Duality Bordism

Poincare Duality for closed, oriented manifolds is a duality statement for the homology or cohomology of the manifold. This duality can be thought of as various perfect pairings on homology, the intersection pairing and the linking pairing. If a closed manifold bounds then Lefschetz duality for the pair produces a self-dual long exact sequence.
Considering chain complexes satisfying duality modulo those that are the `boundary’ term is an exact sequence of complexes satisfying Lefschetz duality, there are exactly two invariants: the signature when the dimension is 4k and the deRham invariant when the dimension is 4k+1.We start by briefly discuss these dualities and invariants.

The rest of the talk is devoted to sketching the construction of a geometric objects whose bordism groups are exactly captured by these two invariants. Using the Goresky-MacPherson idea of intersection homology, we construct a bordism theory whose closed objects are compact stratified spaces with a complex of  sheaves whose hypercohomology satisfies Poincare duality and whose compact objects with boundary are stratified spaces with boundary with a complex of sheaves satisfy Lefschetz duality. Using  work of Cheeger, using L^2 forms and the Hodge * operator of stratified spaces and ideas closely related to these, one can show that, inverting 2, the 4-periodic version of duality bordism of a space X is identified the KO-homology of X. More generally, 4-periodic version duality bordism is the homology theory Anderson dual to the cohomology theory of the 4-periodic space G/TOP, which is the classifying space of surgery theory.





Friday, May 6, 2022






9:00 – 10:00am

Shigefumi Mori (connecting via Zoom from Japan)

The Minimal Model Program in the 1980’s - My Personal View

The Minimal Model Program (MMP) for threefolds began in the early 1980's. It was initiated by M. Reid's work on canonical and terminal singularities and mine on the extremal rays, that is, the cone of curves. In the 1980's, a general framework of MMP was constructed for arbitrary dimensions, and the existence of the minimal model was settled for threefolds. An important portion of these developments took place around the time of my stay at Columbia University. I would like to recall some of them with my personal view and memory of Professor and Mrs. Kuranishi.

10:15 - 11:15am

Yum-Tong Siu

Non-deformability and estimates from Frobenius nonintegrability

Frobenius non-integrability is often the underlying reason for non-deformability and estimates. We discuss two situations.

(1) The global non-deformability of irreducible compact Hermitian symmetric manifolds has been a longstanding open conjecture, known for the complex projective space and the hyperquadric or for Kähler deformation. Confirmation is expected from the contradiction of integrability of certain tangent subspaces when there is global deformation. We discuss a new technique to handle the Grassmannians and the other open cases.

(2) For ¯ estimates, Frobenius non-integrability takes the form of the boundary Levi-form and the gradient term. We discuss its use in the Thullen-type extension across codimension 1 for holomorphic vector bundles with Hermitian metric whose curvature is Lp for some p > 1.

06:00 - 07:00pm






Continuation of conference here.



Columbia University

2990 Broadway

Mathematics Hall, room 312

New York, N.Y. 10027


Co-sponsored by the following

National Science Foundation-DMS 1564497 FRG

Simons Senior Fellow 328185

Fernholz Foundation


NSF Logo | NSF - National Science Foundation








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