DEPARTMENT OF MATHEMATICS
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)
To access the registration form, please scan the QR code.
Join Zoom Meeting
https://columbiauniversity.zoom.us/j/93020053589?pwd=KzRHWHBaeDIyazBJRHpaSDZZYUI1Zz09
Meeting ID: 930 2005 3589
Passcode: 572093
Tuesday, May 3, 2022
|
TIME |
SPEAKER |
|
TITLE |
ABSTRACT |
|
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. |
|
12:40-2:20pm |
LUNCH |
|
|
|
|
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 |
RECEPTION |
|
|
|
Wednesday, May 4, 2022
|
TIME |
SPEAKER |
TITLE |
ABSTRACT |
||
|
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". |
||
|
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. |
||
|
12:40-2:20pm |
LUNCH |
|
|
||
|
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
|
TIME |
SPEAKER |
TITLE |
ABSTRACT |
|||||
|
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. |
|||||
|
12:40-2:20pm |
LUNCH |
|
|
|||||
|
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. |
|||||
Friday, May 6, 2022
|
TIME |
SPEAKER |
TITLE |
ABSTRACT |
|
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 |
BANQUET |
|
|
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
