Speakers:
 Chang, ShuCheng (Tsing Hua
University, Taiwan)

Chen, Jiun Cheng (Tsing
Hua University, Taiwan)
 Hassett, Brendan (Rice
University, USA)
 Li, Peter (University of
California, Irvine, USA)
 Li, Weiping (Hong Kong
University of Science and Technology)
 Sheng, Mao (Johannes Gutenberg
Universitat Mainz, Germany)
 Wang, ChinLung (National
Central University, Taiwan)
 Wang, Jiaping (University of
Minnesota, USA)
 Wu, Siye (University of Hong
Kong)
 Zhang, DeQi (National
University of Singapore)
 Zhang, Qi S. (UC Riverside,
USA)
 Zhou, Jian (Tsing Hua
University, China)
 Zhu, XiPing (Sun YatSen
University, China)


Information:
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Program:
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Titles:


Characteristic
subvarieties and disproof of modularity of moduli space of certain
CalabiYau manifolds

Sheng, Mao

Characterizing
projective spaces

Chen, Jiun Cheng

Filtered
geometry of formal path spaces

Zhou, Jian

Geometrical
approach to dynamics of automorphisms on algebraic manifolds

Zhang, DeQi

Incidence
Hilbert schemes and infinite dimensional algebras

Li, Weiping

Invariance
of big quantum ring under simple flops

Wang, ChinLung

Number
of minimal graphs with disjoint support

Wang, jiaping

Projective
flatness in the geometric quantisation of bosons and fermions

Wu, Siye

Sharp
estimate on the spectrum of a complete Kahler manifold

Li, Peter

Some
new estimates for the conjugate heat equation under Ricci flow

Zhang, Qi S.

Sections
of rationally connected fibrations through prescribed points

Hassett, Brendan

Subgradient
estimate and LiYau Hamilton inequality in pseudohermitian 3manifold

Chang, ShuCheng

The
Existence of Type II Singularity to the Ricci Flow on Compact Manifolds

Zhu, Xi Ping

Characteristic subvarieties and disproof of modularity
of moduli space of certain CalabiYau manifolds
Sheng, Mao
16:00 – 17:00, 21^{st}, December, 2006
This talk is a
joint work with Ralf Gerkmann and Kang Zuo. In this work, we study the
modularity property of the moduli space of CalabiYau manifolds as
resolutions of double covers of P^n branched along 2n+2 hyperplanes in
general positions. Different from n<=2 case, we disproved the modularity
of the moduli space for n=3.
We found a new
type of invariants defined over the moduli space pointwisely, which we call
characteristic subvarieties. In the case of bounded symmetric domain, for
certain CalabiYau like variation of Hodge structures, the characteristic
subvarieties are identified back with the characteristic bundles of N.Mok.
The possiblity of the identification was firstly observed by K.Zuo.
We were able to use the identification to disprove the modularity in the
claimed case by comparing the characteristic subvarieties with the
characteristic bundles at one point.
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Characterizing projective spaces
Chen, Jiun Cheng
17:05 – 18:05, 20^{th}, December, 2006
I will present my recent work on characterizing
projective spaces for possibly singular varieties. We use
the space of twisted stable maps (into a DeligneMumford stack) and study
possible degeneration types. I will also discuss applications in studying
Mukai conjecture.
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Filtered geometry of formal path spaces
Zhou, Jian
11:15 – 12:15, 21^{st}, December, 2006
We study some
natural local coordinates on formal path spaces.
Their
transformations indicate that the tangent spaces of formal path spaces have
natural structures of filtered vector spaces. Furthermore, the spaces of
differential forms and cohomology on formal path spaces have very rich
algebraic structures related to vertex operator algebras and elliptic genera.
This study is inspired by nonlinear supersymmetric sigma models.
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Geometrical approach to dynamics of automorphisms on
algebraic manifolds
Zhang, DeQi
16:00 – 17:00, 20^{th}, December, 2006
We show that
dynamics of automorphisms on projective complex manifolds are built up from
those on only three kinds of manifolds: Complex Tori, Weak CalabiYau
Manifolds and Rationally Connected Manifolds.
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Incidence Hilbert schemes and infinite dimensional
algebras
Li, Weiping
14:30 – 15:30, 20^{th}, December, 2006
Let S be a
smooth projective surface. Using correspondences, we construct an infinite
dimensional Lie algebra that acts on the direct sum
H=\bigoplus_{m=0}^{+\infty}H^*(S^[m,m+1]) of the cohomology groups of the
incidence Hilbert schemes S^[m,m+1]. The algebra is related to an extension
of an infinite dimensional Heisenberg algebra. The space $H$ is a highest
weight representation of this algebra. Our result provides a
representationtheoretic interpretation of Cheah's generating function of
Betti numbers of the incidence Hilbert schemes. As a consequence, an additive
basis of H^*(S^[m,m+1]) is obtained. We also prove similar results for some
DonaldsonThomas moduli spaces of ideal sheaves on a threefold X admitting a
locally trivial fibration to S. This is the joint work with Zhenbo Qin.
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Invariance of big quantum ring under simple flops
Wang, ChinLung
10:00 – 11:00, 20^{th}, December, 2006
While
Kequivalent manifolds are known to share with the same Hodge numbers and
complex elliptic genera, it is wellknown that the homotopy types are in
general different and the ordinary cohomology ring structure is not preserved
even under the simplest type of Kequivalent maps, namely flops. It has
however been conjectured that the quantum product should be preserved under
Kequivalence in a certain sense.
In this talk I
will explain some of my recent progress (joint work with Y.P. Lee and H.W.
Lin) on this problem. For ordinary flops, the correspondence defined by the
graph closure is shown to give equivalence of Chow motives and to preserve
the Poincare pairing. In the case of simple ordinary flops, this
correspondence preserves the big quantum cohomology ring after an analytic
continuation over the extended Kaehler moduli space. The main tools used are
localizations and degenerations.
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Number of minimal graphs with disjoint support
Wang, Jiaping
10:00 – 11:00, 21^{st}, December, 2006
We explain a
joint result with Peter Li on the finiteness of the number of minimal graphs
with disjoint Euclidean domains.
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Projective flatness in the geometric quantisation of
bosons and fermions
Wu, Siye
14:30 – 15:30, 21^{st}, December, 2006
Geometric
quantization procedure requires choosing a real or complex polarisation.
Quantum physics is independent of the choice if there exists a projectively
flat connection on the vector bundle of Hilbert spaces over the space of
polarizations. In this talk, I will begin with the example of symplectic vector
spaces and study the relation with the Maslov index. Quantisation of
fermionic systems will be studied and is related to Clifford algebra and
spinors.
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Sharp estimate on the spectrum of a complete Kahler
manifold
Li, Peter
11:05 – 12:05, 22^{nd}, December, 2006
In this talk, we
will discuss a sharp upper estimate for the greatest lower bound of the
spectrum of the Laplacian on a complete Kahler manifold. The case when the
estimate is realized will also be discussed.
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Some new estimates for the conjugate heat equation under
Ricci flow
Zhang, Qi S.
16:00 – 17:00, 19^{th}, December, 2006
We present a
proof for an upper bound for the fundamental solution of the conjugate heat
equation and a gradient estimate for all positive solutions of the same
equation. This contrasts to the Perelman gradient estimate which does not
work for all positive solutions.
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Sections of rationally connected fibrations through
prescribed points
Hassett, Brendan
11:05 – 12:05, 20^{th}, December, 2006
Let X>B be
a projective variety fibered over a smooth complex curve. `Weak approximation
holds' if, given an
arbitrary collection of horizontal jet data, there exists a section s:
B>X with these jets. We prove this provided
the fibers are smooth and rationally connected. This builds on previous work
of Koll'arMiyaokaMori and
GraberHarrisStarr. We shall also discuss results at places of bad reduction
in special cases like cubic surfaces.
(joint work with Yuri Tschinkel)
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Subgradient estimate and LiYau Hamilton inequality in
pseudohermitian 3manifold
Chang, ShuCheng
10:00 – 11:00, 22^{nd}, December, 2006
In this talk I
will report on the joint work with H.L. Chiu and C.T. Wu. First we get a
subgradient estimate on a pseudohermitian 3manifold which is served as the
CR analogue of Yau.s gradient estimate. With its application, it is shown
that the natural analogue of Liouvilletype theorems holds for the sub
Laplacian on a complete noncompact pseudohermitian 3manifold. Second, we
deform the contact form by the amount of the TanakaWebster curvature minus
its mean value on a closed spherical CR 3manifold. We show that if a contact
form evolves with free torsion from initial data with positive TanakaWebster
curvature, then we obtain a timereversed LiYauHamilton inequality for the
TanakaWebster
curvature. As a consequence, we get the global existence and asymptotic
convergence of the Yamabe .ow on a closed spherical CR 3manifold of zero
torsion.
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The Existence of Type II Singularity to the Ricci Flow
on Compact Manifolds
Zhu, Xi Ping
14:30 – 15:30, 19^{th}, December, 2006
According to
Hamilton, the singularities of the Ricci flow on compact manifolds can be
divided into two types. All known existence results are forming Type I
singularities. In this talk we will prove the existence of Type II
singularities to the Ricci flow on S^n for each n > 2. This is a joint
work with Huiling Gu.
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Organizers:
 Conan Leung (The Institute of Mathematical
Science, CUHK, Hong Kong)
 S.T. Yau (Director, The Institute of
Mathematical Science, CUHK, Hong Kong)
Enquiry:
For
enquiry, please contact Prof. Conan Leung at ncleung@math.cuhk.edu.hk
