### Abstracts

Dynamics of pseudo-automorphisms
of birational maps of complex 2- and 3-space

Eric Bedford

We will present examples of compact,
complex manifolds which carry automorphisms of
positive entropy. Such automorphisms carry rich
dynamical behaviors. The mappings are easy to write down since the
coordinate functions are rational, and the corresponding manifolds are obtained
by blowing up projective space. Our presentation will be organized around
an extended discussion of the map f(x,y) = (y, (y+a)/ (x+b)) and its 3-dimensional analogue.

Classifications of gradient Ricci solitons

Huai-Dong Cao（曹怀东）

Ricci solitons
are natural extensions of Einstein metrics. They are also special solutions to
Hamilton's Ricci flow and often arise as dilation limits of singularities in the
Ricci flow. In this talk we will discuss geometry of gradient Ricci solitons and survey some recent progress on their
classifications.

Heat kernels of a class of degenerate
elliptic operators

Der-Chen Chang（张德健）

In this talk, we first discuss the
geometry induced by a class of second-order subelliptic
operators. This class contains degenerate elliptic and hypoelliptic
operators (such as the Grushin operator and the Baouendi-Goulaouic operator). Given any two points in the
space, the number of geodesics and the lengths of those geodesics are
calculated. We find modified complex action functions and show that the
critical values of these functions will recover the lengths of the
corresponding geodesics. We also find the volume elements by solving transport
equations. Then heat kernels for these operators are obtained. Finally we link
these heat kernels to sharp estimates for Kohn Laplacian
on a family of pseudo-convex hypersurfaces.

Strict
positivity of new numerical invariants of singularities and complex Plateau
problem

Rong Du（杜荣）

We introduce some new invariants for
complex manifolds. These invariants measure in some sense how far the complex
manifolds are away from having global complex coordinates. As an application,
we relate one of these invariants with a CR invariant to solve complex Plateau
problem.

Extension of line bundles

John Erik Fornaess

This is joint work with Sibony and Wold. We study extension
of holomorphic line bundles defined outside compact subsets in open complex
manifolds.

The Mather-Yau
Theorem and homogeneous isolated hyper-surface Singularities

Alexander Isaev

By the famous Mather-Yau
theorem, a complex hypersur-face germ V
with isolated singularity is completely determined by its moduli algebra A(V). The proof of the
theorem does not provide an explicit procedure for recovering V from
A(V), and finding
such a pro-cedure is a long-standing open problem. In
this talk, I will present an explicit way (discovered jointly with N. Kruzhilin) for reconstructing V
from A(V) up to biholomorphic equivalence under the assumption that the
singularity of V is homogeneous, in which case A(V)
coincides with the Milnor algebra of V. Furthermore, I
will discuss a characterization of the Milnor algebras of homogeneous singularities
in the class of all complex commutative associative algebras.

Tropical algebraic curves and outer automorphism groups of free groups

Lizhen Ji（季理真）

The outer automorphism
group of the free group Out(F_{n})
is one of the most important and basic groups in combinatorial and geometric
group theory. It acts on the outer space of metric graphs. This action is an
analogy of the action of arithmetic groups on symmetric spaces, and the action
of the mapping class group on the Teichmuller spaces,
and has played a fundamental role in understanding Out(F_{n}). Both symmetric spaces and Teichmuller spaces admit many complete metrics which are
invariant under the action and they are important for many applications. A
longstanding open problem is to construct complete metrics on the outer space
which are invariant under the action of Out(F_{n}).

In this talk, I will describe a solution
to this problem by using the tropical algebraic geometry: the tropical curves
and the tropical Jacobian varieties of tropical
curves.

The analogy with symmetric spaces and Teichmuller spaces also plays an important role.

Noether's problem and unramified Brauer groups

Ming-chang
Kang（康明昌）

Let
k be
any field, G be a finite group acting on the rational function field k (x_{g}: g G)
by h·x_{g} = x
_{hg} for any h，gG. Define k (G) = k (x_{g}_{
}: g G)^{G}.
Noether’s problem asks whether k (G) is rational (=
purely transcendental) over k. It is known
that, if C (G) is rational over C, then B_{0}(G)
= 0 where B_{0}(G) is the unramified Brauer group associated to G, which is a subgroup of H^{2}
(G,Q/Z). Bogomolov proves that for any prime number p,
there is a p -group G of order p^{
6}
such that B_{0} (G) is non-trivial and therefore C(G)
is not rational over C. He also shows that, if G is a p
-group of order p^{5},
then B_{0}(G) = 0. The latter result was disproved by Moravec for p = 3,5,7 by the computer computing. The case for groups of order
32 and 64 was solved by Chu, Hu, Kang, Kunyvskii and
Prokhorov.

We
will prove the following theorems. Theorem 1 (Hoshi, Kang and
Kunyavskii). Let p
be any odd prime number and G be a group of order p^{5}. Then B_{0}
(G)0 if and only G belongs to the isoclinism
family Φ_{10}. Theorem 2 (Chu, Hoshi, Hu and Kang).
Let G be a group of order 243 with exponent e. Let k
be a field containing a primitive e-th root of unity. Then the followings are equivalent, (i) k
(G) is rational over k, (ii) B_{0} (G) = 0, (iii) G is
not isomorphic to G(243, i )
with 28 i 30. Theorem 3. For any odd prime number p,
any positive integer n, there is a p-group
G such that B_{0}(G) contains an elementary abelian
p-group
of order p^{n} ; similarly,
there is also a p-group G such that B_{0}(G)
contains the cyclic group of order p^{n}.

Biholomorphic equivalence: a
journey from Poincaré to today

Bernhard Lamel

We
will review work on local biholomorphic equivalence
of real-analytic CR manifolds, starting from Poincaré's
observation in 1907 that the problem is highly nontrivial, and going on to
more modern work on the problem.

On the rigidity problem and the frst positive eigenvalues estimate

Song-Ying Li（李松鹰）

In
Riemannian geometry, there is a well-known theorem called Lichnorwicz
and Obata theorem: Let (M^{n},g)
be a compact, complete Rie-mannian manifold
satisfying Ric ≥ (n-1)K for some positive constant K.
Then the first positive eigenvalue of Laplace-Beltrami operator λ_{1 }≥
nK and the equality holds if and only if M is iso-metric to the sphere S^{n}
of radius 1/.

I
will talk about a joint work with Xiaodong Wang and a
joint work with My-An Tran on the CR-analogies of the above theorem.

Period integrals and tautological systems

Bong Lian（连文豪）

This will be a survey of recent
constructions of a new class of partial differential equations arising from the
study of the period mappings of certain CY and general complete intersection
families in a complex manifold X. These PDE systems, which we call tautological
systems, are important in that they often determine the period integrals of
those families of varieties and that they are amenable to explicit
descriptions. We give one in terms of representation theory, in case X is a
projective homogeneous manifold.

Global Torelli
theorems for projective manifolds

Kefeng Liu（刘克峰）

I will discuss our recent works on the injectivity of the period maps for a class of projective
manifolds of Calabi-Yau type.

A dilogarithm
identity on moduli spaces of surfaces

Feng Luo（罗锋）

Given any closed hyperbolic surface of a
fixed genus,we establish an
identity involving dilogarithm of lengths of simple
closed geodesics in all embedded pairs of pants and one-holed tori in the
surface. One may consider this as a counter part of McShane’s
identity for closed hyperbolic surfaces. This is a joint work with Ser Peow Tan.

On a geometric analogue of the Andre-Oort conjecture for the Zariski closure
of positive-dimensional totally geodesic subvarieties

Ngaiming Mok（莫毅明）

Let Ω be a bounded symmetric domain, Γ Aut(Ω) be
a torsion-free lattice, X:= Ω/Γ. Let ZX be an irreducible quasi-projective variety such that
Z is the Zariski closure of an infinite family of
totally geodesic complex subvarieties S_{a}Z, aA. Under certain
non-degeneracy conditions one expects Z to be also totally geodesic, so that Z is
in particular again uniformized by a bounded
symmetric domain. This set-up is related to the Andr´e-Oort
Conjecture since (positive-dimensional) special varieties in the context of the
latter conjecture are known to be totally geodesic.

In the case where Ω is the complex unit
ball B^{n}, Z can be proven to be totally geodesic
without any additional non-degeneracy hypothesis. A generalization of the argument
to bounded symmetric domains Ω leads to the study of holomorphic isometric immersions
of complex unit balls B^{m}
into Ω. In an earlier work the author established that the graph of any such a
germ of holomorphic isometric immersion must be algebraic. Using this, we
solved the problem in the case where Z is a complex surface and S_{a}Z are totally geodesic curves.

The Cauchy-Riemann operator in complex
manifolds

Mei-Chi Shaw

The Cauchy-Riemann operator for domains
in the complex Euclidean space or a Stein manifold is well understood. Much
less is known for the solvability or regularity for the Cauchy-Riemann operator
in a complex manifold which is not C^{n} or
Stein. We will discuss some recent progress on the L^{2} theory of the
Cauchy-Riemann equations on domains in complex manifolds. Comparison of coholmology groups in different function spaces will also
be presented (joint work with Debraj Chakrabarti).

Etale fundamental
groups and D-modules in char. p>0

Xiaotao Sun（孙笑涛）

I will show how a theorem of Hrushovski can be used to prove Gieseker's
conjecture completely, which relates etale
fundamental groups with D-modules. This is a joint work with H. Esnault.

The Best bound of abelian
Automorphism groups of surfaces of general type

Sheng-Li Tan（谈胜利）

We prove that the order of the abelian automorphism group G of a minimal complex surface S of general type is bounded from above by 12.5K+100 provided the geometric genus of the surface is at least 7. The upper bound is reached for infinitely many families of surfaces whose geometric genus can be arbitrarily large. We will present also an example to show that the lower bound 7 on the geometric genus can not be replaced by 3. This is a joint work with Xin Lv.

Geometry of smooth metric measure spaces

Jiaping Wang（王嘉平）

We will present some joint work with Ovidiu Munteanu concerning the
geometry of a smooth manifold with its Bakry-Emery
curvature bounded from below. The issues to be discussed include volume growth
estimates, function theory and topology at infinity.

The affine Plateau problem

Xu-jia Wang（汪徐家）

With Neil Trudinger,
we study the Plateau problem for affine maximal hypersurfaces,
which is the affine invariant analogue of the classical Plateau problem for
minimal surfaces. We formulate the affine Plateau problem as a geometric variational problem for the affine area functional, and
prove the existence and interior regularity of maximizers.
As a special case, we obtain corresponding existence and regularity results for
the variational Dirichlet
problem for the fourth order affine maximal surface equation. Our approach
makes use of techniques and results from the theory of the Monge-Ampère
equation.

Dynamics over Berkovich
projective space

Yuefei Wang（王跃飞）

In recent years there is a considerable
interest in Non-Archimedean dynamics. We will talk about the dynamical systems
over Berkovich projective space with applications to
the Julia sets of commuting p-adic dynamics.

Curvature and rational curves on
projective varieties

Bun Wong（王彬）

Some connections between curvature
tensors of Kaehler metrics with rational connectivity
of projective varieties will be discussed.

On mixed-type equation and degeneracy

Zhouping Xin（辛周平）

In this talk, I will discuss some recent
progress on the existence of smooth transonic flows in a nozzle with variable
sections. The flow will be governed by the well-known steady compressible full
potential flow equation which is a degenerate elliptic-hyperbolic equation for
such flows. We will discuss properties of smooth transonic flows in a de Laval
nozzle and give the first general existence of such solutions with physically reasonable boundary conditions in the 2-dimensional
space.

Concept of space adapted to modern physics

Shing Tung Yau

U(n) invariant Kaehler metrics with posiitve
curvature and Ricci flow

Fangyang Zheng（郑方阳）

In this talk we will report on
a joint work with Bo Yang, in which we examined U(n)-invariant
complete Kaehler metrics on C^{n}
with various degree of positivity in curvature. We also examine the short
time existence problem for Kaehler-Ricci flow when
the inital metric has unbounded curvature. The
underlying problem in this topic is Yau's conjecture
which states that any complete non-compact Kaehler
manifold with positive bisectional curvature is biholomorphic
to C^{n}. It was studied and partailly confirmed by the work of Wan-Xiong Shi, Bing-Long Chen, Xi-Ping Zhu, Lei Ni, Luen-fai Tam, Albert Chau,
etc.

Optimal constant problem in L^{2}
extension

Xiangyu Zhou（周向宇）

In this talk, we'll present our recent
results about our solutions of two different problems, i.e., optimal constant
problem in L^{2 }extension theorem with arbitrary negligible weight on
Stein manifolds, and rigidity problem for automorphism groups for invariant domains in Stein
homogeneous spaces.