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(Fn) 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(Fn). 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(Fn).

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 (xg: g G) by h·xg = x hg for any hgG. Define k (G) = k (xg : 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 B0(G) = 0 where B0(G) is the unramified Brauer group associated to G, which is a subgroup of H2 (G,Q/Z). Bogomolov proves that for any prime number p, there is a p -group G of order p 6 such that B0 (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  p5, then B0(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 p5. Then B0 (G)0 if and only G belongs to the isoclinism family Φ10. 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) B0 (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 B0(G) contains an elementary abelian p-group of order pn ; similarly, there is also a p-group G such that B0(G) contains the cyclic group of order pn.

 

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 (Mn,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 λ1 nK and the equality holds if and only if M is iso-metric to the sphere Sn 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 SaZ, 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 Bn, 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 Bm 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 SaZ 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 Cn or Stein. We will discuss some recent progress on the L2 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 Cn 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 Cn. 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 L2 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 L2 extension theorem with arbitrary negligible weight on Stein manifolds,  and rigidity problem for automorphism groups for invariant domains in Stein homogeneous spaces.

 

 

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