#### Awardees for PhD Theses

Note: All prizes and awards are subjected to the awardee's presence at the Award Ceremony in 2010. The tentative date for the ceremony is December 17, 2010.

Gold Awards (No order)

It is decided that the following three persons will be awarded the title of Gold Awardees in the 2010 ceremony.
They will equally share the sum of the two announced gold award prizes.

Name: Chen-Yu Chi (ÆëÕðÓî)

Title: Pseudonorms and Theorems of Torelli Type for Birational Equivalence

Affiliation: Harvard University

Abstract: Our research interest has been in the birational classification of complex projective varieties using invariants. A classical set of birational invariants of a variety are its pluricanonical spaces and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally invariant. These vector spaces so metrized will be referred to as the pseudonormed spaces of the original varieties.

A fundamental question is the following: given two mildly singular projective varieties with some of the first variety's pseudonormed spaces being isometric to the corresponding ones of the second variety's, can one construct a birational map between them which induces these isometries? We are able to give a positive answer to this question for varieties of general type. Our results can be thought of as theorems of Torelli type for birational equivalence.

Name: Yongquan Hu(ºúÓÀÈª)

Title: On $p$-adic and modulo $p$ local Langlands program

Affiliation: Universit¨¦ Paris-sud

Abstract:This thesis is a part of the $p$-adic and modulo $p$ Langlands program which has been proposed by Breuil. Let $p$ be a prime number and F be a complete discrete valuation field with a finite residuel field of characteristic $p$. In the first chapter, we prove a part of a conjecture of Breuil and Schneider on the existence of stable lattices inside certain locally algebraic representations of $GL_n(F)$, where $F$ is supposed to be of characteristic 0. The second and third chapters treat modulo $p$ representations of $GL_2(F)$. In the second, we associate a ¡°canonical diagram¡± to an irreducible smooth modulo $p$ representation of $GL_2(F)$, such that this diagram determines the isomorphism class of the original representation. In the third, we construct new supersingulier representations of $GL_2(F)$ when $F$ is unramified over $\Q_p$.

Name: Jun Yin(Òü¿¥)

Title: QUANTUM MANY-BODY SYSTEMS WITH SHORT-RANGE INTERACTIONS

Affiliation: Princeton University

Abstract:In this dissertation, the central theme is evaluation of the ground energy and the first few excited energies of Bose or Fermi gas system in continuum $\R^3$ or lattice $\Z^3$ with short-range interactions. In Chapter 2 and 3, we carry out an analysis on low-dimensional behaviors of dilute Bose gas in traps. In Chapter 4 and 5, we generalize the results on the ground state energies of dilute Bose or Fermi gases in thermodynamic limit.

Silver Awards(No order)

It is decided that the following seven persons will be awarded the title of Silver Awardees in the 2010 ceremony. They will equally share the sum of the four announced silver award prizes.

Name: Kwokwai Chan(³Â¹úÍþ)

Title:Strominger-Yau-Zaslow Transformations in Mirror Symmetry

Affiliation: Chinese University of Hong Kong

Abstract:We study mirror symmetry via Fourier-Mukai-type transforma- tions, which we call SYZ mirror transformations, in view of the ground-breaking Strominger-Yau-Zaslow Mirror Conjecture which asserted that the mirror symmetry for Calabi-Yau manifolds could be understood geometrically as a T-duality modified by suitable quantum corrections. We apply these transformations to investigate a case of mirror symmetry with quantum corrections, namely the mirror symmetry between the A-model of a toric Fano manifold X and the B-model of a Landau-Ginzburg model (Y;W). Here Y is a noncompact Kaehler manifold and W is a holomorphic function. We construct an explicit SYZ mirror transformation which realizes canonically the isomorphism between the quantum cohomology ring of X and the Jacobian ring of the function W. We also show that the symplectic structure of X is transformed to the holomorphic volume form (Y;W). Concerning the Homological Mirror Symmetry Conjecture, we exhibit certain correspondences between A- branes on X and B-branes on (Y;W) by applying the SYZ philosophy.

Name: Ke Chen(³Â¿Â)

Title:Special subvarieties of mixed Shimura varieties

Affiliation: Universit¨¦ Paris-Sud XI

Abstract:We study aspects of the Andr¨¦-Oort conjecture for mixed Shimura varieties. We generalize the homogeneous equidistribution theorem of C-special subvarieties proved by L.Clozel, E.Ullmo and A.Yafaev to the case of mixed Shimura varieties: let M be a mixed Shimura variety, Z an closed subvariety in M, then the set of maximal C-special subvarieties of M contained in Z is finite, where C is any fixed rational torus in the generic Mumford-Tate group of M. The proof relies on Ratner's theory on the ergodic properties of unipotent flows on homogeneous spaces. We also adapt a minoration of the degree of the Galois orbit of a special subvariety to the mixed case. Finally, inspired by the Andr¨¦-Oort conjecture, we prove the Manin-Mumford conjecture in the relative case of characteristic zero: let A be an abelian scheme of characteristic zero, then the schematic closure of a sequence of torsion subschemes in A remains a finite union of torsion subschemes.

Name: Kai-Wen Lan(À¶¿­ÎÄ )

Title:Arithmatic compactifications of PEL-type Shimura varieties

Affiliation: Harvard University

Abstract: In this thesis, we constructed minimal (Satake-Baily-Borel) compactifications and smooth toroidal compactifications of integral models of general PEL-type Shimura varieties (defined as in Kottwitz [79]), with descriptions of stratifications and local structures on them extending the well-known ones in the complex analytic theory. This carries out a program initiated by Chai,
Faltings, and some other people more than twenty years ago. The approach we have taken is to redo the Faltings-Chai theory [37] in full generality, with as many details as possible, but without any substantial case-by-case study. The essential new ingredient in our approach is the emphasis on level structures, leading to a crucial Weil pairing calculation that enables us to avoid unwanted boundary components in naive constructions.

Name: Weidong Liu(ÁõÎÀ¶« )

Title:Limit Properties for a Class of Stationary Processes

Affiliation: Zhejiang University

Abstract: The first part of this dissertation concerns some limit properties of a class of stationary processes (i.e. $X_{n}=g(\cdots,\varepsilon_{n-1},\varepsilon_{n})$), including strong invariance principles for partial sums, the maximum of periodograms and asymptotics of the spectral density estimation. These subjects are very important in probability and statistics, and have been explored in many classical textbooks. Because of a surge of interest in nonlinear time series, people proposed many new questions on the subjects referred above. For example, can we consider similar problems for nonlinear time series ? It is well known that martingale approximation is an effective method to deal with stationary processes. However, martingale approximation seems not very suitable for the problems above. For this reason, we use $m$ dependence approximation and obtain the optimal rates for strong invariance principles, the asymptotic distributions of the maxima of periodograms of nonlinear time series and the maximum deviation of the spectral density estimation. Meanwhile, we solve some open questions proposed by some previous papers. The second part of this paper concerns the test on independence between components of a high dimensional vector. The high dimension problem is very popular recently. Since the independence is usually assumed in many statistical problem, test on independence is an important problem. Based on some previous work, we propose a new statistic to test whether components are independent. We also prove that the limit distribution of this statistic is the extreme distribution of type I with a rate of convergence $O((\log n)^{5/2}/\sqrt{n})$. This is much faster than $O(1/\log n)$, a typical convergence rate for this type of extreme distribution. The third part of this paper considers LIL for independent B valued random variables. Based on the previous literature, we prove some LIL for independent B valued random variables when their variances are infinite. Some previous results are extended.

Name: LOK MING, LUI(À×ÀÖÃú )

Title:Computational Conformal Geometry and its Applications to Human Brain Mapping

Affiliation: University of California at Los Angeles (UCLA)

Abstract: Analyzing the data and performing computation effectively on surfaces with complicated geometry is an important research topic, especially in Human Brain Mapping. In this work, we are interested in computing the conformal structure of the Riemann surface and applying it to Human Brain Mapping. In order to analyze the brain data efficiently, the complicated brain cortical surface is usually parameterized to a simple parameter domain such as the sphere or 2D rectangles. This allows us to transform the 3D problems into 2D problems. In order to compare data more effectively, the parameterization has to preserve the geometry of the brain structure while aligning the important anatomical features consistently. Conformal parameterization, that preserves the local geometry, is often used. In our work, we propose algorithms to compute the optimized conformal parameterization of the brain surface which aligns the anatomical features consistently while preserving the conformality of the parameterization as much as possible. With the conformal parameterization, we can solve variational problems and partial differential equations on the surface easily by solving the corresponding equations on the 2D parameter domain. The computation is simple because of the simple Riemannian metric of the conformal map. Finally, we develop an automatic landmark tracking algorithm to detect the sulcal landmarks on the brain cortical surface, which involves solving variational problems on the brain surface.

Name: Jilong Tong(Í¯¼ÍÁú )

Title: I Applications d¡¯Albanese pour les courbes et contractions II diviseur th¨ºta et formes diff¨¦rentielles

Affiliation: University of Paris 11

Abstract: We consider two different problems concerning the geometry of curves. The first part is devoted to the study of Albanese morphism for a curve with mild singularities. As an application, we improve a recent result of Deninger-Werner. The second (and also the main part of this thesis) is a geometric study of theta divisor defined by Raynaud, and we find some applications in the study of the variations of algebraic fundamental groups for a curves.

Name: Xinwen Zhu

Title: Gerbal representations of double loop groups

Affiliation: UC Berkeley

Abstract: A crucial role in representation theory of loop groups of reductive Lie groups and their Lie algebras is played by their non-trivial second cohomology classes which give rise to their central extensions (the affine Kac¨CMoody groups and Lie algebras). Loop groups embed into the group GL\infty of continuous automorphisms of C((t)), and these classes come from a second cohomology class of GL\infty. In a similar way, double loop groups embed into a group of automorphisms of C((t))((s)), denoted by GL\infty,\infty, which has a non-trivial third cohomology. In this paper we explain how to realize a third cohomology class in representation theory of a group: it naturally arises when we consider representations on categories rather than vector spaces. We call them ¡°gerbal representations.¡± We then construct a gerbal representation of GL\infty,\infty (and hence of double loop groups), realizing its non-trivial third cohomology class, on a category of modules over an infinite-dimensional Clifford algebra. This is a two-dimensional analogue of the fermionic Fock representations of the ordinary loop groups.