Bảo Châu Ngô
 (University of Chicago & VIASM)
July 1(Mon) 11:00~11:50

Title: Arithmetic of certain integrable system

Topology of integrable system impose very strong constraints on the cohomology of level sets, in particular in singular levels. We will explain the nature of these constraints with the language of perverse sheaves, and summarize applications in the study of certain moduli spaces and in number theory.
Seung Yeal Ha
(Seoul National University)
July 1(Mon) 13:30~14:20

Title: From particle to kinetic and hydrodynamic descriptions of flocking
In this talk, we will review recent progress on the hierarchical modeling of the Cucker-Smale flocking in terms of particle, kinetic and hydrodynamic models. We will also discuss precise sufficient conditions for flocking to occur and couplings with fluid models and relations with the Kuramoto model for synchronization arising from the statistical physics.
Takuro Mochizuki
(RIMS, Kyoto University)
July 2(Tue) 11:00~11:50

Title: Wild harmonic bundles and some applications
Harmonic bundles with wild singularities have played important roles in the differential geometric study of holonomic D-modules on algebraic varieties. In this talk, after a brief review of the theory of wild harmonic bundles, we present some applications to related subjects.

One is a type of Toda equations. We can classify the real valued solutions of the Toda equations from the viewpoint of the Kobayashi-Hitchin correspondence for wild harmonic bundles. We also have a criterion when the associated semi-infinite variation of Hodge structure has an integral structure. One of the key ingredients is the limit mixed twistor structure with an induced torus action.

The other is the Nahm transforms for some types of instantons. Nahm transforms are a differential geometric analogue of Fourier-Mukai transforms. They are procedures to make a type of instantons satisfying some periodicity and boundary conditions from a different type of instantons. Various versions of the Nahm transforms have been studied intensively by many people, but it looks still interesting to clarify the more details on the boundary conditions.

We explain our refinement of the Nahm transforms for harmonic bundles and doubly periodic instantons.
Shige Peng
(Shandong Univ.)
July 2(Tue) 13:30~14:20

Title : Bridge: Backward stochastic differential equation, nonlinear expectation and path-dependent PDE 
Backward stochastic differential equation (BSDE) theory is a powerful tool in quantitative analysis and calculation in financial risk measurements and many other situations. A BSDE defined in a probability space can be considered as a nonlinear Girsanov transformation and that the solution of the BSDE is a nonlinear martingale under a nonlinear expectation, called g-expectation. The corresponding nonlinear Feynman-Kac formula tells us that, once the coefficients depend only on the state of the Brownian path, then the BSDE is a quasi linear partial differential equation (PDE) of parabolic types. This reveals that, a general (non-Markovian) BSDE is in fact a PDE in which the Brownian path plays the role of state variable. The above Brownian motion can also be replaced by a general and possibly degenerate Markovian process.

For a fully nonlinear parabolic PDE, can one still establish the corresponding BSDE, nonlinear expectation and path-dependence PDE? The answer of this deep problem is: the Wiener probability measure and/or any other probability measures corresponding can no longer be the reference probability space of the corresponding nonlinear expectation since the latter is essentially 'fully nonlinear'. A direct solution to this problem is to use a fully nonlinear PDE to construct the corresponding nonlinear expectation, called G-expectation. The nonlinear martingale under this expectation, called ich can be regarded as the corresponding fully nonlinear path-dependence PDE. It turns out that a well-designed e a large set of fully nonlinear expectations, or BSDE. In this framework, the corresponding canonic process is called G-Brownian motion which is a continuous process with independent and stable increments. The increments are proved to be distributed which coincides with the limit distribution of the central limit theorem under a nonlinear expectation. This approach also gives us a pedagogically direct and simple access to the theory of G-expectation, G-Brownian motion and the corresponding stochastic calculus of Ito's type.

It turns out that a new notion of G-expectation-weighted Sobolev spaces, or in short, G-Sobolev spaces, is naturally introduced and each Ito's process under G-expectation can be written as a generalized Ito's formula. Consequently, a backward SDEs driven by G-Brownian motion is in fact a path dependent PDEs in the corresponding G-Sobolev spaces. For the linear case of G corresponding the classical Wiener probability measure P, we have established 1-1 correspondence between BSDE and such new type of quasi linear PDE in the corresponding P-Sobolev space. When G is nonlinear, we also provide such 1-1 correspondence between a fully nonlinear PDE in the corresponding G-Sobolev space and BSDE driven by G-Brownian. Consequently, the existence and uniqueness of such type of fully nonlinear path-dependence PDE in G-Sobolev space have been obtained via a recent results of BSDE driven by G-Brownian motion.
Zuowei Shen
(National University of Singapore)
July 3(Wed) 10:20~11:10

Title: Image Restoration: Wavelet Frame Approach, Total Variation and Beyond
This talk is about the wavelet frame-based image and video restorations. We start with some of main ideas of wavelet frame based image restorations. Some of applications of wavelet frame based models image analysis and restorations will be shown. Examples of such applications include image and video inpainting, denoising, decomposition, image deblurring and blind deblurring, segmentation, CT image reconstruction and etc. In all of these applications, spline wavelet frames derived from the unitary extension principle are used. This allows us to establish a connection between wavelet frame base method and the total variation based method. In fact, we will show that when spline wavelet frames are used, a special model of a wavelet frame method can be viewed as a discrete approximation at a given resolution to the total variation based method. A convergence analysis in terms of objective functionals and their approximate minimizers as image resolution increases will be discussed.
Takeshi Saito
(University of Tokyo)
July 3(Wed) 11:30~12:20

Title: Wild ramification and the cotangent bundlev
Following an analogy with D-modules, we introduce the characteristic cycle of an l-adic sheaf on a smooth variety of characteristic $p>0$ with wild ramification along the boundary, as a cycle on the cotangent bundle, under a certain condition. The construction is compatible with the pull-back by non-characteristic morphisms and is characterized by the restrictions to curves meeting the boundary transversely. Further it controls the nearby cycles and computes the Euler characteristic. The construction is based on a groupoid structure of partial blow-up of the self product at the ramification divisor embedded diagonally.
Minhyong Kim
(University of Oxford and Seoul National University)
July 4(Thu) 11:20~12:10

Title: Diophantine geometry and non-abelian motives
A substantial portion of mainstream research in number theory over the past several decades has been concerned with the theory of motives, as manifested in the conjectures of Birch and Swinnerton-Dyer, Bloch, Beilinson, and Kato. In this lecture, we will interpret this theory as a linearized, `virtual' theory of Diophantine equations, and describe recents attempts to refine it using ideas of homotopy theory.