•Time: September 25-28, 2019 (Arrival on September 24, Tuesday)

•Location: Institute for Advanced Study in Mathematics, Zijingang Campus, Zhejiang University (ZJU), Hangzhou, China

•Participating Institutions: Ecole Polytechnique, Sorbonne Universite, Ecole Normale Superieure, MINES ParisTech and Zhejiang University

•Organize Committee (in alphabetical order):

Gang BAO (Zhejiang University)

Thierry CAZENAVE (Sorbonne University）

Gaëlle LE GOFF (Ecole Polytechnique)

Chuanhou GAO (Zhejiang University)

Min LI (Zhejiang University)

Weimin SHENG (Zhejiang University)

Xiang XU (Zhejiang University)

Ting ZHANG (Zhejiang University)

•Goal: French mathematics ranks among the top in the world. By 2018, French mathematicians account for 12 of the total 60 Fields Medal Winners. Almost all of the Fields Laureates have studied or taught in Paris, making the top institutions there the cradle of the world’s best mathematicians. ZJU is one of the top universities in China, with excellent students and a distinguished tradition of mathematical research. ZJU’s "Chen-Su School" in the 1940s enjoys a high reputation in the international mathematical community. Currently, ZJU is striving to become a world-class university with first-class mathematics. This forum aims to promote all-round, strategic cooperation and exchange between ZJU and the top institutions in Paris in mathematical education and research.

•Invited Speakers (in alphabetical order):

Alessandro Chiodo (Sorbonne University)

François Gay-Balmaz (Ecole Normale Superieure)

Vincent Giovangigli (Ecole Polytechnique)

Jing Rebecca Li (Ecole Polytechnique)

Yvon Maday (Sorbonne University)

Frank Pacard (Ecole Polytechnique)

Yongbin RUAN (Zhejiang University, University of Michigan)

Speakers from Zhejiang University (to be confirmed)

•Agenda

•Title & Abstract

Speaker: François Gay-Balmaz (Ecole Normale Superieure)

francois.gay-balmaz@lmd.ens.fr

Title: Geometric discretization and geometric modeling in continuum mechanics

Abstract: In this talk we review some recent progresses made in the discretization and modelling of fluid dynamics by using geometric variational methods.

In the first part, we use the geometric variational formulation of reversible fluid dynamics on diffeomorphism groups to derive a structure preserving finite element discretisation for several fluid models used in geophysical fluid dynamics. Given a triangulation on the fluid domain, the discrete group of diffeomorphisms is defined as a certain subgroup of the group of linear isomorphisms of a finite element space of functions. As a consequence of their structure preserving nature, the resulting schemes exhibit an excellent long term energy behavior and respect several conservation laws of the continuous system. We illustrate these properties with some examples from geophysical fluid dynamics.

In the second part, we extend the variational formulation of reversible fluids to the setting of nonequilibrium thermodynamics. The resulting variational formulation extends the classical Hamilton principle to include irreversible processes such as viscosity, diffusion, and heat transfer. We illustrate the efficiency of our variational formulation as a modeling tool by treating some examples from geophysical fluid dynamics.

Speaker: Qirui Li (Zhejiang University)

qi-rui.li@zju.edu.cn

Title:Regularity in Monge’s mass transport problem

Abstract: The optimal transportation problem was introduced by a French mathematician Gaspard Monge in 1781. Since then the problem has been extensively studied and more general costs are allowed. But for Monge's original cost, very little is known about the regularity. In this talk, we discuss the regularity in Monge's problem, and in particular show that, in two dimensional case, the optimal mapping is continuous. The talk is based on joint works with F. Santambrogio and X.-J. Wang.

Speaker: Xiaoguang Wang (Zhejiang University)

wxg688@163.com

Title: Newton's methods for polynomials: a dynamical system viewpoint

Abstract: The talk consists of two parts. In the first part, I will give a brief introduction to the research works of our dynamical system research group.  In the second part, I will discuss the Newton's methods for finding roots of polynomials, from its history to recent progress.

Speaker: Vincent Giovangigli (Ecole Polytechnique)

vincent.giovangigli@polytechnique.edu

Title: Relaxation of internal energy and volume viscosity
Abstract: We investigate the fast relaxation of translational and internal temperatures in nonequilibrium gas models derived from the kinetic theory. Strong solutions are investigated in the fast relaxation limit for ill prepared initial data. In the fast relaxation limit the difference between the translational and equilibrium temperatures becomes asymptotically proportional to the divergence of the velocity field. This yields the volume viscosity term of the limiting one-temperature equilibrium fluid model. Numerical simulations are finally presented of the impact of volume viscosity during a shock/hydrogen bubble interaction.

Speaker: Wei Wang (Zhejiang University)

wangw07@zju.edu.cn

Title: On the stability of current-vortex sheets in ideal incompressible magneto-hydrodynamics

Abstract: In the first part of this talk, we will give a brief introduction to the members and research works of the PDE group in ZJU. In the second part, we will discuss the stability of current-vortex sheets in ideal incompressible magneto-hydrodynamics. It is well-known that vortex sheets for incompressible Euler equations are not stable (called Kelvin-Helmholtz instability). However, in 1953, Syrovatskij derived a stability condition which indicates that when the magnetic field is sufficiently strong, current-vortex sheets for magneto-hydrodynamics could probably be stable. We will present the local-in-time existence result of the solution for the incompressible current-vortex sheets under Syrovatskij's stability condition, which gives a rigorous confirmation of the stabilizing effect of the magnetic field on the Kelvin-Helmholtz instability.

Speaker: Gang Bao (Zhejiang University)

baog@zju.edu.cn

Title &Abstract: TBA

Speaker: Jing-Rebecca Li (Ecole Polytechnique)

jingrebecca.li@inria.fr

Title: Mathematical methods for diffusion magnetic resonance imaging (dMRI)

Abstract: The complex-valued transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium can be modeled by the Bloch-Torrey partial differential equation with discontinuous interface conditions.  The diffusion MRI modeling problem is about quantifying tissue cell structure and membrane permeability from voxel level signals in multiple diffusion directions, diffusion times, and gradient magnitudes.  I will describe a Matlab-based simulation toolbox called SpinDoctor that we developed to solve the forward problem and our recent progress towards the inverse problem and parameters estimation.

Speaker: Jun Lai (Zhejiang University)

laijun6@zju.edu.cn

Title & Abstract: TBA

Speaker: Qinghai Zhang (Zhejiang University)

qinghai@zju.edu.cn

Title: MARS: An Analytic and Computational Framework for Incompressible Flows with Moving Boundaries

Abstract: Current methods such as VOF methods and level-set methods avoid geometry and topology by converting them into problems of numerical PDEs. In comparison, we try to tackle geometric and topological problems with tools in geometry and topology. The first part of our MARS framework is the Yin space, a mathematical model for physically meaningful material regions. Each element in the Yin space is a Yin set, a regular open semianalytic set with bounded boundaries.  Each Yin set is represented by a poset of oriented Jordan curves so that its topological information (such as the Betti numbers of a material region) can be extracted in constant time. We further equip the Yin space with a simple Boolean algebra that is efficient and complete for arbitrarily complex topology.
In particular, non-manifold points on the fluid boundary, a key problem in studying topological changes, are handled naturally. The second part of MARS is the donating region theory in the context of hyperbolic conservation laws. For a fixed simple curve in a nonautonomous flow, the fluxing index of a passively advected Lagrangian particle is the total number of times it goes across the curve within a given time interval. Such indices naturally induce donating regions, equivalence classes of the particles at the initial time. Under the MARS framework, many explicit methods such as VOF methods and fronting tracking methods can be unified and proved to be second-order accurate. MARS also leads to new methods of fourth- and higher-order accuracy for interface tracking and curvature estimation.
The MARS framework can be further expanded with a fourth-order projection method called GePUP for numerically solving the incompressible Navier-Stokes equations (INSE). We have augmented GePUP to irregular domains and are currently working on coupling GePUP with our new interface tracking methods to form a fourth-order solver for INSE with moving boundaries.

Speaker: Yongbin Ruan (Zhejiang university/ University of Michigan)

ruan@umich.edu

Title: Verlinde/Grassmannian correspondence and quantum K-theory

Abstract: More than twenty years ago, Witten proposed an equivalence of two quantum fields governing Verlinde algebra (or the theory of stable bundles over a curve) and the quantum cohomology of Grassmannian. Motivated by Witten’s physical work and recent revival of quantum K-theory, we proposed a K-theoretic version of so-called Verlinde/Grassmannian correspondence. Furthermore, the recent interpretation of quantum K-theory as a 3d quantum field theory opens a door

to much larger area of physics and mathematics. We will first review the new ingredient of level structure in quantum K-theory and surprising appearance of mock theta function. Then, we will present an approach to the proof of correspondence using wall-crossing technique. This is a joint work with Ming Zhang.

Speaker: Alessandro Chiodo (Sorbonne University)

alessandro.chiodo@imj-prg.fr

Title: Spin graphs and quadratic forms

Abstract: Graphs are elementary objects in combinatorics for which a deep theory of divisors, ranks and Riemann-Roch formulae has been developed in full analogy with the theory of Riemann surfaces. In many ways spin graphs lack an analogous treatment. For Riemann surfaces the rank of spin structures exhibit a beautiful dichotomy between even and odd structures governed by a quadratic form. For graphs, the picture so far only exhibited one distinguished spin structure, but the quadratic form does not generalize. We study thick graphs (graph with thickened edges) which shed new light on the theory of ranks of graph. They allow us to provide new formulae for the ranks in the classical case. Finally they allow us to single out a class of (hyperelliptic) graphs where the theory works exactly as it does for Riemann surfaces. This is work in progress with Marco Pacini.

Speaker: Wenshuai Jiang (Zhejiang University)

wsjiang@zju.edu.cn

Title: On the manifolds with Ricci curvature bounds

Abstract: In the first part of the talk, we will introduce our differential geometry group(W. Sheng, Q. Xia, J. Wu, F. Wang, Q. Li ) and briefly discuss some works of them.  In the second part, we will discuss the study of manifolds with Ricci curvature bounds which is based on jointed work with Jeff Cheeger and Aaron Naber.