•**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)

•**Title &
Abstract**

** ****Speaker: Frank Pacard
(Ecole Polytechnique)**

**Title:** Solutions without any symmetry for some nonlinear problems arising from Physics
and Geometry.

**Abstract:** I will present the
construction of solutions for some nonlinear problem from Physics and Geometry
which have few or no symmetry. I will present examples of such
constructions for compact constant-mean curvature surfaces and minimal
surfaces with finite total curvature in Euclidean 3-space, solutions of the
nonlinear Schrodinger equation, the magnetic Ginzburg-Landau equations or the
Chern-Simons-Higgs equations in 2-dimension.

** **

**Speaker: François
Gay-Balmaz (Ecole Normale Superieure)**

**Title &Abstract: **TBA

** **

**Speaker: Xiaoguang
Wang (Zhejiang University)**

**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: Yvon Maday
(Sorbonne University)**

**Title & Abstract:** TBA

**Speaker: Gang Bao
(Zhejiang University)**

**Title &Abstract: **TBA

**Speaker: Wei Wang
(Zhejiang University)**

**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: Vincent
Giovangigli (Ecole Polytechnique)**

**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: Jing-Rebecca
Li (Ecole Polytechnique)**

**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: Qinghai
Zhang (Zhejiang University)**

**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)**

**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)**

**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.

**Wenshuai Jiang (Zhejiang University)**

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.

•**Agenda****:**