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# Second French-Korean Conference in Mathematics University of Bordeaux, Institut de Mathématiques July   7th -- 12th   2016 Summer school    July 4th -- 6th 2016

Program of the summer school

July 4 July 5 July 6
9:00 -- 9:30 Opening
 9:30 -- 11:00
 Main courseYoung-Hoon KiemSalle de Conférence
 CourseBergerSalle 1 CourseGérard-VaretSalle 2
 CourseKimSalle 2 CourseKiemSalle 1
11:00 -- 11:15 break break break
 11:15 -- 12:45
 Main courseYong-Jung KimSalle de Conférence
 ExercisesKimSalle 2 ExercisesKiemSalle 1
 CourseBergerSalle 1 CourseGérard-VaretSalle 2
12:45 -- 14:00 lunch lunch lunch
 14:00 -- 15:30
 Main courseLaurent BergerSalle de Conférence
 CourseKiemSalle 1 CourseKimSalle 2
 ExercisesKimSalle 2 ExercisesBergerSalle 1
15:30 -- 16:00 break break break
 16:00 -- 17:30
 Main courseDavid Gérard-VaretSalle de Conférence
 ExercisesGérard-VaretSalle 2 ExercisesBergerSalle 1
 ExercisesKiemSalle 1 ExercisesGérard-VaretSalle 2

Monday: every participant can attend all courses, no parallel sessions are planed. Tuesday and Wednesday: in parallel to the main courses, sessions of commented examples and exercises are organized ; you are strongly encouraged to interact with the lecturers.

Algebraic Geometry and Number Theory

1. Young-Hoon Kiem (Seoul National University):

Donaldson-Thomas invariants and critical virtual manifolds

The Donaldson-Thomas invariant (or the holomorphic Casson invariant) is a virtual count of simple holomorphic vector bundles on a Calabi-Yau 3-fold Y. This virtual counting is achieved either by integrating differential forms over a special cycle on the moduli space X of simple holomorphic vector bundles, called the virtual fundamental class, or by computing the topological Euler characteristic of X, weighted by the Behrend function. It is well known that the moduli space X is locally the critical locus of a holomorphic function f defined on a finite dimensional complex manifold V and the value of the Behrend function is given by the Milnor numbers of f. In this lecture series, I will talk about a recent joint work with Jun Li (arXiv: 1212.6444) where we showed that the local charts (V,f) for X can be glued to a geometric structure that we call a critical virtual manifold. The perverse sheaves of vanishing cycles for the local charts (V,f) glue to a globally defined perverse sheaf P on X which underlies a polarizable mixed Hodge module when the critical virtual manifold is orientable. As a consequence, we obtain a cohomology theory of X whose Euler characteristic equals the Donaldson-Thomas invariant and which gives us a mathematical theory of the Gopakumar-Vafa invariants. First I will show that the notion of a critical virtual manifold is a natural generalization of a complex manifold. After discussing various properties of critical virtual manifolds such as the Behrend function, weighted Euler characteristic and Donaldson-Thomas type invariant, I will talk about perverse sheaves of vanishing cycles and the gluing problem. Next, by gauge theory, we will see that any moduli space of simple holomorphic vector bundles is a critical virtual manifold. In the remaining time, I will talk about applications to Donaldson-Thomas theory and mirror symmetry.

2. Laurent Berger(Ecole Normale Supérieure, Lyon):

The study of p-adic dynamical systems was started around 1994 by Lubin, who proposed to "lay out some techniques for the study of the behavior of the iterates of a general p-adic analytic transformation". In this course, I will give some p-adic background, review some of Lubin's constructions and results, and discuss the relationship with the theory of formal groups.

Partial Differential Equations and Applications

1. Yong-Jung Kim (KAIST & NIMS, Daejeon):

Mathematics in Fokker-Planck type diffusion (with a focus on math biology)

Diffusion is one of the most fundamental processes in science and engineering. It also provides the most fundamental ingredient in PDE (partial differential equation). In this five hours course, 3X90minutes, we will discuss how Fokker-Planck type diffusions appear and what consequences emerge if the linear diffusion is replaced by a Fokker-Planck type diffusion. The course consists of the following items. 1) ODE versus PDE. 2) Derivations of a Fokker-Planck type diffusion as a diffusion limit or as an approximation of finite difference scheme. 3) Starvation driven diffusion as a survival strategy of biological organisms. 4) Chemotactic traveling waves and aggregation with Fokker-Planck type diffusions. 5) Random diffusion with food metric, but not with Euclidean metric.

2. David Gérard-Varet (Université Paris Diderot):

Mathematical analysis of boundary layers in viscous flows

We shall review in this course recent progress on the mathematical aspects of boundary layer theory. Background motivation is the understanding of Navier-Stokes flows near a rigid wall. At high Reynolds number, these flows exhibit concentration phenomena near the boundary. A formal asymptotics of such boundary layer flow was proposed by L. Prandtl more than a century ago, and has now become a standard in fluids mechanics. Still, the range of validity of this asymptotics is not well understood, due to many instability phenomena. These lectures will describe the mathematical approach on this problem, discussing notably the well-posedness of the boundary layer equations, as well as the stability of boundary layer flows in the Navier-Stokes evolution.

Program of the Conference

July 7 July 8 July 9
9:00 -- 9:50 Opening
10:00 -- 10:50
 Jong-HaeKeumAmphitheater
 Hyung JuHwangAmphitheater GuyHenniartSalle 1
 Young-PilChoiAmphitheater Sung RakChoiSalle 1
11:00 -- 11:50
 ClaudeZuilyAmphitheater
 Yong-JungKimAmphitheater JeehoonParkSalle 1
12:00 -- 13:30 lunch lunch lunch
13:30 -- 14:20
 ClaireVoisinAmphitheater LaurentDesvillettesSalle 1
 OlivierDebarreAmphitheater DonghoChaeSalle 1
14:30 -- 15:20
 DongSeonHwangAmphitheater ChristopheBesseSalle 1
 YoungJuChoieAmphitheater KyudongChoiSalle 1
15:30 -- 16:00 break break
16:00 -- 16:50
 HwajongYooAmphitheater RodolpheTurpaultSalle 1
 MinLeeAmphitheater Sung-JinOhSalle 1
17:00 -- 17:50
 Michelvan Garrel Amphitheater SunhoChoiSalle 1
 DavidHyeonAmphitheater ClairPoignardSalle 1

20:00 -- 22:00 Dinner banquet

### July 10: Visit of a Chateau

July 11 July 12
9:00 -- 9:50
10:00 -- 10:50
 GaëtanChenevierAmphitheater FrédériqueCharlesSalle 1
 MichelBrionAmphitheater DanielHan-KwanSalle 1
11:00 -- 11:50
 SijongKwakAmphitheater JihoonLeeSalle 1
 JunehyukJungAmphitheater OhsangKwonSalle 1
12:00 -- 13:30 lunch lunch
13:30 -- 14:20
 Seung YealHaAmphitheater FarrellBrumleySalle 1
14:30 -- 15:20
 DavidLannesAmphitheater Yong-GeunOhSalle 1
15:30 -- 16:00 break
16:00 -- 16:50
 Seok-BaeYunAmphitheater Ju-LeeKimSalle 1
17:00 -- 17:50
Christophe
Prange
Amphitheater
 Jinhyung Park Seung-Jo Jung Salle 1

Titles and abstracts (Algebraic Geometry and Number Theory)

1. Denis Benoit (Université de Bordeaux):

Iwasawa theory and $(\varphi, \Gamma)$-modules

We discuss some applications of Fontaine's theory of $(\varphi, \Gamma)$-modules to Iwasawa theory of p-adic representations.

2. Michel Brion (Institut Fourier, Grenoble):

Commutative algebraic groups up to isogeny

The commutative algebraic groups over a field $k$ are the objects of an abelian category $C$, with morphisms being the homomorphisms of algebraic $k$-groups. If $k$ is algebraically closed, then by work of Serre and Oort, the homological dimension of $C$ is 1 in characteristic 0 and 2 in positive characteristics. The talk will address the isogeny category of commutative algebraic groups, defined as the quotient of $C$ by the Serre subcategory $F$ of finite algebraic groups. In particular, we will show that the homological dimension of the isogeny category is 1 for any field $k$, and we will discuss analogies with some other categories of homological dimension 1 arising in algebraic geometry.

3. Farrell Brumley (Université Paris 13):

Periods and asymptotic growth of arithmetic eigenfunctions

Given a compact locally symmetric space $Y$ we are interested in the localization properties of sequences of eigenfunctions of the ring of invariant differential operators. When $Y$ is of non-compact type, quantum chaos suggests that such eigenstates should be delocalized. One concrete expression of this is that a generic sequence of $L^2$ normalized eigenfunctions should have small sup norm. We call these nicely behaved sequences tempered'', in analogy with the Ramanujan conjecture from the theory of automorphic forms. We would like to know under what conditions Y admits non-tempered sequences of eigenfunctions, i.e., those whose sup norm grows with a power of the eigenvalue. We answer this question in the arithmetic case, in terms of the recurrence properties of Hecke operators. Our techniques actually pick out the size of certain discrete periods through trace formula methods, and the criterion assuring growth can be read off from the Plancherel measure of an underlying symmetric space $G/H$. This is joint work with Simon Marshall.

4. Gaetan Chenevier (Université Paris-Sud, Orsay):

Algebraic modular forms of small weight for $SL_n(\mathbb{Z})$

In this talk, I will explain a classification of the modular forms for $SL_n(\mathbb{Z})$, for an arbitrary integer $n>0$, which are algebraic of motivic weight'' $<$ 23. I will also briefly discuss some consequences, either proved or conjectural, to other kind of objects, such as Siegel modular forms of weight $<$ 13, Niemeier lattices, or the Hasse-Weil zeta function of the moduli stack (over the integers) of stable curves of genus $g$ and $n$ marked points (for $3g-3+n< 23$). This is a joint work with Jean Lannes.

5. Sung Rak Choi (Yonsei University, Seoul):

Okounkov bodies associated to pseudoeffective divisors

The Okounkov bodies are convex bodies that encode rich information of divisors. I will explain how to extract some of them from the Okounkov bodies.

6. YoungJu Choie (POSTECH, Pohang):

Period of modular forms on $\Gamma_0(N)$ and products of Jacobi Theta functions

We give a closed formula for the sum of all Hecke eigenforms on $\Gamma_0(N)$ multiplied by their odd period polynomials in two variables, as a single product of Jacobi theta series for any squarefree level $N$. We also show that $N= 2, 3$ and $5$ this formula completely determines the Fourier expansions all Hecke eigenforms of all weights on $\Gamma_0(N)$. This is a Generalizing a result of Zagier in 1991 for modular forms of level one.

7. Olivier Debarre (Ecole Normale Supérieure, Paris):

Fake projective spaces and fake tori

Hirzebruch and Kodaira proved in 1957 that when $n$ is odd, any compact Kähler manifold $X$ which is homeomorphic to ${\mathbf P}^n$ is isomorphic to ${\mathbf P}^n$. This holds for all $n$ by Aubin and Yau's proofs of the Calabi conjecture. We discuss the conjecture that it should be sufficient to assume that the integral cohomology rings $H^\bullet(X,{\mathbf Z})$ and $H^\bullet({\mathbf P}^n,{\mathbf Z})$ are isomorphic.

In another direction, Catanese recently observed that complex tori are characterized among compact Käahler manifolds $X$ by the fact that their integral cohomology rings are exterior algebras on $H^1(X,{\mathbf Z})$ and asked whether this remains true under the weaker assumptions that the rational cohomology ring is an exterior algebra on $H^1(X,{\mathbf Q})$. We give a negative answer to Catanese's question by producing explicit examples. This is work in collaboration with Z. Jiang, M. Lahoz, and W. F. Sawin. If time permits, I will also discuss recent work of J. Chen, Z. Jiang, and Z. Tian on compact Käahler manifolds which have the same Hodge numbers as those of an abelian variety of the same dimension.

8. Mladen Dimitrov (Université de Lille 1):

Albanese of Picard modular surfaces and rational points

In a joint work with Dinakar Ramakrishnan, we prove that various arithmetic quotients of the unit ball in $\mathbb{C}^n$ are Mordellic, in the sense that they have only finitely many rational points over any finitely generated field extension of $\mathbb{Q}$. In the previously known case of compact hyperbolic complex surfaces, we give a new proof using their Albanese in conjunction with some key results of Faltings, but without appealing to the Shafarevich conjecture. In higher dimension, our methods allow us to solve an alternative of Ullmo and Yafaev. Our strongest result uses in addition Rogawski's theory and establishes the Mordellicity of the Baily-Borel compactifications of Picard modular surfaces of some precise levels related to the discriminant of the imaginary quadratic field.

9. Michel van Garrel (KIAS, Seoul):

Log BPS state count of del Pezzo surfaces

We introduce these counts and relate them to the famous local BPS state counts arising from physics. This is joint work with J. Choi and N. Takahashi.

10. Guy Henniart (Université Paris-Sud, Orsay):

Higher ramification and the local Langlands correspondence for $GL(n)$

This is joint work with C. Bushnell. Let $F$ be a locally compact non Archimedean field. Class field theory identifies characters of the absolute Galois group $G_F$ of $F$ and finite order characters of the multiplicative group of $F$. Two characters of $G_F$ with the same restriction to the higher ramification subgroup of $G_F$ with upper index $i$ correspond to two characters of $F*$ having the same restriction on the unit group $U_F^i$. We extend those results to the setting of the Langlands correspondence for $GL(n)$, which associates to an irreducible degree n representation of $G_F$ a supercuspidal representation of $GL(n,F)$. In that more general setting appears a new function relating ramification on the Galois side and ramification on the side of $GL(n)$.

11. DongSeon Hwang (Ajou University, Suwon):

Cascade structure on log del Pezzo surfaces of rank one

Every nonsingular del Pezzo surface is either the projective plane, the quadric surface, or the blow-up of the projective plane in at most 8 general points. In particular, every nonsingular del Pezzo surface admits a birational morphism to the projective plane or the quadric surface, which is a sequence of compositions of blow-downs. The resulting diagram, roughly, is called a cascade. The aim of the talk is to find a similar cascade structure for log del Pezzo surfaces of rank one. First, we completely describe the remaining cases of Miyanishi-Zhang's theory. By applying the Miyanish-Zhang's theory, we prove that every log del Pezzo surface of rank one admits a similar diagram, also called a cascade. As an application, we present an explicit method to enumerate all such surfaces.

12. David Hyeon (Seoul National University):

Commuting nilpotents modulo simultaneous conjugation and Hilbert scheme

Pairs of commuting nilpotent matrices have been extensively studied, especially from the view point of quivers. But the space of commuting nilpotents modulo simultaneous conjugation has not received any attention at all although it has a definite moduli theory flavor. Unlike the case of commuting nilpotents paired with a cyclic vector, the GIT is not well behaved in this case. I will explain how a moduli space'' can be constructed as a homogeneous space, and show that it is isomorphic to an open subscheme of a punctual Hilbert scheme. Over the field of complex numbers, thus constructed space is diffeomorphic to a direct sum of twisted tangent bundles over a projective space. Time permitting, I will also explain how the new development in GIT (of affine spaces modulo solvable groups) might possibly treat this case and produce a moduil space as a GIT quotient. This is a joint work with W. Haboush.

13. Junehyuk Jung (KAIST, Daejeon):

On nodal domains of eigenfunctions in chaotic quantum systems

It has been known for decades that on a flat torus or on a sphere, there exist sequences of eigenfunctions having bounded number of nodal domains. In contrast, for a manifold with chaotic geodesic flow, the number of nodal domains of eigenfunctions is expected to grow with the eigenvalue. In this talk, I explain how one can prove that this is indeed true for the Maass-Hecke eigenforms on a compact arithmetic triangles. This talk is based on joint work with S. Zelditch and S. Jang.

14. Seung-Jo Jung (KIAS, Seoul):

The Craw--Ishii Conjecture

In this talk, I propose a conjecture on quotient singularities and the moduli spaces of $G$-constellations. For a finite group $G$ in $GL_n$, a $G$-equivariant sheaf $F$ on $C^n$ is called a $G$-constellation if $H^0(F)$ is isomorphic to the regular representation of $G$ as a $G$-representation. In [Craw and Ishii, Duke 2004], Craw and Ishii proved that for a finite abelian group $G$ in $SL_3(C)$, every projective crepant resolution of $C^3/G$ is isomorphic to the fine moduli space of theta-stable $G$-constellations for some GIT parameter theta. The (generalised) Craw-Ishii conjecture says that for $G$ in $GL_3$, every relative (projective) minimal model of $C^3/G$ has a moduli interpretation using $G$-constellations. In this talk, I prove this conjecture in some cases.

15. JongHae Keum (KIAS, Seoul):

$K3$ surfaces are $2$-dimensional Calabi-Yau manifolds, generalizing elliptic curves

I will start with standard examples of $K3$ surfaces, then review basic result and recent progress on automorphisms of K3 surfaces, e.g., how to determine the finiteness of the full automorphism group of a given $K3$ surface, how to compute the automorphism group for some nice classes of $K3$ surfaces, what are the possible orders of automorphisms, etc. If time permits, dynamics on $K3$ surfaces by an automorphism will be discussed.

16. Ju-Lee Kim (MIT, Cambridge):

Asymptotic behavior of supercuspidal representations and Sato-Tate equidistribution for families

We establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at a given finite set of primes. For the tame supercuspidals, we prove the limit multiplicity property with error terms. Thereby we obtain a Sato-Tate equidistribution for the Hecke eigenvalues. The main new ingredient is to show that the orbital integrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connected reductive $p$-adic group tend to zero uniformly for every noncentral semisimple element. This is a joint work with Shin and Templier.

17. Sijong Kwak (KAIST, Daejeon):

Characterization of ACM varieties with linear resolutions and classification of varieties with extremal graded Betti numbers

I'd like to characterize the ACM varieties with $d$-linear resolution with respect to the degree, the number of defining equations and syzygies. In addition, varieties with extremal graded Betti numbers can also be classified . This is a generalization of some results due to Castelnuovo and Fano

18. Min Lee (University of Bristol):

Selberg trace formula as a Dirichlet series

We explore the idea of Conrey and Li of presenting the Selberg trace formula for Hecke operators, as a Dirichlet series. We explore the idea of Conrey and Li presenting the Selberg trace formula for Hecke operators, as a Dirichlet series. We enhance their work in few ways and present several applications of our formula. This is a joint work with Andrew Booker.

19. Yong-Geun Oh (IBS-CGP, Pohang):

Lipschitz-exact Lagrangian submanifolds and Tonelli Hamiltonian

In this talk, we will introduce the notion of Lipschitz-exact Lagrangian submanifolds and prove that any such Larangian admits a graph selector. Then we explain How this can be used to generalize Arnaud's result to the class of Lipschitz-exact Lagrangians: any such Lagrangian submanifold must be a graph provided it is invariant under a Tonelli Hamiltonian. This is based on the joint work with Amorim and Oliveira Dos Santos.

20. Jeehoon Park (POSTECH, Pohang):

Deformations for period matrices of smooth projective complete intersections

Period matrices of algebraic varieties are important invariants which are defined as the matrices of the integrals of their de-Rham cohomology classes over singular homology classes. In this talk, we will present an explicit algorithmic formula between the period matrices of two algebraically different smooth projective complete intersection varieties with the same degree and same dimension (topologically same). Our method is based on the theory of algebraic Dwork complexes, DGBV (differential Gerstenhaber Batalin-Vilkovisky) algebras, and homotopy Lie theory (so called, $L_\infty$-algebras and $L_\infty$-morphisms). This is a joint work with Yesule Kim (POSTECH).

21. Jinhyung Park (KIAS, Seoul):

Moving Seshadri constants via Okounkov bodies

Seshadri constants measure local positivity of divisors, and bounding such constants is an interesting problem. The moving Seshadri constant of a divisor at a point on a variety is an important asymptotic invariant, which is a generalization of the usual Seshadri constant of an ample divisor. In this talk, I give lower and upper bounds for moving Seshadri constants by the size of simplexes contained in Okounkov bodies. This is joint work with Sung Rak Choi, Yoonsuk Hyun, and Joonyeong Won.

22. Claire Voisin (Collège de France, Paris):

New stable birational invariants

One classical question in algebraic geometry is whether a unirational variety is rational. This has been solved negatively in the 70's. For the stable variant of this problem, a topological invariant was exhibited by Artin-Mumford. This invariant unfortunately is trivial for most classes of Fano varieties. I used recently the notion of decomposition of the diagonal as a stable birational invariant and a degeneration method in order to solve negatively the stable version of Lüuroth problem in these remaining cases.

23. hwajong Yoo (IBS-CGP, Pohang):

The kernel of Eisenstein ideals on modular Jacobian varieties

For an Eisenstein prime $m$ of the Hecke algebra of level $N$, we give a recipe to compute the dimension of its kernel, $J_0(N)[m]$ (under a mild assumption).

Titles and abstracts (Partial Differential Equations and Applications)

1. Christophe Besse (Université Toulouse 3):

Transparent boundary conditions for the linearized Benjamin-Bona-Mahony equation

The Benjamin-Bona-Mahony (BBM) equation is a classical nonlinear, dispersive equation which model the unidirectional propagation of weakly nonlinear, long waves in the presence of dispersion. It is usually proposed as an analytically advantageous alternative to the well-known Korteweg-de Vries equation. We consider various approximations of transparent boundary conditions (TBC) for linearized BBM equation. In this talk, we derive explicit TBCs both continuous and discrete for the linearized BBM equation. The equation is discretized with the Crank Nicolson time discretization scheme and we focus on the difference between the upwind and the centered discretization of the convection term. We use these boundary conditions to compute solutions with compact support in the computational domain and also in the case of an incoming plane wave which is an exact solution of the linearized BBM equation.

2. Dongho Chae (Chung-Ang University, Seoul):

On the Hall-MHD equations

In this talk we discuss the the Cauchy problem of the Hall-magnetohydrodynamic system, where the "Hall term" is added to the usual incompressible MHD equations. This is a mathematical modeling equation of the motion of plasma with strong shears in the magnetic field in such case of motion of the solar flares in the astrophysics. After surveying recent results of the problem, we focus on the issue of the finite time singularity when the resistivity constant is zero. In this case we show that there exists a smooth initial data for which either the Cauchy problem is locally ill-posed, or it is locally well-posed but the apparition of finite time singularity happens.

3. Frédérique Charles (UPMC, Paris):

From particle methods to hybrid semi-Lagrangian schemes for transport equations

Particle methods for transport equations consist in pushing forward particles along the characteristic lines of the flow, and to describe then the transported density as a sum of weighted and smoothed particles. Conceptually simple, standard particle methods have the main drawback to produce noisy solutions or to require frequent remapping. In this talk we present two classes of particle methods which aim at improving the accuracy of the numerical approximations with a minimal amount of smoothing.
The idea of the Linearly Transformed Particle method is to transform the shape functions of particles in order to follow the local variation of the flow. This method has been adapted and analyzed for the Vlasov- Poisson system (joint with M. Campos-Pinto) and for a compressible aggregation equation (joint work with M. Campos-Pinto, J-A Carrillo, Y-P Choi). In both cases the error estimate is improved compared to classical particle methods, with the gain of a strong convergence of the numerical solution.
However, for long remapping periods, shapes of particles could become to much stretched out. The second method solve this problem of locality by combining a backward semi-Lagrangian approach and local linearizations of the flow. The convergence properties are improved and validated by numerical experiments (joint with M. Campos-Pinto) .

4. Kyudong Choi (UNIST, Ulsan):

On the Finite-Time Blowup of 1D Models for the 3D Axisymmetric Euler Equations

In connection with the recent proposal for possible singularity formation at the boundary for solutions of 3d axi-symmetric incompressible Euler's equations (Luo and Hou, PNAS, 2014), we study models for the dynamics at the boundary and show that they exhibit a finite-time blow-up from smooth data. This is joint work with T. Hou, A. Kiselev, G. Luo, V. Sverak, and Y. Yao.

5. Sunho Choi (Kyunghee University, Seoul):

Traveling wave solution on a kinetic model derived by food metric

I will present a mesoscopic scale model for chemotaxis induced by food metric. This model consists of separated density distributions for discrete velocities and this is a counter part of kinetic model corresponding to the diffusive model. A distance system that counts the amount of resources such as food could be meaningful in many cases. The migration distance of biological organisms that is measured by the amount of food between two points lead a new chemotaxis model. In the last work, It was shown that, if the length of the random walk is given by such a metric, the well-known traveling wave phenomena of the chemotaxis theory can be obtained without the typical assumption that microscopic scale bacteria may sense the macroscopic scale gradient chemical concentration. In this talk, I present the same traveling wave phenomena in kinetic model. This is joint work with Yong-Jung Kim.

6. Young-Pil Choi (Technische Universität, Munchen):

Mathematical modeling of flocking behavior

Emergent aggregation and flocking phenomena appearing in many biological systems are simple instances of collective behavior. Recently, they have been an active research in applied mathematics, biology, engineering, and physics. In this talk, we present several different types of mathematical models describing flocking behaviors from microscopic to macroscopic descriptions. For kinetic models, we discuss the well-posedness, large-time behavior of solutions and mean-field limits. We also study the critical thresholds for hydrodynamic models.

7. Thierry Colin (Université de Bordeaux):

Inverse problem for tumor growth modeling with clinical data

A huge number of mathematical/numerical models of tumor growth are available in the literature. Most of them aim at integrating an increasing amount of biological/medical knowledges. These models are able to account at least qualitatively for several complex phenomenas (angiogenesis, influence of particular molecular pathway, effects of targeted therapies, ...). They could be useful for clinical applications in order to help to understand the evolution of the disease or the response to the treatment in a personalized clinical context. The challenge is therefore to be able to obtain a parametrization of the models with the available data. If we restrict ourself to a clinical context the information is scarce. It consists mainly in the nature of the cancer that is known thanks to biological exams (blood samples, biopsies) and also to imaging data (CT-scans, MRI, PET-scans). The model has therefore to be designed according to the nature of the cancer, its localization but also according to the available imaging data. The images will give information on the volume, but also on the shape and the metabolism of the tumor (thanks to functional imaging technics like perfusion MRI or CT-scans). Moreover, for a particular patient, we often have several successive exams at different times. We therefore have to solve a complex inverse problem in order to be able to give a forecast of the progression of the disease or of the answer to a treatment. As far as in vivo or in vitro experiments are concerned, the same kind of problems appear except that it is easier to have homogenous populations of tumors and also more precise quantitative informations. In this talk, I will present three examples of such inverse problems in a clinical context. For lung metastases, meningiomas, the challenge is to be able to give some forecast of the evolution of the disease for patients that have no treatment in order to help to understand what could be the best moment for starting some therapy. The third case concern liver metastases of gastro-intestinal stromal tumors in the presence of targeted therapies or anti-angiogenic drugs.

My collaborators in this project are: The Institut Bergonié in Bordeaux: Dr. Xavier Buy, Guy Kantor, Michèle Kind, Jean Palussière. The data concerning lung metastases, liver metastases are provided by Institut Bergonié. CHU of Bordeaux (Bordeaux University Hospital): Pr. Hugues Loiseau, Dr. François Cornelis. The data concerning meningiomas are provided by the CHU of Bordeaux

8. Laurent Desvillettes (Université Paris Diderot):

Large time behavior of reaction-diffusion equations coming out of chemical reaction networks

We present in this talk the results obtained lately in collaboration with Klemens Fellner and Bao Quoc Tang on the study of chemical reaction networks for species diffusing in a domain. We present results specific to the case of so-called "complex balance equilibria" which naturally appear in many chemical networks, and we focus on the treatment of so-called "boundary equilibria".

9. Seung-Yeal Ha (Seoul National University):

Recent progress on the classical and quantum synchronization

In this talk, I will review on the complete synchronization problem for the classical Kuramoto model and quantum Lohe model which can be regarded as a non-abelian generalization of the Kuramoto model. We discuss several sufficient conditions for the complete synchronization of Kuramoto and Lohe oscillators in terms of coupling strength and initial configurations.

10. Daniel Han-Kwan (Ecole Polytechnique):

On the quasineutral limit of the Vlasov-Poisson system

We are interested in the behaviour of solutions to the Vlasov-Poisson system in the regime of small Debye length (a problem referred to as the quasineutral limit). The formal limit is a singular Vlasov equation exhibiting strong instabilities that may lead to ill-posedness. We shall present a joint work with F. Rousset in which we justify the limit for data that are stable in some sense (more precisely, we require a pointwise Penrose stability condition).

11. Hyung Ju Hwang (POSTECH, Pohang):

Regularity vs singularity in kinetic equations

We describe the structure of solutions of kinetic equations in domains with boundaries near the grazing set. Representative equations are the Vlasov-Poisson, the kinetic Fokker-Planck equations. We discuss in particular regularity, singularity, and the behavior of the solutions of these equations with various boundary conditions.

12. Yong-Jung Kim (KAIST & NIMS, Daejeon):

Introducing Fokker-Planck type diffusion in population models

If a space variable is introduced to an ODE model, the migration of species should be introduced to give it a meaning. If the migration is a directed movement, an advection term is usually introduced. If not, a diffusion is introduced. In this talk we will introduce another option that can combine the both, which is given by a Fokker-Planck type diffusion. Using it we may obtain a better and simpler analysis. Some of examples are given related to chemotactic aggregation and traveling wave phenomena.

13. Ohsang Kwon (Chungbuk National University):

Evolution of dispersal with starvation measure and coexistence

Many biological species increase their dispersal rate if starvation starts. To model such a behavior, we need to understand how organisms measure starvation and response to it. In this talk, we compare three di erent ways of measuring starvation by applying them to starvation-driven di usion. The evolutional selection and coexistence of such starvationmeasures are studied within the context of LotkaVolterra-type competition model of two species. We will see that, if species have di erent starvation measures and di erent motility functions, both the coexistence and selection are possible. This is the joint work with Y.-J. Kim.

14. David Lannes (Université de Bordeaux):

On the dynamics of floating structures

The goal of this talk is to derive some equations describing the interaction of a floating solid structure and the surface of a perfect fluid. This is a double free boundary problem since in addition to the water waves problem (determining the free boundary of the fluid region), one has to find the evolution of the contact line between the solid and the surface of the water. The so-called floating body problem has been studied so far as a three-dimensional problem. Our first goal is to reduce it to a two-dimensional problem that takes the form of a coupled compressible-incompressible system. We will also show that the hydrodynamic forces acting on the solid can be partly put under the form of an added mass-inertia matrix, which turns out to be affected by the dispersive terms of the equations.

15. Jihoon Lee (Chung-Ang University, Seoul):

Global existence and asymptotic behaviors of solutions for an aerotaxis model coupled to fluid equations

In this talk, we consider a coupled system of Keller-Segel type equations and the incompressible Navier-Stokes equations in spatial dimension two. We consider unique existence of a local-in-time solution, a blowup criterion, and the existence of a global-in-time solution with small initial values. We show temporal decay estimates of solutions with small initial data and obtain their asymptotic profiles as time tends to infinity. This is the joint work with Myeongju Chae and Kyungkeun Kang.

16. Luc Mieussens (Université de Bordeaux):

Local velocity grids for deterministic simulations of rarefied flows

Most deterministic numerical methods for rarefied gas dynamics are based on a common idea: the Boltzmann equation is discretized with a finit set of discrete velocities. This set is generally given by a global Cartesian velocity grid, which is the same grid for every point in the physical domain, and for every time. The advantages of this approach is its simplicity, and the strong mathematical properties that inherits the discrete model from the continuous one (stability, positivity, etc.). This is due to the fact that all the distribution functions are discretized on the same grid. However, for some practical problems with strong variations of macroscopic temperature and velocity fields, like in atmospheric re-entry flows, this approach is very expensive: the discrete velocity grid must be very large and very thin in order to capture all the different distribution functions.

In this talk, I will present a method designed for unsteady flows in which it is even difficult a priori to know which grid should be used. This method uses a discretization of the kinetic equation on local velocity grids: these grids dynamically adapt in time and space to the variations of the width of the distribution functions. Contrary to other recent works, even the bounds of each local grids can vary in time and space. This work is a collaboration with S. Brull and L. Forestier-Coste.

17. Sung-Jin Oh (KIAS, Seoul):

Linear instability of the Cauchy horizon in subextremal Reissner-Nordström spacetime under scalar perturbations

We consider the linear scalar wave equation on a fixed subextremal Reissner-Nordström spacetime with non-vanishing charge. We show that generic smooth and compactly supported initial data on a Cauchy hypersurface give rise to solutions with infinite nondegenerate energy near the Cauchy horizon in the interior of the black hole. This instability is related to the celebrated blue shift effect in the interior of the black hole. The problem is motivated by the strong cosmic censorship conjecture and it is expected that for the full nonlinear Einstein-Maxwell system, this instability leads to a singular Cauchy horizon for generic small perturbations of Reissner-Nordström spacetime. This is a joint work with J. Luk (Cambridge).

18. Clair Poignard (Université de Bordeaux):

Free-boundary problem for cell protrusion formation

In our talk, we present a free boundary problem for cell protrusion formation in which the cell membrane is precisely described thanks to a level set function, whose motion is due to specific signalling pathways. The aim is to model the chemical interactions between the cell and its environment, in the process of invadopodia or pseudopodia formation. The model consists in Laplace equation with Dirichlet condition inside the cell coupled to Laplace equation with Neumann condition in the outer domain. The actin polymerization is accounted for as the gradient of the inner signal, which drives the motion of the interface. We prove the well-posedness of our free boundary problem under a sign condition on the datum. This criterion ensures the consistency of the model, and provides conditions to focus on for any enrichment of the model. We then propose a new second order Cartesian finite-difference method to solve the problem. We eventually exhibit the main biological features that can be accounted for by the model: the formation of thin and elongated protrusions as for invadopodia, or larger protrusion as for pseudopodia, depending on the source term in the equation. Joint work with O. Gallinato, M. Ohta (TUS, Tokyo), T. Suzuki (Osaka Univ.)

19. Christophe Prange (Université de Bordeaux):

Couches limites en homogénéisation

In this talk we will review some recent results about the homogenization of elliptic equations with oscillating Dirichlet boundary conditions.

20. Rudolphe Turpault (Université de Bordeaux):

High-order asymptotic and admissibility preserving schemes for systems of conservation laws with diffusive limit on unstructured meshes

The aim of this talk is to introduce a numerical procedure which allows to obtain numerical schemes suited for a class of systems of conservation laws with diffusive limit. Such schemes are expected to preserve both the set of admissible states and the asymptotic in the diffusive limit. Furthermore, one usually requires high order in order to improve the accuracy of the approximation. In 1D, several techniques have been proposed during the last 10-15 years to adress this issue. In higher dimension however, the problem is much more difficult. I will present a technique which is generic for any system in the relevent class and show its behavior on various exemples (Euler with friction, M1 model for radiative transfer, etc.)

21. Seok-Bae Yun (Sungkyunkwan University, Seoul):

Convergence of a semi-Lagrangian scheme for the ellipsoidal BGK model

The ellipsoidal BGK model is a generalized version of the original BGK model, designed to reproduce the correct Prandtl number in the Navier-Stokes limit. In this talk, we propose an implicit semi-Lagrangian scheme for this model, which, utilizing the special structure of the ellipsoidal Gaussian, can be computed in an almost explicit manner. We then show that the discrete solution computed from this scheme converges to the continuous solution in a weighted L^{\infty} norm. This is a joint work with Giovanni Russo.

22. Claude Zuily (Université Paris-Sud, Orsay):

Concentration of Laplace eigenfunctions and stabilization of weakly damped wave equation

In this talk, we shall discuss some universal bounds on the speed of concentration on small (frequency-dependent) neighbourhoods of submanifolds of $L^2$-norms of quasi modes for Laplace operators on compact manifolds. We deduce new results on the rate of decay of weakly damped wave equations. This is a joint work with Nicolas Burq.

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