Program of the summer school

July 4

July 5

July 6

9:00  9:30 
Opening 


Main course YoungHoon Kiem Salle de Conférence


Course Berger Salle 1 
Course GérardVaret Salle 2 

 Course Kim Salle 2 
Course Kiem Salle 1 

11:00  11:15 
break 
break 
break 

Main course YongJung Kim Salle de Conférence


 Exercises Kim Salle 2 
Exercises Kiem Salle 1 

Course Berger Salle 1 
Course GérardVaret Salle 2 

12:45  14:00 
lunch 
lunch 
lunch 

Main course Laurent Berger Salle de Conférence


Course Kiem Salle 1 
Course Kim Salle 2 

 Exercises Kim Salle 2 
Exercises Berger Salle 1 

15:30  16:00 
break 
break 
break 

Main course David GérardVaret Salle de Conférence


 Exercises GérardVaret Salle 2 
Exercises Berger Salle 1 

Exercises Kiem Salle 1 
Exercises GérardVaret Salle 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

YoungHoon Kiem (Seoul National University):
DonaldsonThomas invariants and critical virtual manifolds
The DonaldsonThomas invariant (or the holomorphic Casson invariant) is a virtual count of simple holomorphic vector bundles on a CalabiYau 3fold 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 DonaldsonThomas invariant and which gives us a mathematical theory of the GopakumarVafa 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 DonaldsonThomas 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 DonaldsonThomas theory and mirror symmetry.

Laurent Berger(Ecole Normale Supérieure, Lyon):
Padic dynamical systems
The study of padic 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 padic analytic transformation". In this course, I will give some padic background, review some of Lubin's constructions and results, and discuss the relationship with the theory of formal groups.
Partial Differential Equations and Applications

YongJung Kim (KAIST & NIMS, Daejeon):
Mathematics in FokkerPlanck 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 FokkerPlanck type diffusions appear and what consequences emerge if the linear diffusion is replaced by a FokkerPlanck type diffusion. The course consists of the following items. 1) ODE versus PDE. 2) Derivations of a FokkerPlanck 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 FokkerPlanck type diffusions. 5) Random diffusion with food metric, but not with Euclidean metric.

David GérardVaret (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 NavierStokes 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 wellposedness of the boundary layer equations, as well as the stability of boundary layer flows in the NavierStokes evolution.
Program of the Conference

July 7

July 8

July 9

9:00  9:50

Opening


10:00  10:50


Hyung Ju Hwang Amphitheater

Guy Henniart Salle 1


YoungPil Choi Amphitheater

Sung Rak Choi Salle 1


11:00  11:50


YongJung Kim Amphitheater

Jeehoon Park Salle 1


Thierry Colin Amphitheater

Mladen Dimitrov Salle 1


12:00  13:30

lunch

lunch

lunch

13:30  14:20

Claire Voisin Amphitheater

Laurent Desvillettes Salle 1


Olivier Debarre Amphitheater

Dongho Chae Salle 1



14:30  15:20

DongSeon Hwang Amphitheater

Christophe Besse Salle 1


YoungJu Choie Amphitheater

Kyudong Choi Salle 1


15:30  16:00

break

break

16:00  16:50

Hwajong Yoo Amphitheater

Rodolphe Turpault Salle 1


Min Lee Amphitheater

SungJin Oh Salle 1


17:00  17:50

Michel van Garrel Amphitheater

Sunho Choi Salle 1


David Hyeon Amphitheater

Clair Poignard Salle 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

Gaëtan Chenevier Amphitheater

Frédérique Charles Salle 1


Michel Brion Amphitheater

Daniel HanKwan Salle 1


11:00  11:50

Sijong Kwak Amphitheater

Jihoon Lee Salle 1


Junehyuk Jung Amphitheater

Ohsang Kwon Salle 1


12:00  13:30

lunch

lunch

13:30  14:20

Seung Yeal Ha Amphitheater

Farrell Brumley Salle 1



14:30  15:20

David Lannes Amphitheater

YongGeun Oh Salle 1


15:30  16:00

break

16:00  16:50

SeokBae Yun Amphitheater

JuLee Kim Salle 1


17:00  17:50

Christophe Prange Amphitheater

Jinhyung Park 
SeungJo Jung 
Salle 1 


Titles and abstracts (Algebraic Geometry and Number Theory)

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 padic representations.

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.

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 noncompact 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 nontempered 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.

Gaetan Chenevier (Université ParisSud, 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 HasseWeil zeta function of the moduli stack (over the integers) of stable curves of genus $g$ and $n$ marked points (for $3g3+n< 23$). This is a joint work with Jean Lannes.

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.

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.

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.

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 BailyBorel compactifications of Picard modular surfaces of some precise levels related to the discriminant of the imaginary quadratic field.

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.

Guy Henniart (Université ParisSud, 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 $GL(n)$.

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 blowup 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 blowdowns. 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
MiyanishiZhang's theory. By applying the MiyanishZhang'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.

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.

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 MaassHecke eigenforms on a compact arithmetic triangles. This talk is based on joint work with S. Zelditch and S. Jang.

SeungJo Jung (KIAS, Seoul):
The CrawIshii 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 thetastable $G$constellations for some GIT parameter theta. The (generalised) CrawIshii 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.

JongHae Keum (KIAS, Seoul):
$K3$ surfaces are $2$dimensional CalabiYau 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.

JuLee Kim (MIT, Cambridge):
Asymptotic behavior of supercuspidal representations and SatoTate 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 SatoTate 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.

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

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.

YongGeun Oh (IBSCGP, Pohang):
Lipschitzexact Lagrangian submanifolds and Tonelli Hamiltonian
In this talk, we will introduce the notion of Lipschitzexact
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 Lipschitzexact 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.

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 deRham 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 BatalinVilkovisky) algebras,
and homotopy Lie theory (so called, $L_\infty$algebras and $L_\infty$morphisms).
This is a joint work with Yesule Kim (POSTECH).

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.

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 ArtinMumford. 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.

hwajong Yoo (IBSCGP, 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)

Christophe Besse (Université Toulouse 3):
Transparent boundary conditions for the linearized BenjaminBonaMahony equation
The BenjaminBonaMahony (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 wellknown Kortewegde 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.

Dongho Chae (ChungAng University, Seoul):
On the HallMHD equations
In this talk we discuss the
the Cauchy problem of the Hallmagnetohydrodynamic 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 illposed, or it is locally wellposed but the apparition of finite time singularity happens.

Frédérique Charles (UPMC, Paris):
From particle methods to hybrid semiLagrangian 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. CamposPinto) and for a compressible aggregation equation (joint work with M. CamposPinto, JA Carrillo, YP 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 semiLagrangian approach and local linearizations of the flow. The convergence properties are improved and validated by numerical experiments (joint with M. CamposPinto) .

Kyudong Choi (UNIST, Ulsan):
On the FiniteTime 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 axisymmetric incompressible Euler's equations (Luo and Hou, PNAS, 2014), we study models for the dynamics at the boundary and show that they exhibit a finitetime blowup from smooth data. This is joint work with T. Hou, A. Kiselev, G. Luo, V. Sverak, and Y. Yao.

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 wellknown 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 YongJung Kim.

YoungPil 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 wellposedness, largetime behavior of solutions and meanfield limits. We also study the critical thresholds for hydrodynamic models.

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 (CTscans, MRI, PETscans). 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 CTscans). 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 gastrointestinal stromal tumors in the presence of targeted therapies or antiangiogenic 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

Laurent Desvillettes (Université Paris Diderot):
Large time behavior of reactiondiffusion 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 socalled "complex balance equilibria" which naturally appear in many chemical networks, and we focus on the treatment of socalled "boundary equilibria".

SeungYeal 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 nonabelian 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.

Daniel HanKwan (Ecole Polytechnique):
On the quasineutral limit of the VlasovPoisson system
We are interested in the behaviour of solutions to the VlasovPoisson 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 illposedness. 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).

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 VlasovPoisson, the kinetic FokkerPlanck equations. We discuss in particular regularity, singularity, and the behavior of the solutions of these equations with various boundary conditions.

YongJung Kim (KAIST & NIMS, Daejeon):
Introducing FokkerPlanck 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 FokkerPlanck 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.

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 starvationdriven di usion. The
evolutional selection and coexistence of such starvationmeasures are studied
within the context of LotkaVolterratype 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.

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 socalled floating body problem has been studied so far as a threedimensional problem. Our first goal is to reduce it to a twodimensional problem that takes the form of a coupled compressibleincompressible system. We will also show that the hydrodynamic forces acting on the solid can be partly put under the form of an added massinertia matrix, which turns out to be affected by the dispersive terms of the equations.

Jihoon Lee (ChungAng 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 KellerSegel type equations and the incompressible NavierStokes equations in spatial dimension two. We consider unique existence of a localintime solution, a blowup criterion, and the existence of a globalintime 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.

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 reentry 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. ForestierCoste.

SungJin Oh (KIAS, Seoul):
Linear instability of the Cauchy horizon in subextremal ReissnerNordström spacetime under scalar perturbations
We consider the linear scalar wave equation on a fixed subextremal ReissnerNordström spacetime with nonvanishing 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 EinsteinMaxwell system, this instability leads to a singular Cauchy horizon for generic small perturbations of ReissnerNordström spacetime. This is a joint work with J. Luk (Cambridge).

Clair Poignard (Université de Bordeaux):
Freeboundary 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 wellposedness 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 finitedifference 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.)

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.

Rudolphe Turpault (Université de Bordeaux):
Highorder 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 1015 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.)

SeokBae Yun (Sungkyunkwan University, Seoul):
Convergence of a semiLagrangian 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 NavierStokes limit. In this talk, we propose an implicit semiLagrangian 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.

Claude Zuily (Université ParisSud, 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 (frequencydependent) 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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