The 4th Korea-France Conference on Mathematics

09:30-10:20 (Korean Standard Time)

JongHae Keum, KIAS 

Title: Mori dream surfaces

Abstract: The Cox ring of a variety is the total coordinate ring, i.e., the direct sum of all spaces of global sections of all divisors.

When this ring is finitely generated, the variety is called Mori dream (MD). A necessary condition for being MD is the finite generatedness of Pic(X), i.e., the vanishing of the irregularity. 

Smooth rational surfaces with big anticanonical divisor are MD. Thus all del Pezzo surfaces of any degree are.

A K3 surface or an Enriques surface is MD iff its automorphism group is finite.

In this talk I will consider the case of surfaces of general type with $p_g=0$, and provide several examples that are MD.

I will also provide non-minimal examples that are not MD.

This is a joint work with Kyoung-Seog Lee.

10:20-11:10

Alessandra Sarti, Université de Poitiers

Title: Complex Reflection Groups, K3 surfaces and Lehrer-Springer theory

Abstract: I will talk about a long project in collaboration with Cédric Bonnafé, that relates K3 surfaces and complex reflection groups. This generalizes and explains some results of 2003 by W. Barth and myself. In fact by using complex reflection groups and Lehrer-Springer theory we obtained the following three main results :

1. Classification of some finite groups of maximum order acting on K3 surfaces.

2. Classification of all K3 surfaces that one can obtain as quotient of surfaces in complex projective three space by certain subgroups of finite   complex reflection groups of rank four.

3. Description of elliptic fibrations on the previous K3 surfaces.

After an overview of the results, I will explain more in detail point 2., in particular I will introduce some Lehrer-Springer theory from the theory of complex reflection groups, which is a fundamental tool to avoid a case-by-case analysis in the classification.

11:10-12:00

Samuel Boissiere, Laboratoire de Mathématiques et Applications, Université de Poitiers, UMR 7348 CNRS, France

Title: The Fano variety of lines of a cuspidal cyclic cubic fourfold

Abstract: In the framework of the compactification of the moduli spaces of prime order non-symplectic automorphisms of irreducible holomorphic symplectic manifolds, a key question is to understand the geometry of limit automorphisms. Starting from a nodal degeneration of cubic threefold, the general member of the family of Fano varieties of lines of the triple covering branched over the cubic is an IHS manifold equiped with the automorphism induced by the covering. It degenerates to a variety whose singular locus is a K3 surface.

I will present recent results obtained in collaboration with Chiara Camere and Alessandra Sarti that explain how the geometry of this K3 surface permits to define a limit automorphism in a suitable moduli space parametrizing pairs of IHS manifolds with automorphism.

13:30-14:20

YoungJu Choie, POSTECH

Title: Arithmetics on Herglotz-Zagier-Novikov function

Abstract: In this talk, we will revisit the functions $J(x) = \int_{0}^{1} \frac{\log(1+t^{x})}{1+t}dt$ and $T(x)=\int_{0}^{1} \frac{\tan^{-1}(t^{x})}{1+t^{2}} dt$, exploring their connection with the Herglotz-Zagier function. These functions are closely related to the Kronecker limit formula of real quadratic fields. We will introduce a more general function with cohomological relations.

This is a joint work with R. Kumar at Penn State University.

14:20-15:10

Emmanuel Peyre, Université Grenoble Alpes

Title: Counting the uncountable

Abstract: In the 1990’s, Yu Manin launched a vast program towards a better understanding of the asymptotic behaviour of the rational solutions of polynomial equations. Using the classical analogy between function fields in one variable and number fields this has recently led to new insights concerning stabilisation phenomena for the moduli space of morphisms from a curve as degrees go to infinity. In particular, this motivated a rather precise conjecture, which was proven for several families of varieties by M. Bilu, T. Browning and L. Faisant. This talk will present a survey of this program.

15:50-16:40

Keunyoung Jeong, Chonnam National University

Title: Nonvanishing of the special value of some Hecke $L$-functions

Abstract: Yang gave a relation between the special $L$-value of certain Hecke character and theta lifting. This is very explicit, hence it is possible to show the nonvanishing of $L$-values of Hecke character by computing some local factors on theta lifting sides, including the action of the Weil representation on the space of Schwartz functions. Following this strategy, Stoll–Yang gave a formula for $L$-values of the Hecke characters attached to the hyperelliptic curves $y^{2}=x^{5}+A$, and showed that the analytic rank of the hyperelliptic curve $y^2=x^5+1$ is zero. By examining the local computations at split primes and prime above 2, we find infinitely many twists of the hyperelliptic curve $y^{2}=x^{5}+1, y^{2}=x^{11}+1$ and the Fermat curves whose analytic rank is zero. This is a joint work with Junyeong Park, Donggeon Yhee, and Yeong-Wook Kwon.

09:30-10:20 (Korean Standard Time)

Gye-Seon Lee, SNU

Title: Convex real projective structures on reflection orbifolds

Abstract: Let $\mathcal{O}$ be a compact reflection n-orbifold whose underlying space is homeomorphic to a truncation $n$-polytope, i.e. a polytope obtained from an $n$-simplex by successively truncating vertices. In this talk, I will give a complete description of the deformation space of convex real projective structures on the orbifold $\mathcal{O}$ of dimension at least 4. Joint work with Suhyoung Choi and Ludovic Marquis.

10:20-11:10

Hervé Gaussier, Université Grenoble Alpes - Institut Fourier

Title: Metric properties and geometry of domains in the complex Euclidean space

Abstract: Invariant metrics, under the action of biholomorphic maps, are important objects in the study of domains in the complex Euclidean space; for instance, they encode in their asymptotic behavior some of the geometric properties of domains, such as curvature. I will try to explain how some properties of such invariant metrics may characterize model domains, like the unit ball.

11:10-12:00 (Online: Zoom)

Dami Lee, Indiana University

Title: On infinite octavalent polyhedral surfaces

Abstract: In this talk, we find octavalent triply periodic polyhedral surfaces that have identifiable underlying Riemann structures. We focus our study to eightfold cyclic covers over the thrice punctured sphere. There are three such surfaces up to equivalency. In this project we include a polyhedral surrogate of a cover of Fermat’s quartic, a polyhedral surrogate of Schwarz minimal CLP surface and a polyhedral surface that is not a cover of the Bolza surface. Part of this work is joint work with Charles Camacho.

13:30-14:20

Sang-hyun Kim, KIAS

Title: First order rigidity of manifold homeomorphism groups

Abstract: Two groups are elementarily equivalent if they have the same sets of true first order group theoretic sentences. We prove that if the homeomorphism groups of two compact connected manifolds are elementarily equivalent, then the manifolds are homeomorphic. This generalizes Whittaker’s theorem on isomorphic homeomorphism groups (1963) without relying on it. Joint work with Thomas Koberda (U Virginia) and Javier de la Nuez-Gonzalez (KIAS).

14:20-15:10

Susanna Zimmermann, Institut de mathématiques d’Orsay, Université Paris Saclay

Title: Non-conjugate involutions of the plane fixing birational curves.

Abstract: The Cremona group Bir($\mathbb{P}_{\bf{k}}^{2}$) is the group of birational map of some projective plane $\mathbb{P}_{\bf{k}}^{2}$ defined over the field $\bf{k}$. It is a very large group, it is not finite dimensional in any sense and it contains many finite subgroups. The classification of the finite subgroups up to conjugation has a long history, starting with Bertini and involutions, and it has been accomplished for $\bf{k}=\mathbb{C}$ by I. Dolgachev and V. A. Iskovskikh and also J. Blanc in 2009; the classification of involutions inside Bir($\mathbb{P}_{\bf{k}}^{2}$) was finalised by L. Bayle and A. Beauville in 2000. Over a non-closed field $\bf{k}$, only partial classifications of finite subgroups of Bir($\mathbb{P}_{\bf{k}}^{2}$) exist. In this talk I will present the classification up to conjugation of involutions in Bir($\mathbb{P}_{\mathbb{R}}^{2}$).

Two involutions in Bir($\mathbb{P}_{\mathbb{C}}^{2}$) that fix irrational curves are conjugate if and only if they fix birational curves. The main aim of this talk is to explain that Bir($\mathbb{P}_{\mathbb{R}}^{2}$) contains uncountably many non-conjugate involutions fixing birational curves. This is a collaboration with I. Cheltsov, F. Mangolte and E. Yasinsky.

15:50-16:40

Inhyeok Choi, KIAS

Title: Genericity of contracting elements in groups

Abstract: A finitely g enerated group $G$ comes with a homogeneous and locally compact simplicial graph called the Cayley graph of $G$. Since $G$ naturally acts on its Cayley graph, one can discuss the dynamics of each element of $G$. When $G$ is hyperbolic in the sense of Gromov, the action of an element of $G$ is either elliptic or loxodromic, the latter case being generic. In this talk, I will explain the converse of this statement. Namely, we deduce the global hyperbolicity of $G$ from the genericity of its contracting elements, which is a natural notion that generalizes loxodromic elements. This is based on joint work with Kunal Chawla and Giulio Tiozzo.

09:30-10:20 (Korean Standard Time)

Cécile Gachet, Institut für Mathematik, Humboldt Universität zu Berlin

Title: Smooth projective surfaces with infinitely many real forms

Abstract: A common undergraduate exercise is to classify quadratic forms over the real and complex numbers. Its conclusion could be that the two non-isomorphic real conics $x^{2}+y^{2}+z^{2}=0$ and $x^{2}+y^{2}-z^{2}=0$ become isomorphic, if considered as complex curves. In fact, this complex curve is $\mathbb{P}_{\mathbb{C}}^{1}$, and it admits only the afore-mentioned two non-isomorphic real forms.

Although it is quite common to find complex projective varieties admitting several real forms, the first example of a variety with infinitely many non-isomorphic real forms can be found in a 2018 paper by Lesieutre. More examples of varieties with infinitely many real forms have been found later, for instance as rational surfaces and as surfaces birational to $K3$ surfaces, see the 2022 paper by Dinh, Oguiso, and Yu.

This talk, reporting on joint work with Tien-Cuong Dinh, Hsueh-Yung Lin, Keiji Oguiso, Long Wang, and Xun Yu, completes the picture for smooth projective surfaces. It features the following two results: First, if a smooth projective surface admits infinitely many real forms, then it is rational, or birational to a $K3$ surface and non-minimal, or birational to an Enriques surface and non-minimal. Second, there are surfaces obtained by blowing-up one point in an Enriques surface, which admit infinitely many non-isomorphic real forms. I will explain the key ideas involved in the proofs of the two results, and try to give an idea of the construction used for the second one. This will feature a fair share of group theory, group actions, and dynamics.

10:20-11:10

Yeansu Kim, Chonnam National University

Title: The generic Arthur packet conjecture for unitary similitude and unitary groups

Abstract: We establish the generic local Langlands correspondence for unitary similitude groups. One main strategy in the generic case is through so-called Langlands-Shahidi method. We also describe L-packets that contain generic representations, i.e., generic $L$-packets. We further study its application on the properties of $L$-packets, namely, the generic Arthur packet conjecture for unitary similitude groups. The conjecture states that certain generic $L$-packet consists of tempered representations. Note that this conjecture is the converse to so-called Shahidi’s conjecture and is also considered local version of the generalized Ramanujan conjecture. This is a joint work with Muthu Krishnamurthy and Freydoon Shahidi.

11:10-12:00

Anna Cadoret, IMJ-PRG, Sorbonne Université

Title: On toric points of $p$-adic local systems arising from geometry

Abstract: For a smooth variety over a number field and a $p$-adic local systems arising from geometry on it, classical conjectures on algebraic cycles predict that the toric points should fit with the CM points of the associated variation of Hodge structure; in particular they should have similar properties in terms of sparcity. I will discuss results in this direction. This is a joint work with Jakob Stix.

09:30-10:20 (Korean Standard Time)

Olivier Fouquet, Laboratoire de mathématiques de Besançon

Title: The Iwasawa Main Conjecture and finiteness of Sha

Abstract: The Tate-Shafarevic group is a mysterious generalization to abelian varieties of the Class group of ring of integers of number field. Thanks to notable results of Rubin, Gross-Zagier, Kolyvagin, Kato... the finiteness of Sha is known for a rational elliptic curve $E$ when the $L$-function of $E$ does not vanish or vanishes at order exactly 1 at $s = 1$. However, not a single example of a rational elliptic curve whose $L$-function vanishes at higher order with a provably finite Tate-Shafarevic group is known at present. I will explain how a combination of theoretical consequences of the Iwasawa Main Conjecture and explicit arguments allow to prove the finiteness of the p-part of Sha for some elliptic curves whose $L$-function vanishes at higher order and to show that the order of the $p$-part is exactly the one predicted by the Birch and Swinnerton-Dyer.

10:20-11:10

Wontae Hwang, Jeonbuk National University

Title: Automorphism groups of simple abelian fourfolds over finite fields and Jordan constants

Abstract: In this talk, we give a classification of maximal elements of the set of finite groups that can be realized as the full automorphism groups of simple polarized abelian fourfolds over finite fields. As an application, we also compute the Jordan constants of the automorphism groups of simple abelian fourfolds over finite fields. Compared to a similar result for the case of simple abelian surfaces in positive characteristic, the Jordan constants that we obtain are rather “small”.

11:10-12:00

Michel Brion, Institut Fourier, université Grenoble

Title: Minimal rational curves on complete symmetric varieties

Abstract: Minimal rational curves play a prominent role in the geometry of uniruled projective algebraic varieties. Their tangent directions at a general point form the variety of minimal rational tangents (VMRT), a local invariant which has important applications to rigidity questions. The description of the families of minimal rational curves and their VMRTs is an open problem for almost homogeneous varieties. It has been solved for homogeneous varieties (Hwang–Mok), toric varieties (Chen–Fu–Hwang), and wonderful compactifications of adjoint semisimple groups (Brion–Fu). The talk will present a solution to this problem for the class of complete symmetric varieties, which contains the two latter classes. In particular, we will see that the VMRTs are homogeneous varieties, and admit an explicit description in terms of the restricted root system of the associated symmetric space.

13:30-14:20

Sijong Kwak, KAIST

Title: On the generalized gonality conjecture for secant varieties of curves

Abstract: The well-known gonality conjecture of smooth curves was raised by Green-Lazarsfeld(1984) and proved by Ein-Lazarsfeld in 2016. In this talk, we’d like to generalize this conjecture to the first non-trivial strand of syzygies in the category of higher secant varieties of curves and further investigate the shape of the Betti table of secant varieties when a curves is embedded in a projective space with degree large enough.

It can be shown that the vanishing and non-vanishing of Betti numbers of secant varieties is completely determined by the gonality sequence of an original curve. This is a joint work with Junho Choe and Jinhyung Park.

14:20-15:10

Seung-Jo Jung, JBNU

Title: Briançon-Skoda exponents and minimal exponents

Abstract: The Briançon-Skoda exponent of a holomorphic function $f$ on a complex manifold $X$ is the smallest integer $k$ such that $f^{k}$ is in the Jacobian ideal. Roughly, the Brian¸con-Skoda exponent measures how far $f$ is from being quasi-homogeneous. This talk relates it with the minimal exponent of $f$ which is one of invariants of hypersurface singularities. This talk is based on the joint work with I.-K. Kim, Morihiko Saito, and Y. Yoon.

15:50-16:40

Insong Choe, Konkuk University

Title: Simplicity of tangent bundles on the moduli of symplectic and orthogonal bundles

Abstract: In this talk, we show that the tangent bundles on the moduli of symplectic and orthogonal bundles over a curve are simple. To prove this, we use the symplectic and orthogonal Hecke curves which are known to have minimal degree among the rational curves passing through a general point of the moduli space. The key idea is to check the non-degeneracy of the associated variety of minimal rational tangents(VMRT). This is based on a joint work with Jaehyun Hong and George H. Hitching.

09:30-10:20 (Korean Standard Time)

Bo-Hae Im, KAIST

Title: MZV’s in positive characteristic I: Zagier-Hoffman’s conjectures

Abstract: Zagier-Hoffman’s conjectures in the classical setting on multiple zeta values over $\mathbb{Q}$ of Euler and Euler sums are still open. As analogues of the classical case, multiple zeta values and alternating multiple zeta values in positive characteristic were introduced by Thakur and Harada. In this talk, we determine the dimension and a basis of the span of all alternating multiple zeta values over the rational function field by finding all linear relations among them. As a consequence, we completely establish Zagier-Hoffman’s conjectures in positive characteristic formulated by Todd and Thakur which predict the dimension and an explicit basis of the span of multiple zeta values of Thakur of fixed weight.

This is a joint work with Hojin Kim, Khac Nhuan Le, Tuan Ngo Dac, and Lan Huong Pham.

10:20-11:10

Tuan Ngo Dac, CNRS and université de Caen Normandie

Title: MZV’s in positive characteristic II: algebraic structures

Abstract: Multiples zeta values (MZV’s for short) in positive characteristic were introduced by Thakur as analogues of classical multiple zeta values of Euler. In this talk we present a systematic study of algebraic structures of MZV’s in positive characteristic. We construct both the stuffle algebra and the shuffle algebra of these MZV’s and equip them with algebra and Hopf algebra structures. In particular, we completely solve a problem suggested by Deligne and Thakur and establish Shi’s conjectures. This is a joint work with Bo-Hae Im, Hojin Kim, K. N. Le and L. H. Pham.

09:30-10:20 (Korean Standard Time)

Philippe Thieullen, University of Bordeaux

Title: Zero temperature convergence of Gibbs measures for locally a finite potential in a 2-dimensional lattice

Abstract: We consider the space of all configurations over a 2-dimensional lattice and over a finite alphabet. A Gibbs measure is a spatially invariant measure on the space of configurations associated to a given potential at a certain temperature. A potential is locally finite if it depends on a finite index set. We want to understand the limit of these Gibbs measures when the temperature goes to zero. It may happen that the limit does not exist and several accumulation measures appear for complex potentials. For 1-dimensional lattices and locally finite potentials, the limit does exist. We show that it is not any more true for 2-dimensional lattices. This result extends a similar result proved by Chazottes and Hochman for lattices in dimension 3 or higher.

This result is a joint work with S. Barbieri, R. Bissacot, and Gregorio Dalle Vedove.

10:20-11:10

Nam-Gyu Kang, KIAS

Title: Conformal field theory for multiple SLEs and its classical limit

Abstract: Multiple SLEs describe several random interfaces consistent with conformal symmetries. I will explain a version of conformal field theory constructed from background charge modifications of Gaussian free field and insertion of $N$-leg operators (with screening) to show that this version produces a collection of martingale-observables for commuting multiple SLEs. I will also explain how this theory is related to its classical limit. Based on joint work with Tom Alberts, Sung-Soo Byun, and Nikolai Makarov.

11:10-12:00

Marc Arnaudon, Institut de Mathématiques de Bordeaux

Title: Coupling of Brownian motions with set valued dual processes on Riemannian manifolds

Abstract: In this talk we will motivate and explain the evolution by renormalized stochastic mean curvature flow, of boundaries of relatively compact connected domains in a Riemannian manifolds. We will construct coupled Brownian motions inside the moving domains, satisfying a Markov intertwining relation. We will prove that the Brownian motions perform perfect simulation of uniform law, when the domain reaches the whole manifold. We will investigate the example of evolution of discs in spheres, and of symmetric domains in the Euclidean plane. Skeletons of moving domains will play a major role.

13:30-14:20

KyeongSik Nam, KAIST

Title: Universality of Poisson-Dirichlet law for log-correlated fields

Abstract: It is widely conjectured that the Poisson-Dirichlet behavior appears universally in low-temperature disordered systems. However, this principle has been verified only for the particular models which are exactly solvable. In this talk, I will talk about the universal Poisson-Dirichlet behavior for the general log-correlated Gaussian fields. This is based on the joint work with Shirshendu Ganguly.

14:20-15:10

Helge Dietert, Université Paris-Cité and Sorbonne Université, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche (IMJ-PRG), F-75013, Paris, France

Title: Mixing and enhanced dissipation with velocities on the sphere

Abstract: Motivated by a popular kinetic model by Saintillan and Shelley for the dynamics of suspensions of active elongated particles, we study phase-mixing and enhanced dissipation with velocities on the sphere. In particular, we show that, up to log errors, the phase mixing estimate persists until the enhanced dissipation takes over. This is proved by combining an optimised hypocoercive approach with the vector field method.

15:50-16:40

Mouad Ramil, SNU

Title: Stability characterization, cutoff phenomenon and Eyring-Kramers estimates for the Langevin dynamics

Abstract: In this talk we will present some recent results obtained on the Langevin dynamics. First we shall detail a criterion ensuring global stability for the Langevin dynamics with general non-conservative forces. We will see that this criterion is almost-sharp as it is a necessary and sufficient condition in the case wh en the force is li near wi th a no rmal ma tr ix. As suming this criterion, we will be able to extend previous work by Barrera, Jara on the overdamped Langevin dynamics regarding the existence of a cutoff phenomenon in the convergence to equilibrium, to the case of the Langevin dynamics. Finally, we will conclude this presentation by extending the well-known Eyring-Kramers estimates to the case of the Langevin dynamics. This is a joint work with Seungwoo Lee and Insuk Seo.

09:30-10:20 (Korean Standard Time)

Kyungkeun Kang, Yonsei University

Title: Local regularity of weak solutions for the Stokes and Navier-Stokes equations near boundary in the half space

Abstract: In this talk, we discuss local boundary regularity of weak solutions for Stokes system and the Navier-Stokes equations.. We construct weak solutions with no-slip boundary conditions of the Stokes system whose normal derivatives are unbounded near boundary caused by localized singular force in a half space. Similar construction can be made to the Navier-Stokes equations as well. This is a joint work with Dr. Tongkeun Chang.

10:20-11:10

Raoul Gael, École polytechnique  

Title: TBA

Abstract: TBA

11:10-12:00 

Sung Woong Cho, KAIST

Title: Estimating Asymptotic Spreading Speed for A Predator-Prey System with Two Predators Via Neural Network Approach

Abstract: This study presents a neural network-based approach for estimating asymptotic spreading speeds in a three-species predator-prey system with two distinct predators (weak and strong) and one prey. Our model approximates species density, taking into account various biological parameters including predation rate, intrinsic growth rate, competition, and diffusion coefficients. In contrast, traditional numerical methods like the Finite Difference Method (FDM) require repetitive computations for each parameter pair and can be inefficient over prolonged time periods. Moreover, they can introduce inaccuracies in complex systems such as our system.

To validate our approach, we employed the well-known Classical Logistic Equation as a benchmark and further tested on the Predator-Prey system. While the spreading speed of the strong predator is known, the speeds for the weak predator and prey are not. Our model consistently estimated the spreading speeds for all species, showcasing its robustness and reliability.

13:30-14:20

Francois Golse, CMLS, École polytechnique

Title: The Regularity Problem for the Landau Equation

Abstract: It is well known that the dynamics of particles interacting through the Coulomb potential cannot be described by the Boltzmann equation. In this case, the Boltzmann collision integral must be replaced with the Landau operator. In the late 1990’s, Villani defined a notion of global, space-homogeneous solutions to the Landau equation, called $H$-solutions (in view of the importance of Boltzmann’s $H$-Theorem in the definition of such solutions). This talk will review some recent progress on the regularity of Villani solutions of the Landau equation. (Based in particular on joint work with M.P. Gualdani, C. Imbert and A. Vasseur [1, 2].)

References

[1] F. Golse, M.-P. Gualdani, C. Imbert, A. Vasseur, Partial Regularity in Time for the Space Homogeneous Landau Equation with Coulomb Potential, Ann. Scient. Éc. Norm. Sup. 4e série, 55 (2022) 1575–1611.

[2] F. Golse, C. Imbert, A. Vasseur, Local regularity for the space-homogeneous Landau equation with very soft potentials, preprint arXiv:2206.05155 [math.AP].

14:20-15:10

Stéphane Brull, Institut de Mathématiques de Bordeaux

Title: Approximation of the bitemperature Euler system

Abstract: This talk is devoted to the numerical approximation of the bitemperature Euler system. This model appears in the context of plasma physics in order to describe a mixture of ions and electrons in the quasi-neutral regime. This model is a nonconservative hyperbolic system describing an out of equilibrium plasma in a quasi-neutral regime. In a first part, we consider the case where only the electric field is considered. This system is non conservative because it involves products between velocity and pressure gradients that cannot be transformed into a divergential form. We develop a second order numerical scheme by using a discrete BGK relaxation model. The second order extension is based on a subdivision of each cartesian cell into four triangles to perform affine reconstructions of the solution. Such ideas have been developed in the litterature for systems of conservation laws. We show here how they can be used in our nonconservative setting. Next we show how a DG method can be developped in order to discretize the space variable. Therefore we obtain high order schemes. In a next part, we derive a bitemperature model including magnetic fields and we show how the magnetic field can be integrated into the discrete BGK formulation. In a last part, we finish by simulations illustrating the method.

15:50-16:40

Sungsoo Byun, SNU

Title: Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials

Abstract: In this talk, I will discuss two-dimensional Coulomb gases at a specific temperature with free or Neumann boundary conditions, which can be realised as eigenvalues of random normal matrices or planar symplectic ensembles. I will present the asymptotic expansion of the partition functions when the underlying field is radially symmetric. Notably, our findings stress that the expansion contains topological data of the associated droplet. This is based on joint work with Nam-Gyu Kang and Seong-Mi Seo.

09:30-10:20 (Korean Standard Time)

François Bouchut, National Center of Scientific Research, France

Title: An entropy satisfying explicit second-order scheme for incompressible Navier-Stokes equations

Abstract: I will show how we can use a numerical low Mach limit to build explicit schemes for the incompressible Navier-Stokes equations (artificial compression method). At the level of the compressible approximation we use a kinetic flux decomposition method. The scheme verifies a discrete entropy inequality under a CFL condition of parabolic type and a stability condition involving a cell Reynolds number, that ensures that the viscosity dominates advection at the cell level. This ensures the robusteness of the method, with uniform bounds on the solution. By choosing well the parameters we obtain second-order accuracy in space. The method is evaluated on classical test cases with attachment points and in moderately turbulent regime for Reynolds numbers of a few hundreds.

10:20-11:10

Jeongho Kim, Kyung Hee University

Title: Quantified asymptotic analysis for the relativistic quantum mechanical system with electromagnetic fields

Abstract: We study asymptotic analysis of the relativistic quantum mechanical system interacting with a self-consistent electromagnetic fields. In particular, our analysis foucses on the MaxwellKlein-Gordon (MKG) model, wherein the Klein-Gordon scalar field is coupled with the Maxwell equations for electromagnetic fields. To derive an asymptotic analysis, we consider the semiclassical and non-relativistic regime at the same time, by introducing a single scaling parameter that simultaneously parametrizes the Planck constant and the speed of light. As the scaling parameter vanishes, we derive rigorous and quantified estimates regarding the asymptotic convergence of the MKG system towards the classical Euler-Poisson system. Our analytical framework relies on the modulated energy estimate.

11:10-12:00

Kevin Guillon, University of Bordeaux , France

Title: Modelization and asymptotic preserving scheme for the 2D bi-temperature dimensionless Euler system in a plasma

Abstract: Surging interests and concerns for alternative reliable sources of energy have led to consider fusion energy as a trustful candidate for tomorrow, as may witness the current ITER project. However, important costs and precise parameters of such a process require preliminary knowledge, modelization and numerical simulation of high temperature plasmas. The present work follows the path of the previous results from [1], [2] and [3] . First will be derived the dimensionless fluid bi-temperature Euler equations for an inert plasma composed with electrons and cations, and submitted to a transverse electromagnetic field. Through the c omputations, we will be discussing the role and magnitude of crucial plasma parameters. This derivation will result from the hydrodynamic limit of BGK-Vlasov-Maxwell equations. Then will be written an asymptotic preserving kinetic scheme relying on the general Aregba-Natalini procedure [4]. In particular, we will address nonconservative terms from the energies equations through a particular splitting procedure, and retrieve relevant physical properties. Finally will be discussed perspective and limitations for possible generalizations of the model derived above.

References

[1] E. Estibals, H. Guillard, A. Sangam, Derivation and numerical approximation of twotemperature Euler plasma model , J. Comput. Phys. 444 , 2021.

[2] D. Aregba-Driollet, S. Brull, C. Prigent, A discrete velocity numerical scheme for the two-dimensional bitemperature Euler system, SIAM J. Numer. Anal. 60, 2022.

[3] S. Brull, B. Dubroca, X. Lh´ebrard, Modelling and entropy satisfying relaxation scheme for the nonconservative bitemperature Euler system with transverse magnetic field, Com-put. Fluids 214, 2021.

[4] D. Arebga-Driollet, R. Natalini, Discrete kinetic schemes for multidimensional systems of conservation laws, SIAM J. Numer. Anal. 37, 2000.

09:30-10:20 (Korean Standard Time)

Seok-Bae Yun, SKKU

Title: Weak solutions to the stationary BGK model in a slab

Abstract: We consider stationary flows b etween two condensed phases that emerge from the evaporation and condensation process on the two phases in the framework of the stationary BGK model in a slab. Under the physically minimum conditions on the inflow functions, namely the finite m ass flux, energy flux and entropy flux , the exis tence of weak solu tion is derived. The main difficulties ar e, am ong ot hers, (1) the im possibility of truncation of the relaxation operator in the vanishing velocity region in the last limit process, and (1) the control of the velocity distribution functions using the macroscopic fields near vanishing velocity r egion. This is joint work with Stephane Brull.

10:20-11:10

JaeYong Lee, KIAS

Title: Deep learning approach for solving kinetic equations

Abstract: Recently, deep learning-based methods have been developed to solve PDEs with many advantages. In this talk, I introduce our recent results on the deep neural network solutions to the kinetic equation. We study Vlasov-Poisson-Fokker-Planck equation and its diffusion limit via the deep learning approach. Also, we propose a new framework to approximate the solution to Fokker-Planck-Landau equation which has a nonlinearity and a high dimensionality of variables.

11:10-12:00

In-Jee Jeong, SNU

Title: On vorticity supported on logarithmic spirals

Abstract: We study logarithmic spiraling solutions to the 2d incompressible Euler equations which solve a nonlinear transport system on S. We show that this system is locally well-posed in $L^{p},p\geq1$ as well as for atomic measures, that is logarithmic spiral vortex sheets. Logarithmic spiral vortex sheets introduced by Prandtl and Alexander are just particular examples in this well-posedness class which satisfy an additional scaling symmetry involving time. Moreover, we realize the dynamics of logarithmic vortex sheets as the well-defined limit of solutions which are smooth in the angle. Furthermore, our formulation not only allows for a simple proof of existence and bifurcation for non-symmetric multi-branched logarithmic spiral vortex sheets, but also provides a framework for studying asymptotic stability of self-similar dynamics.

For logarithmic spiraling solutions, we make an observation that the local circulation of the vorticity around the origin is a strictly monotone quantity of time, which allows for a rather complete characterization of the long-time behavior. We prove global well-posedness for bounded logarithmic spirals as well as data that admit at most logarithmic singularities. We are then able to show a dichotomy in the long time behavior, solutions either blow up or completely homogenize. In particular, bounded logarithmic spirals should converge to constant steady states. For logarithmic spiral sheets, the dichotomy is shown to be even more drastic, where only finite time blow up or complete homogenization of the fluid can and does occur.

This is based on a joint work with Ayman Said.

13:30-14:20

Marwa Shahine, University of Bordeaux

Title: Fredholm Property of the Linearized Boltzmann Operator for a Mixture of Polyatomic Gases

Abstract: In this talk, we consider the Boltzmann equation that models a mixture of polyatomic gases assuming the internal energy to be continuous. Under some convenient assumptions on the collision cross-section, we prove that the linearized Boltzmann operator $L$ is a Fredholm operator. For this, we write $L$ as a perturbation of the collision frequency multiplication operator. We prove that the collision frequency is coercive and that the perturbation operator is Hilbert-Schmidt integral operator.

14:20-15:10

Donghyun Lee, POSTECH

Title: Geometric effects for the regularity of the Boltzmann boundary problems

Abstract: Regularity and singularity of the solution according to the shape of domains is a challenging research theme in the Boltzmann theory. We review some recent developments for these problems. In particular, we study (optimal) $C^{\frac{1}{2}-}$ regularity of the hard-sphere Boltzmann equation exterior of an uniformly convex obstacle with specular reflection boundary condition. This is joint work with Chanwoo Kim.

15:50-16:40

Dongnam Ko, The Catholic University of Korea

Title: Emerging asymptotic patterns in a Winfree ensemble with higher-order couplings

Abstract: The Winfree model is a phase-coupled synchronization model which simplifies pulse-coupled models such as the Peskin model on pacemaker cells. It is well-known that the Winfree ensemble with the first-order coupling exhibits discrete asymptotic patterns such as incoherence, locking and death depending on the coupling strength and variance of natural frequencies. In this talk, we breifly look over the phase-locking phenomenon in the Winfree model with the standard setting, and further discuss higher-order couplings which makes the dynamics more close to the behaviors of the Peskin model. For this, we propose several sufficient conditions on the coupling strength, natural frequencies and initial data, where the conditions are independent of the number of oscillators so that they can be applied to the corresponding mean-field PDE model. We also provide several numerical simulations and compare them with analytical results. This talk is based on the work with Professor Seung-Yeal Ha and Jaeyoung Yoon in Seoul National University.

09:30-10:20 (Korean Standard Time)

Insuk Seo, SNU

Title: Resolvent equations and Metastability of Langevin dynamics

Abstract: In this talk, we will discuss the connection between the behavior of solutions of certain class of resolvent equations and the metastability of Langevin dynamics. This presentation is based on recent collaboration with Claudio Landim, Diego Marcondes, and Jungkyoung Lee.

10:20-11:10

Gi-Chan Bae, SNU

Title: Large amplitude solution of BGK model

Abstract: Bhatnagar–Gross–Krook (BGK) equation is a relaxation model of the Boltzmann equation which is widely used in place of the Boltzmann equation for the simulation of various kinetic flow problems. In this work, we study the asymptotic stability of the BGK model when the initial data is not necessarily close to global equilibrium pointwisely. The main difficulty of the BGK equation comes from the highly nonlinear structure of the relaxation operator. To overcomes this issue, we derive refined control of macroscopic fields to guarantee the system enters a quadratic nonlinear regime, in which the highly nonlinear perturbative term relaxes to essentially quadratic nonlinearity after a long but finite time. This is joint work with G.-H. Ko, D.-H. Lee and S.-B. Yun.