Analysis, PDE & Probability Seminar

2025 Seminars

2025. 10. 15 (Wed) 16:00-17:30  Room 1423
Gianluca Crippa (University of Basel)

Weak, renormalized, and vanishing-viscosity solutions of the two-dimensional Euler equations

ABSTRACT. Let us consider the Euler equations modeling the behavior of an incompressible, homogeneous, inviscid fluid. In the two-dimensional case, the Euler equations can be written in vorticity form as a continuity equation, in which the advecting velocity depends on the vorticity through an integral operator. In my talk, I will introduce several notions of weak solutions for the two-dimensional Euler equations in vorticity form: weak solutions, renormalized solutions, and vanishing-viscosity solutions. Relying on the linear theory for continuity equations with Sobolev velocity field by DiPerna and Lions, I will show that in the subcritical case weak solutions do not exhibit anomalies. In the supercritical case, I will show by means of a duality approach that the same holds for vanishing-viscosity solutions. This has some connections with the two-dimensional theory of turbulence of Kraichnan and Batchelor.

2025. 10. 14 (Tue) 14:00-15:00  Room 1423
Jinsol Seo (KIAS)

DiPerna–Lions Theory and Large Deviations for Stochastic Flows with Applications to Nonlinear PDEs

ABSTRACT. The dynamical algebra of the generic superintegrable model on S_2 will be introduced. It will be shown to provide an algebraic interpretation of the bivariate Jacobi polynomials. This will offer a framework to review the properties of certain algebras of the Askey-Wilson type that are associated to bispectral hypergeometric polynomials. The covariance of the pentagonal scheme that will have emerged shall be described.

2025. 10. 14 (Tue) 10:00-11:30  Room 1423
Luc Vinet (Université de Montréal)

The dynamical algebra of the superintegrable model on the 2-sphere and two-variable Jacobi polynomials

ABSTRACT. The dynamical algebra of the generic superintegrable model on S_2 will be introduced. It will be shown to provide an algebraic interpretation of the bivariate Jacobi polynomials. This will offer a framework to review the properties of certain algebras of the Askey-Wilson type that are associated to bispectral hypergeometric polynomials. The covariance of the pentagonal scheme that will have emerged shall be described.

2025. 10. 10 (Fri) 16:00-17:00  Room 1423
Nam-Gyu Kang (KIAS)

Analysis of Conformal Fields

ABSTRACT. Conformal field theory (CFT) was originally introduced to connect various critical statistical physics models with Virasoro representation theory. Since then, it has found applications in string theory, condensed matter physics, vertex operator algebras, and probability theory. Meanwhile, stochastic Loewner evolution (SLE), introduced by Schramm, emerged as the only possible candidates for scaling limits of interface curves in several critical lattice models. This development has led to rigorous proofs of significant conjectures in statistical physics, including some highly non-trivial predictions of CFT. After reviewing the evolution of these ideas, I will present the connections between CFT and SLE across various conformal frameworks.

2025. 10. 2 (Thu) 11:00-12:00  Room 1423
Minjae Park (Auburn University)

Random surfaces and the six-vertex model

ABSTRACT. In this talk, I will survey Liouville quantum gravity (LQG) surfaces in the supercritical regime, where the matter central charge exceeds 1. Unlike the subcritical case---where many models from mathematical physics (such as percolation and Ising) are known to converge to LQG surfaces---no canonical model is currently available in this regime. I will discuss how the six-vertex model may serve as a candidate whose scaling limit is a supercritical LQG surface, and I will present some preliminary progress and many conjectures in this direction, based on joint work with Ewain Gwynne and Andrew Elvey Price.

2025. 9. 24 (Wed) 16:00-17:00  Room 1423
Taegyu Kim (KIAS)

Classification of Blow-up Dynamics for the Calogero–Moser Derivative NLS

ABSTRACT. I will present recent progress on the study of finite-time blow-up dynamics for the Calogero–Moser derivative nonlinear Schrödinger equation (CM-DNLS). A central objective is to describe and ultimately classify the possible blow-up scenarios for this model. Along the way, several steps have been achieved: the construction of smooth chiral blow-up solutions (joint with Kihyun Kim, Soonsik Kwon), a soliton resolution result (joint with Soonsik Kwon), and the existence of smooth quantized blow-up solutions under symmetry restrictions (joint with Uihyeon Jeong). Building on these developments, in joint work with Uihyeon Jeong, Kihyun Kim, and Soonsik Kwon, we establish a classification of finite-time blow-up scenarios involving a single soliton for CM-DNLS. I will describe this classification result in detail and also briefly introduce some of the main ideas of the proof.

2025. 9. 15 (Mon) 11:00-12:00  Room 8309
Donghan Kim (KAIST)

Roughness in finance via Schauder Representation

ABSTRACT. This presentation will explain two distinct concepts for measuring the roughness of financial data. We first introduce the idea of the p-th variation of a real-valued continuous function along a general class of refining partition sequences. We demonstrate that the finiteness of the p-th variation of a given path is closely linked to the finiteness of the lp -norm of the coefficients along a Schauder basis, analogous to how the Hölder exponent relates to the l∞ -norm of the Schauder coefficients. This result establishes an isomorphism between the space of Hölder continuous functions with finite (generalized) p-th variation along a given partition sequence and a subclass of infinite-dimensional matrices, equipped with an appropriate norm, in the spirit of Ciesielski.

2025. 8. 13 (Wed) 16:00-17:00  Room 1423
In-Jee Jeong (Seoul National University)

Sharp Hölder estimates in corner domains and the incompressible Euler equations

ABSTRACT. We consider double Riesz transforms on certain corner domains, namely the domains obtained by intersections of half-spaces in Rn. We prove sharp Hölder estimates for functions not necessarily vanishing on the boundary. This can be applied to the incompressible Euler equations to obtain wellposedness and growth of norms of solutions in time.

2025. 8. 4 (Mon) 16:00-17:00  Room Online
Pêdra Andrade (Paris Lodron University of Salzburg)

Hölder Regularity of the Gradient for Degenerate Fully Nonlinear with Hamiltonian Terms

ABSTRACT. In this talk, we present optimal, quantitative Hölder regularity results for the gradient of solutions to a class of degenerate fully nonlinear elliptic equations. These equations are characterized by a loss of ellipticity as the gradient vanishes and include nonlinear firstorder terms of Hamiltonian type. The presence of such terms introduces additional challenges in the analysis, particularly in regimes where the gradient is very small or very large. Our approach combines perturbation techniques with a maximum principle argument, enabling us to overcome these difficulties and derive sharp interior regularity estimates.

2025. 7. 30 (Wed) 16:00-17:00  Room 1423
Min Jun Jo (Duke University)

Cusp Formation in Vortex Patches

ABSTRACT. Cohen-Danchin (2000) conjectured that any initial vortex patch with acute corners would instantaneously evolve into a cusp structure of order one with logarithmic sharpness. This talk is based on the recent joint work with T. Elgindi where we fully affirm the conjecture.

2025. 7. 29 (Tue) 16:00-17:00  Room 204 (Soorim)
Jinyeop Lee (University of British Columbia)

Derivation of the Chern-Simons-Schrodinger equation from the dynamics of an almost-bosonic-anyon gas

ABSTRACT. TBA

2025. 7. 25 (Fri) 16:00-17:00  Room 1423
Eun Ji Kang (Seoul National University)

A Categorical Interpretation of Continuous Orbit Equivalence for Partial Dynamical Systems

ABSTRACT. We define the orbit morphism of partial dynamical systems and prove that an orbit morphism being an isomorphism in the category of partial dynamical systems and orbit morphisms is equivalent to the existence of a continuous orbit equivalence between the given partial dynamical systems that preserves the essential stabilisers. We show that this is equivalent to the existence of a diagonal-preserving isomorphism between the corresponding crossed products when the essential stabilisers of partial actions are torsion-free and abelian. We also characterize when an étale groupoid is isomorphic to the transformation groupoid of some partial action. Additionally, we explore the implications in the context of semi-saturated orthogonal partial dynamical systems over free groups, establishing connections with Deaconu-Renault systems and the concept of eventual conjugacy.

2025. 7. 25 (Fri) 14:30-15:30  Room 1423
Yoonje Jeong (서울대학교)

Free Probability in Operator Algebras

ABSTRACT. Free probability has proved to be an unexpectedly powerful tool for longstanding questions in operator algebras. In this expository talk, based on the speaker’s personal selection, we will survey three examples that illustrate the theme: • Metric characterisations of freeness: Beginning from Kesten’s result for group von Neumann algebras, we will see how Cadilhac and Collins generalized this result in the setting of free probability, giving a new route to embeddings of free group factors. • Haagerup–Thorbjørnsen’s Strong Convergence Result: We will see how their work for strong convergence of suitable unitary random-matrices was able to solve a problem in operator algebras. • Ultrapowers, free entropy and asymptotic freeness: We discuss how free entropy and asymptotic freeness results can probe the elementary isomorphism problems for II_1 factors.

2025. 7. 22 (Tue) 16:00-17:00  Room 8309
Beomjun Choi (KAIST)

Thom's gradient conjecture for nonlinear evolution equations

ABSTRACT. The Łojasiewicz–Simon inequality is a powerful tool for proving the uniqueness of blowups in equations with gradient structure. It is therefore natural to seek a general theory describing the next-order behavior of solutions near these limits. In joint work with P.-K. Hung, we characterize both the convergence rate and the dominant mode of convergence for nonlinear evolution equations originally studied by Simon. As a consequence, we give an affirmative answer to Thom’s gradient conjecture, concerning the alignment of secant lines for convergent gradient flows, in the context of geometric problems.

2025. 7. 14 (Mon) 17:00-18:00  Room 1424
Sang-Jung Park (CNRS, Laboratoire de Physique Théorique (LPT) Toulouse)

High-dimensional entanglement with positive partial transpose: asymptotic, random, and deterministic approaches

ABSTRACT. Finding PPT (Positive Partial Transpose) quantum states with high Schmidt numbers has recently become a significant mathematical challenge in quantum information theory. In this talk, we introduce and discuss three complementary approaches to address this question: (1) high-dimensional convex geometry, (2) random matrix theory, and (3) duality and representation theory. In the first two parts, we show the existence of sequences of quantum states whose Schmidt number grows linearly with the local dimension. In the third part, by considering the quantum objects having symplectic group symmetry, we provide broad explicit families of d⊗d quantum states with the Schmidt number exactly d/2.

2025. 7. 14 (Mon) 15:30-16:30  Room 1424
Ikhan Choi (University of Tokyo)

A solution to Haagerup’s problem on normal weights

ABSTRACT. We will introduce three open problems related to normal weights posed by Haagerup 50 years ago in his master’s thesis. Among these, we especially discuss how we could solve the last problem and its meaning. This will provide a complete set of the positive Hahn-Banach separation theorems on operator algebras and shed light on why the weak∗ topologies on the duals of C∗-algebras are relatively difficult to work with. (arXiv:2501.16832.)

2025. 7. 11 (Fri) 16:00-17:00  Room 1423
Pilgyu Jung (연세대학교)

Estimates for Half-Time Derivatives and its Application

ABSTRACT. The present talk is devoted to introducing estimates for the half-time derivative in the context of parabolic partial differential equations. We then discuss applications of these estimates to the following problems: (1) the parabolic conormal problem with leading coefficients that are merely measurable in time on Reifenberg-flat domain; (2) the L2-theory for poroelastic equations; (3) the path regularity of stochastic partial differential equations(SPDEs).

2025. 7. 11 (Fri) 15:00-16:00  Room 1423
Kwan Woo (서울대학교 수학연구소)

Stability of Vortex Dipoles in Two-Dimensional Euler Flows

ABSTRACT. In this talk, I will present recent results on the existence and stability of traveling wave solutions to the two-dimensional incompressible Euler equations, which take the form of counter-rotating vortex dipoles symmetric across a horizontal axis. These structures include classical examples such as the Chaplygin–Lamb dipole and the Sadovskii vortex patch, both of which exhibit singular behavior near the symmetry axis. I will describe a variational framework for constructing such solutions and discuss their stability properties. This is joint work with Kyudong Choi and Young-Jin Sim (UNIST).

2025. 7. 9 (Wed) 16:00-17:00  Room 1423
Junhee Ryu (KIAS)

Partial differential equations in weighted Sobolev spaces

ABSTRACT. In this talk, we discuss two types of equations in weighted Sobolev spaces. First, we present degenerate equations on the upper half space. The coefficient matrices of the equations are the product of $x_d^2$ and bounded uniformly elliptic matrices. Next, we consider nonlocal equations on $C^{1,\tau}$ open sets where $\tau\in(0,1)$. The operators we consider are infinitesimal generators of symmetric stable L\'evy processes, whose L\'evy measures are allowed to be very singular. This talk is based on joint works with Jae-Hwan Choi, Kyeong-Hun Kim, and Hongjie Dong.

2025. 7. 9 (Wed) 15:00-16:00  Room 1423
Jae-Hwan Choi (KIAS)

Well-posedness and Large Deviations of Stochastic Lagrangian Flows with Applications to Nonlinear PDEs

ABSTRACT. In this talk, I will present recent results on the well-posedness and large deviation principles for stochastic Lagrangian flows with singular coefficients. As applications, I will discuss the vanishing viscosity limit of the two-dimensional Navier–Stokes equations and the three-dimensional Vlasov–Poisson–Fokker–Planck equation. This talk is based on joint work with Chanwoo Kim, Dohyun Kwon, and Jinsol Seo.

2025. 6. 27 (Fri) 17:00-18:00  Room Online
Kai Zhang (University of Granada)

Boundary $C^{1,\alpha}$, $C^{2,\alpha}$ and higher regularity

ABSTRACT. In this talk, we consider boundary $C^{1,\alpha}$, $C^{2,\alpha}$ and higher regularity. The basic idea is perturbation. We do not use the classical technique of flattening the boundary by some transformation. Instead, we obtain the pointwise regularity based a new notion of the smoothness of the boundary. The regularity is new even for the Laplace equation.

2025. 6. 20 (Fri) 17:00-18:00  Room Online
Kai Zhang (University of Granada)

Boundary $C^{0,1}$ regularity, the Hopf lemma and boundary $C^1$ regularity.

ABSTRACT. In this talk, we regard the boundary $C^{0,1}$ regularity and the Hopf lemma as the same kind of regularity and prove these two kinds of regularity in a unified way. They hold under the boundary $C^{1,dini}$ condition, which is optimal. In addition, we also present the boundary $C^1$ regularity, which is the most complicated boundary regularity.

2025. 6. 18 (Wed) 16:00-17:00  Room 1423
Kihoon Seong (Cornell University)

Central limit theorem for Φ^4 QFT measures in low temperature and thermodynamic limit

ABSTRACT. I will introduce basic concepts such as the concentration and fluctuation of Φ^4 Gibbs type measures from the perspectives of statistical physics, quantum field theory, probability theory, and PDEs. The focus will be on the low temperature behavior and the thermodynamic limit of these probability measures, with particular attention to fluctuations around the soliton manifold.

2025. 6. 17 (Tue) 16:00-17:00  Room 1424
Thomas Alberts (University of Utah)

Large deviations of geodesic midpoint fluctuations in last passage percolation

ABSTRACT. Last passage percolation is a representative model of phenomena that characterize the Kardar-Parisi-Zhang universality class. The model has an interesting integrable structure that allows for the computation of many surprising exact formulas, based on connections with random matrices, representation theory, orthogonal polynomials, and more. The last 5-6 years has even seen the construction of LPP's universal scaling limit, called the directed landscape. This talk focuses on the pre-limiting lattice model, specifically on large deviations for its geodesic paths. As for simple random walk, the large deviations formulas are not universal. Instead they depend on the distribution of the iid inputs, which is an aspect of the general problem that is much less studied. I will describe our characterization of these large deviations formulas in terms of last passage times. For integrable choices of the iid weights these characterizations lead to very explicit formulas, which allow us to directly compare the large deviations for the geodesic to those of random walk paths. Based on joint work with Riddhi Basu, Sean Groathouse, and Xiao Shen.

2025. 6. 13 (Fri) 17:00-18:00  Room Online
Kai Zhang (University of Granada)

Introduction to the boundary regularity for elliptic equations and $C^\alpha$ regularity

ABSTRACT. In this talk, we introduce some motivations and classical results regarding the boundary regularity for the Dirichlet problem of uniformly elliptic equations. Then we present two kinds of boundary $C^\alpha$ regularity: for some $0<\alpha<1$ and for any $0<\alpha<1$. The first regularity is derived based on the strong maximum principle and the scaling invariance. The second regularity is derived based on the perturbation technique.

2025. 5. 30 (Fri) 16:00-17:00  Room 8309
Taehun Lee (Konkuk University)

Allen—Cahn solutions and Hardt—Simon foliations

ABSTRACT. We investigate the Allen--Cahn equation, a reaction-diffusion model emerging in phase transition theory. It is well-established that minimal surfaces can be approximated by the nodal sets of Allen--Cahn solutions. One natural question is to what extent the nodal set determines the Allen--Cahn solution. From this perspective, the strongest statement is that the Allen--Cahn equation with a given nodal set admits a unique solution. We prove the uniqueness of the Allen--Cahn solution in the case where its nodal set is given by a leaf of the Hardt--Simon foliation.

2025. 5. 30 (Fri) 15:00-16:00  Room 7323
Beom-Seok Han (Sungshin Women's University)

Well-posedness and space-time regularity for stochastic reaction-diffusion-advection equations with variable-order nonlocal operators and spatially homogeneous colored noise

ABSTRACT. In this talk, we discuss the well-posedness and H\"older regularity of strong solutions to stochastic reaction-diffusion-advection equations driven by spatially homogeneous colored noise and nonlinear multiplicative noise. The equations involve variable-order nonlocal operators defined via Bernstein functions. While various types of SPDEs with variable-order nonlocal operators and SPDEs with nonlinear terms have been studied individually, we present a unified result that recovers and generalizes many of the existing results. We introduce a condition on the noise - strongly reinforced Dalang's condition, which provides a refined criterion for establishing well-posedness and solution regularity. This talk is based on joint work with Jae-Hwan Choi and Daehan Park.

2025. 5. 29 (Thu) 10:30-11:30  Room 1423
Jung-Kyoung Lee (KIAS HCMC)

Metastability of Langevin Dynamics and Its Applications

ABSTRACT. Metastability is a phenomenon observed in various physical models. While it is an interesting topic in its own right, it can also be applied to various problems. In this talk, we will introduce the metastability of overdamped Langevin dynamics and present results related to the corresponding parabolic equation and slow mixing.

2025. 5. 23 (Fri) 17:00-18:00  Room Online
Yuanyuan Lian (Universidad de Granada)

Rigidity results for the capillary overdetermined problem

ABSTRACT. https://kias.re.kr/kias/activities/seminars/view.do?seqno=PGN1720250510-0001&menuNo=404003&schoolsCd=M¢rspgmsCd=&sdate=2025-04-20&edate=2025-06-20&mjrcdnm=&searchCnd=1&searchWord=&pageIndex=2

2025. 5. 23 (Fri) 16:00-17:00  Room 204 (Soorim)
Mingu Jung (KIAS HCMC)

Nonlinear Banach space theory: Toward the Ribe program

ABSTRACT. This talk provides a brief overview of the Ribe program, the structure of Lipschitz-free spaces (also known as transportation cost spaces), and highlights from some recent advances in the field.

2025. 5. 13 (Tue) 16:00-17:00  Room Online
Ho-Sik Lee (Universität Bielefeld)

Local Hölder regularity for nonlocal porous media equations

ABSTRACT. In this talk, we discuss Holder regularity results for locally bounded, local weak solutions to porous media and fast diffusion equations involving general kernel corresponding to s-fractional Laplacian (0

2025. 4. 28 (Mon) 16:00-17:00  Room 1424
Seongmin Jeon (Hanyang University)

A minimization problem for a superlinear system with free boundary

ABSTRACT. We study vector-valued minimizers of the energy $$\int_{\Omega} (|D u|^2+\frac{2}{p}|u|^p), 0

2025. 4. 4 (Fri) 14:00-15:00  Room 1423
Manwook Han (Chungbuk National University)

On the weak minimizing property

ABSTRACT. The weak maximizing property, introduced by R.M. Aron, D. García, D. Pellegrino, and E.V. Teixeira in [1], is presented as an extension of reflexivity. Indeed, a Banach space $X$ is reflexive if and only if the pair $(X, Y)$ has the weak maximizing property for some nontrivial Banach space $Y$. Later, S. Dantas, M. Jung and G. Martínez-Cervantes proved in [2] that a Banach space $Y$ has the Schur property if and only if the pair $(X, Y)$ has the weak maximizing property for every reflexive Banach space $X$. We study a minimum norm version of the weak maximizing property, namely the weak minimizing property. We show that the pairs $(\ell_p, L_p[0,1])$ for $2 \leq p < \infty$ and $(\ell_s \oplus_q \ell_q , \ell_s \oplus_p \ell_p)$ for $1 < p \leq s \leq q < \infty$ satisfy the weak minimizing property. These cases were previously unknown. Additionally, we prove that the pairs $(\ell_1, \ell_p)$, $(\ell_1, c_0)$, $(\ell_1, \ell_1)$, $(c_0, \ell_p)$ and $(c_0, c_0)$ do not satisfy the weak minimizing property. We also study property strict (m), which is the minimum norm version of property strict (M). We observe for Banach spaces $X$ and $Y$, if $X$ is reflexive and the pair $(X, Y)$ satisfy property strict (m), then $(X, Y)$ has the weak minimizing property. We show that for a Banach space $X$, the pair $(X, X)$ has property strict (m) if and only if $X$ has property strict (M). However, we also exhibit a pair of Banach spaces that satisfies property strict (m) without property strict (M). [1] R. M. Aron, D. Garc´ıa, D. Pellegrino, and E. V. Teixeira, Reflexivity and nonweakly null maximizing sequences, Proc. Amer. Math. Soc. 148 (2020), no. 2, 741-50. [2] S. Dantas, M. Jung and G. Mart´ınez-Cervantes, Some remarks on the weak maximizing property. J. Math. Anal. Appl. 504 (2021), no. 2, 125433.

2025. 3. 28. (Fri) 17:00-18:00  Room Online
Abraham Rueda Zoca (University of Granada)

On some topological properties in tensor products of Banach spaces

ABSTRACT. We will expose the part of the work [1] where we study several topological properties (like weakly compactly generated, weakly Lindelöf determined, property (C) of Corson...) on tensor products of Banach spaces. [1] A. Avil\'es, G. Mart\'inez-Cervantes, J. Rodr\'iguez and A. Rueda Zoca, Topological properties in tensor products of Banach spaces, J. Funct. Anal. 283 (2022), article 109688.

2025. 3. 28. (Fri) 11:00-12:00  Room 1423
Minkyu Lim (KAIST SAARC)

Gradient estimates of very weak solutions to mixed local problems

ABSTRACT. The mixed problem, which involves an elliptic equation with both local and nonlocal terms, has recently gained attention. It arises from the combination of two stochastic processes with different scales, such as a classical random walk and a Lévy process. In this presentation, we provide a sub-natural gradient estimate for a very weak solution to the mixed problem, under the condition that the integrability of the source term is below the natural exponent.

2025. 3. 28. (Fri) 11:00-12:00  Room 204 (Soorim)
Seonwoo Kim (Yonsei University)

$\Gamma$-Expansion of the Measure-Current Large Deviations Rate Functional

ABSTRACT. In this talk, we present the general theory of connecting the hierarchical structure of metastability and the $\Gamma$-expansion of the measure-current (or level-2.5) large deviations rate functional of non-reversible finite-state Markov chains. The first bridge between the metastability and the $\Gamma$-expansion of the (measure, or level-2) rate functional was discovered recently by Bertini, Gabrielli and Landim (Ann. Appl. Probab. 2024). We briefly review the previous works on this topic and present our own contributions. The talk is based on a recent work (arXiv:2412.13515) with Claudio Landim (IMPA).

2025. 3. 24. (Mon) 14:00-15:00  Room Online
Jongmyeong Kim (Academia Sinica)

Existence of one point singular symmetric self-similar solution of fast diffusion equation and Its large time behavior

ABSTRACT. In this talk, I will present the existence of global, forward eternal, one point singular, symmetric, and self-similar solution of fast diffustion equation u_t = \Delta u^m for n>2 and 0

2025. 3. 21. (Fri) 16:00-17:00  Room 1423
Eui Yoo (Seoul National University)

Three topological phases of the elliptic Ginibre ensembles with a point charge

ABSTRACT. In the large N limit, complex and symplectic random matrix models exhibit limiting spectra whose support is called the droplets. In this talk, we will discuss the elliptic Ginibre ensemble conditioned to have real eigenvalue with multiplicity proportional to the dimension of the matrix. We prove that the droplets are either simply connected, doubly connected, or composed of two simply connected components. Moreover, we present the explicit description of the droplet and electrostatic energies for the simply and doubly connected case. Finally, we introduce the asymptotic behavior of the moments of the characteristic polynomials of elliptic Ginibre matrices as an application. This is a joint work with Sung-Soo Byun.

2025. 3. 12. (Wed) 11:00-12:00  Room 1423
Minhyun Kim (Hanyang University)

Removable singularities for nonlocal equations and nonlocal minimal graphs

ABSTRACT. TBA

2025. 2. 27. (Thu) 16:00-17:00  Room 1503
Jeongtae Oh (Seoul National University)

Multiparameter Lacunary Maximal Functions on the Heisenberg Group

ABSTRACT. Ricci and Stein, in their work, considered the multiparameter lacunary maximal operator under suitable regularity conditions on the measure and established L^p estimates. In this talk, we introduce the Lacunary Elliptic Maximal Operator on the Heisenberg group and outline how to prove its L^p boundedness. Finally, under the regularity condition proposed by Ricci and Stein, we discuss how these results can be extended in the Heisenberg group setting.

2025. 2. 20. (Thu) 11:00-12:00  Room 1423
Takwon Kim (Sungshin Women's University)

Higher Regularity of Degenerate or Singular Equations via Generalized Schauder Theory

ABSTRACT. In this presentation, we introduce generalized Schauder theory for degenerate or singular parabolic equations and propose a novel approach to overcoming the limitations of traditional regularity theory. Conventional regularity methods often rely on the bootstrap argument to establish higher-order regularity, but this approach is not applicable to degenerate or singular equations. To address this challenge, we develop a new approximation method based on s-polynomials, which approximate the coefficients of the equation using fractional-order monomials rather than classical polynomials. Using this framework, we establish C_s^{2,α}-regularity for solutions and provide a theoretical foundation for proving higher regularity even in the presence of degeneracy or singularity. Furthermore, we demonstrate that this generalized Schauder theory is applicable to various problems, including Monge–Ampère equations, nonlocal equations, Gauss curvature flow, and Porous medium equations.

2025. 2. 14. (Fri) 16:30-17:30  Room Online
Marí­a del Mar González (Autonomous University of Madrid)

The limiting case of the fractional Caffarelli-Kohn-Nirenberg inequality in dimension one

ABSTRACT. We study the limiting case in dimension one for the fractional Caffarelli-Kohn-Nirenberg inequality, obtaining Onofri’s inequality in the unit disk as a limit. One difficulty here is the lack of an explicit expression for the extremal. An important aspect is the study of solutions of the weighted Liouville equation for the half-Laplacian in dimension one. This is joint work with A. Hyder (TIFR Bangalore) and M. Saez (U. Pontificia Chile).

2025. 2. 6. (Thu) 10:30-11:30  Room 1423
Ho-Sik Lee (Universität Bielefeld)

Regularity results for nonlocal equations

ABSTRACT. By Green's representation formula, the gradient of the solution to Poisson's equation is pointwisely bounded by a Riesz potential of the right-hand side of the equation, the so-called gradient potential estimate. We show gradient potential estimates for nonlinear nonlocal elliptic equations from the joint with Prof. Lars Diening (Bielefeld), Kyeongbae Kim (SNU), and Dr. Simon Nowak (Bielefeld). Next, we consider examples of solutions that such higher differentiability estimates fail when the nonlocal equation has irregular coefficients, even in the homogeneous linear case. Inspired by Meyers' example, which is considered in [Meyers '63] and [Kh. Balci, Diening, Giova, Passarelli di Napoli '22], we give such examples of nonlocal equations. This is the joint work with Prof. Anna Kh. Balci (Charles), Prof. Lars Diening, and Prof. Moritz Kassmann (Bielefeld).

2025. 1. 24. (Fri) 17:00-18:00  Room Online
Colin Petitjean (Gustave Eiffel University)

On curve-flat Lipschitz functions and their linearizations

ABSTRACT. Given two pointed metric spaces $M$, $N$, any basepoint-preserving Lipschitz map $f:M \to N$ can be associated with a canonical linearization $T_f:\mathcal{F}(M) \to \mathcal{F}(N)$, which is a bounded linear operator. The spaces $\mathcal{F}(M)$ and $\mathcal{F}(N)$ are known as Lipschitz free spaces over $M$ and $N$, respectively. A more or less recent trend explores the relationship between the properties of the nonlinear map $f$ and those of the linear operator $T_f$. In this talk, we will characterize the Lipschitz maps $f$ for which $T_f$ is a Dunford-Pettis operator (also referred to as completely continuous). If time permits, we will also discuss additional operator properties equivalent to the Dunford-Pettis property within this specific framework. This work is based on a collaboration with Gonzalo Flores, Mingu Jung, Gilles Lancien, Antonin Prochazka, and Andrés Quilis.

2025. 1. 23. (Thu) 15:00-16:00  Room 204 (Soorim)
Joonil Kim (Yonsei University)

Oscillatory integrals over global domains

ABSTRACT. Since Varchenko's seminal work, the asymptotics of oscillatory integrals and related problems have been effectively analyzed using the Newton polyhedra associated with the phase function $P$. These analyses have traditionally focused on integrals supported in sufficiently small neighborhoods. The purpose of this talk is to introduce the estimates of sub-level sets and oscillatory integrals over global domain $D$ with a primary focus on $D=\mathbb{R}^d$. In particular, we shall discuss (i) Piuseux series on $\mathbb{R}^2$ and resolutions of singularities, (ii) Analogue of Varchenko's theorem on the global domain (iii) Strichartz type inequality for dispersive partial differential equations (PDEs) as a key application. (iv) Phong and Stein's theorem regarding the global oscillatory integral operator (a joint work with Minbeom Kang)

Past Seminars

2024 Seminars
2023 Seminars
2022 Seminars
2021 Seminars
2020 Seminars
2019 Seminars

Organizers

Jae-Hwan Choi   -   Korea Institute for Advanced Study

Yong-Gwan Ji   -   Korea Institute for Advanced Study

Mingu Jung   -   Korea Institute for Advanced Study

Nam-Gyu Kang   -   Korea Institute for Advanced Study

Seonwoo Kim   -   Korea Institute for Advanced Study

Taegyu Kim   -   Korea Institute for Advanced Study

Wontae Kim   -   Korea Institute for Advanced Study

Joonhyun La   -   Korea Institute for Advanced Study

Jung-Kyoung Lee   -   Korea Institute for Advanced Study

Juyoung Lee   -   Korea Institute for Advanced Study

Se-Chan Lee   -   Korea Institute for Advanced Study

Taehun Lee   -   Korea Institute for Advanced Study

Sewook Oh   -   Korea Institute for Advanced Study

Sung-Jin Oh   -   UC Berkeley & KIAS

Hyangdong Park   -   Korea Institute for Advanced Study

Jaehyeon Ryu   -   Korea Institute for Advanced Study

Junehee Ryu   -   Korea Institute for Advanced Study

Jinsol Seo   -   Korea Institute for Advanced Study

Kyeong Song   -   Korea Institute for Advanced Study

Hyungsung Yun   -   Korea Institute for Advanced Study

Directions

Korea Institute for Advanced Study
Buildings 1 & 8, 85 Hoegi-ro, Dongdaemun-gu, Seoul, Korea



June E Huh Center for Mathematical Challenges
2nd Floor, 118 Hongneung-ro, Dongdaemun-gu, Seoul, Korea