Speaker: Nam-Gyu Kang (KIAS) [15:00~15:30] (Chairman: Wanmo Kang)
- Title: Geometry, Analysis, and Probability of Random Conformal Fields
- Abstract: I will explain Ward’s equation in conformal field theory in terms of the insertion of a stress tensor, the Teichmüller term (if any), and Lie derivative operator. As applications, I will outline their relation to SLE theory in various geometric settings.
Speaker: Jaehun Lee (KAIST) [15:30~16:00] (Chairman: Nam-Gyu Kang)
- Title: Stability of heat kernel estimates for symmetric jump processes on metric measure spaces
- Abstract: In this talk, we study symmetric pure jump Markov processes on metric measure spaces under volume doubling conditions. Our focus is on the stability of heat kernel estimates when the jumping kernel exhibits mixed polynomial growth. Unlike classical settings, the growth rate of the jumping kernel may not align with the scale function that determines near-and off-diagonal behaviors of the heat kernel. Assuming the lower scaling index of the scale function exceeds one, we establish the stability of heat kernel estimates under general conditions.
This talk is based on the joint work with Joohak Bae, Jaehoon Kang and Panki Kim.
Speaker: Jongbong An (KAIST) [16:15~16:45] (Chairman: Donghan Kim)
- Title: Optimal Contract Design with Labor-Leisure Choice under Limited Commitment: A Free Boundary Approach
- Abstract: We present a continuous-time optimal contracting problem involving labor-leisure choices under limited commitment. A principal offers a contract to a risk-averse agent whose wage follows a geometric Brownian motion. The agent derives utility from both consumption and leisure, modeled through a Cobb-Douglas utility function. Due to limited commitment, the agent's participation and promised utility constraints must be satisfied throughout the contracting horizon. By employing a dual approach and dynamic programming principles, we transform the problem into a singular stochastic control problem associated with a variational inequality and a free boundary. We provide an explicit closed-form solution to the variational inequality and characterize the optimal contract in terms of consumption and leisure processes. Numerical simulations illustrate the dynamic behavior of the optimal consumption, leisure, and continuation utility processes. Our approach demonstrates the effectiveness of duality methods and singular control techniques in solving nonlinear stochastic optimization problems with state-dependent constraints, contributing to the computational aspects of optimal control and contract theory.
Speaker: Yoonsoo Nam (KAIST) [16:45~17:15] (Chairman: Chulhee Yun)
- Title: Solve layerwise linear model to understand neural dynamical phenomena
- Abstract: While linear networks cannot learn non-linear features, they serve as tractable models for studying the dynamics induced by depth. By analyzing gradient flow through their layered structure (dynamical feedback), we provide a unified explanation for neural collapse, emergence, scaling laws, the lazy/rich regimes, and grokking.
Speaker: Diksha Gupta (KAIST) [17:15~17:45] (Chairman: Jaeyoung Byeon)
- Title: A Nonlinear Choquard Equation in the Hyperbolic Space
- Abstract: In this work, we explore the positive solutions of the following nonlinear Choquard equation involving the Green kernel of the fractional operator (-ΔBN)−α/2 in the hyperbolic space BN :
−ΔBN u − λu = [(-ΔBN)−α/2 |u|p] |u|p−2 u.
Here ΔBN denotes the Laplace–Beltrami operator, λ ≤ (N−1)2/4, 1 < p < 2∗α = (N+α)/(N−2), 0 < α < N, N ≥ 3. The exponent 2∗α is the critical exponent associated with the Hardy–Littlewood–Sobolev inequality. This problem is analogous to the Choquard equation in RN involving the nonlocal Riesz potential. We work in the Sobolev space H1(BN), employing harmonic analysis techniques including the Helgason Fourier transform and the semigroup approach to the fractional Laplacian. The Hardy–Littlewood–Sobolev inequality on complete Riemannian manifolds, as developed by Varopoulos, plays a central role in our analysis. We establish existence of solutions, prove their radial symmetry, and derive regularity properties.
Speaker: Kyeongsik Nam (KAIST) [9:30~10:00] (Chairman: Ji Oon Lee)
- Title: Moments of Critical 2D Stochastic Heat Flow
- Abstract: The stochastic heat equation (SHE) describes the evolution of a field under a random source. It arises as the universal scaling limit for a wide class of microscopic systems, and is a cornerstone of the Kardar-Parisi-Zhang (KPZ) universality class in 1+1 dimensions. In 2+1 dimensions, SHE undergoes a weak-to-strong disorder phase transition depending on the strength of noise. Its scaling limit in the ''critical'' regime, called Critical 2D Stochastic Heat Flow, was conjectured to display extreme intermittency, with h-th moments growing like exp(exp(h)). In this talk, we establish a lower bound of this conjecture. Joint work with Shirshendu Ganguly.
Speaker: Jungkyoung Lee (KIAS) [10:00~10:30] (Chairman: Kyeongsik Nam)
- Title: Metastability of Langevin Dynamics and Its Application
- 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.
This talk is based on joint with Claudio Landim(Instituto de Matemática Pura e Aplicada) and Insuk Seo(Seoul National University).
Speaker: Sungho Han (KAIST) [10:45~11:15] (Chairman: Moon-Jin Kang)
- Title: Stability of Viscous-Capillary Shock Waves for the Navier-Stokes-Korteweg System
- Abstract: In this talk, I will discuss the long-time behavior of the one-dimensional barotropic Navier-Stokes-Korteweg system. For small-amplitude Riemann data generating a viscous–capillary shock profile, we show that solutions converge to this profile up to a time-dependent shift. The analysis is based on the method of $a$-contraction with shifts, adapted to handle the third-order Korteweg term. We also treat Riemann data whose inviscid solution contains a rarefaction wave and a shock wave, and prove the time-asymptotic stability of the associated viscous–dispersive composite wave in the small-amplitude regime. These results are based on joint works
[1] S. Han, M.–J. Kang, J. Kim, and H. Lee, Long-time behavior toward viscous-dispersive shock for Navier-Stokes equations of Korteweg type, J. Differential Equations 426 (2025), 317-387.
[2] S. Han and J. Kim, Time-asymptotic stability of composite waves for the one-dimensional compressible fluid of Korteweg type, to appear in SIAM Journal on Mathematical Analysis.
Speaker: Junhee Ryu (KIAS) [11:15~11:45] (Chairman: Kyeong-Hun Kim)
- Title: $L_p$-estimates for nonlocal equations with general L\'evy measures
- Abstract: In this talk, we consider time-dependent nonlocal operators associated with general L\'evy measures of order $\sigma \in(0,2)$. We allow the class of L\'evy measures to be very singular and impose no regularity assumptions in the time variable. Continuity of the operators and the unique strong solvability of the corresponding nonlocal parabolic equations in $L_p$ spaces are established. We also demonstrate that, depending on the ranges of $\sigma$ and $d$, the operator can or cannot be treated in weighted mixed-norm spaces.
Speaker: Yoochan Han (KAIST)
- Title: Spectral properties and weak detection in stochastic block models
- Abstract: We consider the spectral properties of balanced stochastic block models of which the average degree grows slower than the number of nodes (sparse regime) or proportional to it (dense regime). For both regimes, we prove a phase transition of the extreme eigenvalues of SBM at the Kesten-Stigum threshold. We also prove the central limit theorem for the linear spectral statistics for both regimes. We propose a hypothesis test for determining the presence of communities of the graph, based on the central limit theorem for the linear spectral statistics.
Speaker: Seokun Choi (KAIST)
- Title: Exponentially slow mixing arising from entropic repulsion in $p$-SOS model
- Abstract: We investigate the Glauber dynamics of the generalized (2+1)-dimensional $p$-SOS model under a hard floor constraint ($1<p<\infty$). This setting induces entropic repulsion: the integer-valued interface height is forced to remain above the wall and consequently rises to a typical height $H(p,L)$ that depends on both the parameter $p$ and the system size $L$. In the classical SOS model ($p=1$), [caputo et al., 2014, caputo et al., 2016] derived an exponential lower and upper bound for the mixing time, demonstrating that the Glauber dynamics mixes only after an exponentially long time in the low-temperature regime (large $\beta$, the inverse temperature). However, beyond this case, no rigorous lower bounds were previously known: even for the widely studied Discrete Gaussian model ($p=2$), the metastable slowdown predicted by the entropic repulsion picture had remained an open problem.
Our main contribution is to close this gap by proving that exponentially slow(stretched-exponential) mixing arising from entropic repulsion persists throughout the regime $1<p<\infty$. Specifically, we establish an exponential lower bound, showing that the mixing time satisfies $\tau_{\mathrm{mix}}\ge \exp{\left(cL^{1-o(1)}\right)}$ for some $c>0$ depending on $p$ and $\beta$. In addition, we provide a refined metastability analysis, proving that the hitting time of an intermediate level $aH(p,L)$ is at least $\exp{\left(cL^{a^{d(p)}-o(1)}\right)}$, where $0<a<1$ and $d(p)$ is a positive function depending on $p$. The proof relies on an extension of the classical Peierls-type contour estimates, originally developed for $p=1$, to the nonlinear $p$-SOS setting. Taken together, these results demonstrate that entropic repulsion induces uniformly slow mixing across the entire $p$-SOS family, thereby extending a phenomenon that had previously been established only for $p=1$.
Speaker: Bonsoo Koo (KAIST)
- Title: Hedging ELS via local expected shortfall
- Abstract: Equity-Linked Securities (ELS) are popular in Korea but difficult to hedge. We first review fundamental concepts of financial mathematics regarding risk and hedging. Secondly, we examine specific ELS challenges, such as delta spikes. Finally, we propose a modern numerical approach based on local expected shortfall.
Speaker: Kyungjae Lee (KAIST)
- Title: On the Stability of One-Step Generative Models: Rethinking GAN Training Instability Without Heuristic Regularization
- Abstract: While multi-step generative models such as diffusion have achieved remarkable empirical success, stably learning one-step generative models remains challenging. In practice, training such models---including distilled diffusion and consistency models---often relies on heuristic regularization. Using GANs as a classical one-step generative testbed, we show that training instability is not inherent to one-step generation, but can be addressed by rethinking both the loss landscape and the gradient dynamics of GANs. On the loss side, we challenge the prevailing reliance on the Wasserstein distance, and propose the zero-infinity distance which equals zero when two distributions match exactly and infinity otherwise. On the dynamics side, we show that GAN objectives admit strict non-minimax points---the minimax analogue of strict saddles---and that two-timescale extragradient methods are guaranteed to escape them while remaining stable at global solutions. As a further consequence, we show that among infinitely many global solutions, the learned generator exhibits an implicit bias toward minimum-norm solutions; somewhat unexpectedly, our real-world experiments indicate that this bias alone does not translate into good generalization, in sharp contrast to supervised learning, highlighting a fundamental gap between optimization-induced implicit bias and generalization in one-step generation.
Speaker: Chaewon Moon (KAIST)
- Title: The Cost of Robustness: Tighter Bounds on Parameter Complexity for Robust Memorization in ReLU Nets
- Abstract: We study the parameter complexity of robust memorization for ReLU networks: the number of parameters required to interpolate any given dataset with ϵ-separation between differently labeled points, while ensuring predictions remain consistent within a μ-ball around each training sample. We establish upper and lower bounds on the parameter count as a function of the robustness ratio ρ=μ/ϵ. Unlike prior work, we provide a fine-grained analysis across the entire range ρ∈(0,1) and obtain tighter upper and lower bounds that improve upon existing results. Our findings reveal that the parameter complexity of robust memorization matches that of non-robust memorization when ρ is small, but grows with increasing ρ.
Speaker: Ki Hun Hong (KAIST)
- Title: Deep Latent Variable Model based Vertical Federated Learning with Flexible Alignment and Labeling Scenarios
- Abstract: Federated learning (FL) has attracted significant attention for enabling collaborative learning without exposing private data. Among the primary variants of FL, vertical federated learning (VFL) addresses feature-partitioned data held by multiple institutions, each holding complementary information for the same set of users. However, existing VFL methods often impose restrictive assumptions such as a small number of participating parties, fully aligned data, or only using labeled data. In this work, we reinterpret alignment gaps in VFL as missing data problems and propose a unified framework that accommodates both training and inference under arbitrary alignment and labeling scenarios, while supporting diverse missingness mechanisms. In the experiments on 168 configurations spanning four benchmark datasets, six training-time missingness patterns, and seven testing-time missingness patterns, our method outperforms all baselines in 160 cases with an average gap of 9.6 percentage points over the next-best competitors. To the best of our knowledge, this is the first VFL framework to jointly handle arbitrary data alignment, unlabeled data, and multi-party collaboration all at once.
Speaker: Eunchan Jeon (KAIST)
- Title: The Neumann version of the Caffarelli-Kohn-Nirenberg inequality on bounded domains.
- Abstract: We study the Neumann version of the Caffarelli-Kohn-Nirenberg inequality on bounded domains containing the origin. We investigate the optimal constant of the inequality as a variational problem. Using direct variational methods, we establish sharp positivity criteria for the optimal constant. In the subcritical regime, we obtain almost complete attainability and non-attainability results that hold independently of the geometry of the domain. In contrast, in the critical regime, the attainability and non-attainability results strongly depend on the geometry of the domain. These results and basic direct method will be introduced in this talk.
Speaker: Junyoung Heo (KAIST)
- Title: Fragmentation rate of the 1-dimensional logistic diffusive equation
- Abstract: The logistic diffusive equation is an important model in ecology. Given diffusion rate and resource density, this model describes the population density, assuming that the population grows logistically and the population diffuses. The main concern is to find a resource density which maximizes total population. Such a resource is called optimal control. One important characteristic of an optimal control is that it should be fragmented as the diffusion rate gets smaller. In mathematical language, an optimal control should satisfy , where is the diffusion rate. We prove that in 1-dimensional case, this fragmentation rate is exact. In other words, we show that. The proof consists of carefully estimating population and resource using the Taylor expansion, and then locally constructing a better resource when a part of the resource is too fragmented.
Speaker: HyeonSeop Oh (KAIST)
- Title: Convergence to superposition of stationary solution, rarefaction and shock for the inflow problem of the 1D Navier-Stokes equations
- Abstract: In this talk, we study the asymptotic stability of solutions to the inflow problem for the onedimensional isentropic Navier-Stokes system in the half space. When the boundary value lies in the subsonic region, all possible asymptotic wave patterns were classified in [1].
We consider the most complicated case, the superposition of a stationary solution, a rarefaction wave, and a viscous shock wave. In this superposition, the stationary solution is degenerate and large. We prove that if the initial perturbation and the strengths of the rarefaction wave and the shock wave are small, then the solution converges asymptotically to the superposition up to a dynamical shift of the shock.
This talk is based on a joint work with Sungho Han, Moon-Jin Kang, Jeongho Kim, and Nayeon Kim [2].
[1] Akitaka Matumura. Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas. Methods Appl. Anal. 8 (2001), no. 4, 645-666.
[2] Sungho Han, Moon-Jin Kang, Jeongho Kim, Nayeon Kim, HyeonSeop Oh. Convergence to superposition of boundary layer, rarefaction and shock for the inflow problem of the 1D Navier-Stokes equations. to appear in Commun. Math. Phys.
Speaker: Namhyun Eun (KAIST)
- Title: $L^2$-contraction for viscous-dispersive shock waves of KdV-Burgers equation
- Abstract: It is well known that the Korteweg-de Vries-Burgers (KdVB) equation admits traveling wave solutions, referred to as viscous-dispersive shock waves. When the dispersion dominates viscosity, the associated shock profiles exhibit infinitely many oscillations. Although the KdVB equation serves as a canonical model incorporating nonlinearity, viscosity, and dispersion, the stability of these shock profiles---especially in the oscillatory regime---remains poorly understood.
In this talk, we discuss detailed structural properties of the shock profiles, and then prove an $L^2$-contraction property under arbitrarily large perturbations, up to a time-dependent shift. Finally, we justify the zero viscosity-dispersion limit.
This is a collaborative work with Geng Chen (University of Kansas), Moon-Jin Kang (KAIST), and Yannan Shen (University of Kansas).
Speaker: Beomjong Kwak (KAIST)
- Title: Strichartz estimates and global well-posedness of the cubic NLS on $\mathbb{T}^2$
- Abstract: In this talk, we present an optimal $L^4$-Strichartz estimate for the Schrödinger equation on the two-dimensional rational torus $\mathbb{T}^2$. We first recall the previously known results and counterexamples on the Strichartz estimates on the torus. Then we present our new Strichartz estimate, which has an optimal amount of loss, and the small-data global well-posedness of (mass-critical) the cubic NLS in $H^s,s>0$ as its consequence. Our Strichartz estimate is based on a combinatorial proof. We introduce our key proposition, the Szemerédi-Trotter theorem, and outline the idea of the proof. This is a joint work with Sebastian Herr.
Namhyun Eun (KAIST)
- Title: Stability of a Riemann shock in a physical class: from Brenner-Navier-Stokes-Fourier to Euler
- Abstract: The stability of an irreversible singularity, such as a Riemann shock to the full Euler system, in the absence of any technical conditions on perturbations, remains a major open problem even within one-dimensional framework. A natural approach to justify such stability is to consider vanishing dissipation (or viscosity) limits of physical viscous flows. We prove the existence of vanishing dissipation limits, on which a Riemann shock of small amplitude is stable (up to a time-dependent shift) and unique. We adopt the Brenner-Navier-Stokes-Fourier system, based on the bi-velocity theory, as a physical viscous model. The key ingredient of the proof is the uniform stability of the viscous shock with respect to the viscosity strength. The uniformity is
ensured by contraction estimates of any large perturbations around the shock. We use the method of a-contraction with shifts, but we improve it by
introducing a more delicate analysis of the localizing effect given by viscous shock derivatives. This is the first resolution of the open problem on the unconditional stability and uniqueness of Riemann shock solutions to the full Euler system in a class of vanishing physical dissipation limits.
Eunchan Jeon (KAIST)
- Title: The Neumann version of the Caffarelli-Kohn-Nirenberg inequality on bounded domains.
- Abstract: We study the Neumann version of the Caffarelli-Kohn-Nirenberg inequality on bounded domains containing the origin. We investigate the optimal constant of the inequality as a variational problem. Using direct variational methods, we establish sharp positivity criteria for the optimal constant. In the critical regime, the attainability and non-attainability results strongly depend on the geometry of the domain. In this poster, we mainly present the results and methods for the critical case.
HyeonSeop Oh (KAIST)
- Title: Asymptotic behavior toward viscous shocks for the outflow problem of isentropic Navier-Stokes equations
- Abstract: We study the large-time asymptotic stability of viscous shock profile to the outflow problem of barotropic Navier-Stokes equations on a half line. We consider the case when the far-field state as a right-end state of 2-Hugoniot shock curve belongs to the subsonic region or transonic curve. We employ the method of a-contraction with shifts, to prove that if the strength of viscous shock wave is small and sufficiently away from the boundary, and if a initial perturbation is small, then the solution asymptotically converges to the viscous shock up to a dynamical shift. We also prove that the speed of time-dependent shift decays to zero as times goes to infinity, the shifted viscous shock still retains its original profile time-asymptotically. This is based on a joint work with Moon-Jin Kang, and Yi Wang.
Aro Lee (KAIST)
- Title: Fluctuations of the Largest Eigenvalues of Transformed Spiked Wigner Matrices
- Abstract: We consider a spiked random matrix model obtained by applying a function entrywise to a signal-plus-noise symmetric data matrix. We prove that the largest eigenvalue of this model, which we call a transformed spiked Wigner matrix, exhibits Baik–Ben Arous–P´ech´e (BBP) type phase transition. We show that the law of the fluctuation converges to the Gaussian distribution when the effective signal-to-noise ratio (SNR) is above the critical number, and to the GOE Tracy– Widom distribution when the effective SNR is below the critical number. We provide precise formulas for the limiting distributions and also concentration estimates for the largest eigenvalues, both in the supercritical and the subcritical regimes. This is joint work with Ji Oon Lee.