In this talk, we study nonlinear drift diffusion equations with measure valued external forces, focusing on the porous medium and fast diffusion regimes By introducing a new class of energy estimates that effectively handle both measure data and drift terms, we establish the existence of nonnegative weak solutions with gradient estimates under broad conditions on the drift This is joint work with Sukjung Hwang, Kyungkeun Kang, and Hwa Kil Kim.

We consider a class of nonlinear measure data problems involving non-uniformly elliptic operators. The leading operator of the problem under consideration is characterized by the fact that its growth and ellipticity show drastic change with respect to the position. We present a new approach to local Calderón-Zygmund type gradient estimates for such a problem, identifying a new, natural structural assumption.

This talk is based on joint work with Sungkyung Kang and JungHwan Park. We show that the (2n,1)-cable of the figure-eight knot has infinite order in the smooth concordance group, for any n≥1. The proof relies on the real κ-invariant, which satisfies a real version of the 10/8-inequality, in combination with techniques involving higher-order branched covers of knots and surfaces. Together with earlier work by Hom, Kang, Park, and Stoffregen, this result implies that any nontrivial cable of the figure-eight knot has infinite order in the smooth concordance group.

We prove that for any circle graph $H$ with at least one edge and for any positive integer $k$, there exists an integer $t=t(k,H)$ so that every graph $G$ either has a vertex-minor isomorphic to the disjoint union of $k$ copies of $H$, or has a $t$-perturbation with no vertex-minor isomorphic to $H$. Using the same techniques, we also prove that for any planar multigraph $H$, every binary matroid either has a minor isomorphic to the cycle matroid of $kH$, or is a low-rank perturbation of a binary matroid with no minor isomorphic to the cycle matroid of $H$. This is joint work with Rutger Campbell, J. Pascal Gollin, Meike Hatzel, O-joung Kwon, Rose McCarty, and Sebastian Wiederrecht.

In this talk I will discuss front propagation in the KPP type reaction-diffusion equations with spatially periodic coefficients. Since the pioneering work of Kolmogorov--Petrovsky--Piscounov and Fisher in 1937, front propagations in KPP type reaction-diffusion equations have been studied extensively. Starting around 1950's, KPP type equations have played an important role in mathematical ecology, in particular, in the study of biological invasions in a given habitat. What is particularly important is to estimate the speed of propagating fronts. In the spatially homogeneous case, there is a simple formula for the speed, which was given in the work of KPP and Fisher in 1937. However, if the coefficients are spatially periodic, estimating the front speed is much more difficult, and it involves the principal eigenvalue of a certain operator that is not self-adjoint. In this talk, I will mainly focus on the one-dimensional problem and give an overview of the past research on this theme starting around 1980's. I will also present a work of mine on KPP type equations in 2D in periodically stratified media.

In recent years, the behavior of solution fronts of reaction-diffusion equations in the presence of obstacles has attracted attention among many researchers. Of particular interest is the case where the equation has a bistable nonlinearity. In this talk, I will consider the case where the obstacle is a wall of infinite span with many holes and discuss whether the front can pass through the wall and continue to propagate (“propagation”) or is blocked by the wall (“blocking”). The answer depends largely on the size and the geometric configuration of the holes. This problem has led to a variety of interesting mathematical questions that are far richer than we had originally anticipated. Many questions still remain open. This is joint work with Henri Berestycki and François Hamel.

Lehel's conjecture states that every 2-edge-colouring of the complete graph $K_n$ admits a partition of its vertices into two monochromatic cycles. This was proven for sufficiently large n by Luczak, Rödl, and Szemerédi (1998), extended by Allen (2008), and fully resolved by Bessy and Thomassé in 2010. We consider a rainbow version of Lehel’s conjecture for properly edge-coloured complete graphs. We prove that for any properly edge-coloured $K_n$ with sufficiently large n, there exists a partition of the vertex set into two rainbow cycles, each containing no two edges of the same colour. This is joint work with Pedro Araújo, Xiaochuan Liu, Taísa Martins, Walner Mendonça, Luiz Moreira, and Vinicius Fernandes dos Santos.

TBA

We consider a class of linear estimates for evolution PDEs on the Euclidean space, called Strichartz estimate. Strichartz estimates are well-established for fundamental linear PDEs, such as heat and wave equations. As a simple model of such, we consider the Schrödinger example, introducing classical Strichartz estimates with proofs. Reference Terence Tao, Nonlinear dispersive equations: local and global analysis, Chapter 2.3

Abstract: In this talk, we consider the second-order quasilinear degenerate elliptic equation whose dominant part has the form $(2x - au_x)u_{xx} + bu_{yy} - u_x = 0$, where $a$ and $b$ are positive constants. We first introduce the physical situation that motivates the present analysis in a very brief manner, and then discuss mathematical difficulties involved in the analysis of the problem. The main part of this talk focuses on methods to overcome those difficulties, such as vanishing viscosity approximation and parabolic scaling. - Reference: [1] Chen, G.-Q. and Feldman, M. (2010). Global solutions to shock reflection by large-angle wedges, Ann. of Math. 171: 1019–1134. *Main reference [2] Bae, M., Chen, G.-Q. and Feldman, M. (2009). Regularity of solutions to regular shock reflection for potential flow, Invent. Math. 175: 505–543.

Vortex dipoles are one of the most iconic structures in two-dimensional incompressible flows. In this talk, I will present recent results on the existence and stability of traveling wave solutions to the two-dimensional incompressible Euler equations. These solutions take the form of counter-rotating vortex dipoles symmetric across a horizontal axis. A classical example is the Chaplygin–Lamb dipole, where the two vortex regions are tightly packed near the symmetry axis, leading to intense interaction. I will describe a variational framework for constructing such solutions and discuss their dynamical properties. This is joint work with Kyudong Choi and Young-Jin Sim (UNIST).

6 students (four advanced undergraduate students and two first year graduate students) present their summer study results on the subject of algebraic curves and Riemann surfaces. The topic include: - On sheaves and cohomology over topological spaces. - On relationship between divisors and line bundles on compact Riemann surfaces. - On holomorphic vector bundles on compact Riemann surfaces. - On Cech cohomology of sheaves on compact Riemann surfaces. - On Serre duality theorem on compact Riemann surfaces. - On the Riemann-Roch theorem on compact Riemann surfaces.

Topological Data Analysis (TDA) has emerged as a powerful framework for uncovering meaningful structure in high-dimensional, complex datasets. In this talk, we present two applications of TDA in analyzing patterns, one in the tumor microenvironment (TME) and the other in high-resolution chemical profiling. In the first case, we develop a TDA-based framework to quantify malignant-immune cell interactions in Diffuse Large B Cell Lymphoma using multiplex immunofluorescence imaging. By introducing Topological Malignant Clusters (TopMC) and leveraging persistence diagrams, we capture both global infiltration patterns and local density-based features. This robust approach enables consistent prognostic assessment regardless of tumor region heterogeneity and reveals correlations with patient survival. In the second application, we utilize the Ball Mapper algorithm to simplify and visualize high-dimensional data obtained from 2D Chromatography with high-resolution mass spectrometry. This enables interpretable chemical profiling of complex mixtures and supports tasks such as sample authentication and environmental analysis. Together, these studies demonstrate the versatility and interpretability of TDA for extracting biologically and chemically meaningful information.

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https://scholar.google.com/citations?user=4w2vNhcAAAAJ&hl=en

The matroid intersection problem is a fundamental problem in combinatorial optimization. In this problem we are given two matroids and the goal is to find the largest common independent set in both matroids. This problem was introduced and solved by Edmonds in the 70s. The importance of matroid intersection stems from the large variety of combinatorial optimization problems it captures; well-known examples in computer science include bipartite matching and packing of spanning trees/arborescences. In this talk, we introduce a “sparsifer” for the matroid intersection problem and use it to design algorithms for two problems closely related to streaming: a one-way communication protocol and a streaming algorithm in the random-order streaming model. This is a joint-work with François Sellier.

In physics, the phase transition between localized and delocalized phases in disordered systems, often called the Anderson transition, has attracted significant interest. Several intriguing models display this behavior, including random Schrödinger operators, random band matrices, and sparse random matrices. Heavy-tailed random matrices similarly capture this phase transition, making them a crucial class of models in understanding localization phenomena. In this talk, we will discuss the phase transition of the right singular vector associated with the smallest singular value of a rectangular random matrix. This work is in collaboration with Zhigang Bao (University of Hong Kong) and Xiaocong Xu (University of Southern California).

We study the time evolution of an initial product state in a system of almost-bosonic-extended-anyons in the large-particle limit. We show that the dynamics of this system can be well approximated, in finite time, by a product state evolving under the effective Chern–Simons–Schrödinger equation. Furthermore, we provide a convergence rate for the approximation in terms of the radius of the extended anyons. These results establish a rigorous connection between the microscopic dynamics of almost-bosonic-anyon gases and the emergent macroscopic behavior described by the Chern–Simons–Schrödinger equation.This talk is based on a work with Théotime Girardot

We formulate a geometric version of the Erdős-Hajnal conjecture that applies to finite projective geometries rather than graphs. In fact, we give a natural extension of the ‘multicoloured’ version of the Erdős-Hajnal conjecture. Roughly, our conjecture states that every colouring of the points of a finite projective geometry of dimension $n$ not containing a fixed colouring of a fixed projective geometry $H$ must contain a subspace of dimension polynomial in $n$ avoiding some colour. When $H$ is a ‘triangle’, there are three different colourings, all of which we resolve. We handle the case that $H$ is a ‘rainbow’ triangle by proving that rainbow-triangle-free colourings of projective geometries are exactly those that admit a certain decomposition into two-coloured pieces. This is closely analogous to a theorem of Gallai on rainbow-triangle-free coloured complete graphs. The two non-rainbow colourings of $H$ are handled via a recent breakthrough result in additive combinatorics due to Kelley and Meka. This is joint work with Carolyn Chun, James Dylan Douthitt, Wayne Ge, Matthew E. Kroeker, and Peter Nelson.