The singular limit problem is an important issue in various forms of ODEs and PDEs, and it is particularly known as a fundamental problem in equations derived from fluid dynamics. In this presentation, I will introduce some general phenomena of the singular limit problem through several examples. Subsequently, I will examine how the solution of the Euler-Maxwell equations converges to the MHD equations under the assumption that the speed of light approaches infinity, and how the Boussinesq equations converge to the QG equations in certain regimes.

Individual human cancer cells often show different responses to the same treatment. In this talk I will share the quantitative experimental approaches my lab has developed for studying the fate and behavior of human cells at the single-cell level. I will focus on the tumor suppressor protein p53, a transcription factor controlling genomic integrity and cell survival. In the last several years we have established the dynamics of p53 (changes in its levels over time) as an important mechanism controlling gene expression and guiding cellular outcomes. I will present recent studies from the lab demonstrating how studying p53 dynamics in response to radiation and chemotherapy in single cells can guide the design and schedule of combinatorial therapy, and how the p53 oscillator can be used to study the principles and function of entertainment in Biology. I will also present new findings suggesting that p53’s post-translational modification state is altered between its first and second pulses of expression, and the effects these have on gene expression programs over time.

Wavelets provide a versatile framework for signal representation and analysis, integrating ideas from harmonic analysis, approximation theory, and practical algorithm design. In this talk, we introduce foundational concepts in wavelet theory, focusing on classical results regarding wavelet expansions and approximations. Building on these basics, we explore modern developments and discuss how these approaches can balance theoretical rigor with practical convenience. The presentation aims to offer both a solid introduction to classical wavelet theory and a glimpse into current and future research directions. Part of the talk is based on joint work with Hyojae Lim.

A surface can be decomposed into a union of pairs of pants, a construction known as a pants decomposition. This fundamental observation reveals many important properties of surfaces. By forming a simplicial graph whose vertices represent pants decompositions, connecting two vertices with an edge whenever the corresponding decompositions differ by a simple move, we obtain a graph that is quasi-isometric to the Weil–Petersson metric on Teichmüller space. Meanwhile, topologists often study a structure called a rose, formed by attaching multiple circles at a single point. A rose is homotopy equivalent to a compact surface with boundary. Consequently, we can define a pants decomposition of a rose as the pants decomposition of a surface homotopy equivalent to it. In this talk, we will explore the concept of pants decompositions specifically in the context of roses.

H. Föllmer introduced in 1981 a version of Itô's formula without any probabilistic assumptions. It has been generalized in several aspects, including pathwise Tanaka's formula, high-order, and functional change-of-variable formula. Its drawbacks and a brief application to mathematical finance will also be presented.

In this note, we investigate threshold conditions for global well-posedness and finite-time blow-up of solutions to the focusing cubic nonlinear Klein–Gordon equation (NLKG) on $\bbR^{1+3}$ and the focusing cubic nonlinear Schrödinger equation (NLS) on $\bbR$. Our approach is based on the Payne–Sattinger theory, which identifies invariant sets through energy functionals and conserved quantities. For NLKG, we review the Payne–Sattinger theory to establish a sharp dichotomy between global existence and blow-up. For NLS, we apply this theory with a scaling argument to construct scale-invariant thresholds, replacing the standard mass-energy conditions with a $\dot{H}^{\frac12}$-critical functional. This unified framework provides a natural derivation of global behavior thresholds for both equations.

In this talk, we will discuss about smooth random dynamical systems and group actions on surfaces. Random dynamical systems, especially understanding stationary measures, can play an important role to understand group actions. For instance, when a group action on torus is given by toral automorphisms, using random dynamics, Benoist-Quint classified all orbit closures. In this talk, we will study non-linear actions on surfaces using random dynamics. We will discuss about absolutely continuity and exact dimensionality of stationary measures as well as classification of orbit closures. This talk will be mostly about the ongoing joint work with Aaron Brown, Davi Obata, and Yuping Ruan.

Our present healthcare system focuses on treating people when they are ill rather than keeping them healthy. We have been using big data and remote monitoring approaches to monitor people while they are healthy to keep them that way and detect disease at its earliest moment presymptomatically. We use advanced multiomics technologies (genomics, immunomics, transcriptomics, proteomics, metabolomics, microbiomics) as well as wearables and microsampling for actively monitoring health. Following a group of 109 individuals for over 13 years revealed numerous major health discoveries covering cardiovascular disease, oncology, metabolic health and infectious disease. We have also found that individuals have distinct aging patterns that can be measured in an actionable period of time. Finally, we have used wearable devices for early detection of infectious disease, including COVID-19 as well as microsampling for monitoring and improving lifestyle. We believe that advanced technologies have the potential to transform healthcare and keep people healthy.

Hyperbolicity is a fundamental concept that connects differential geometry and algebraic geometry. It is in general very hard to determine whether a given manifold or variety is hyperbolic or not. A key tool for verifying hyperbolicity is symmetric differentials; more precisely, the positivity of the cotangent bundle. In this talk, I will introduce various notions of hyperbolicity and explore their geometric properties. I will also discuss how the cotangent bundle, or more generally the syzygy bundle, plays a crucial role in this context.

We establish the generic local Langlands correspondence by showing the equality of Langlands-Shahidi L-functions and Artin L-functions in the case of even unitary similitude groups. As an application, with one assumption on L-function, we prove both weak and strong versions of the generic Arthur packet conjectures in the cases of even unitary similitude groups and even unitary groups. Furthermore, we describe and define generic L-packets and therefore we were able to remove the above assumption. With our definition of L-packets, we recently prove its expected properties such as Shahidi's conjecture and finiteness of L-packets. This is in preparation and joint work with Muthu Krishnamurthy and Freydoon Shahidi.

The Langlands program, introduced by Robert Langlands, is a set of conjectures that attempt to build bridges between two different areas: Number Theory and Representation Theory (Automorphic forms). The program is also known as a generalization of a well-known theorem called Fermat’s Last Theorem. More precisely, when Andrew Wiles proved Fermat’s Last Theorem, he proved a special case of so-called Taniyama-Shimura-Weil Conjecture, which states that every elliptic curve is modular. And as a corollary, he was able to prove Fermat’s Last Theorem since Taniyama-Shimura-Weil Conjecture implies that certain elliptic curves associated with Fermat-type equations must be modular, leading to a contradiction. Note that the Langlands program is a generalization of the Taniyama-Shimura-Weil conjecture. In the first part of the colloquium, we briefly go over the following subjects: (1) Fermat’s Last Theorem (2) Taniyama-Shimura-Weil conuecture And then, in the remaining of the talk, we start to explain a bit of the Langlands program (3) Langlands program and L-functions (4) (If time permits) Recent progress This colloquium will be accessible to graduate students in other fields of mathematics (and undergraduate students who are interested in Number theory) at least in the first part.

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심사위원장: 백형렬, 심사위원: 김우진, 박정환, 김상현(고등과학원), 이상진(건국대학교)

The dimension of a poset is the least integer $d$ such that the poset is isomorphic to a subposet of the product of $d$ linear orders. In 1983, Kelly constructed planar posets of arbitrarily large dimension. Crucially, the posets in his construction involve large standard examples, the canonical structure preventing a poset from having small dimension. Kelly’s construction inspired one of the most challenging questions in dimension theory: are large standard examples unavoidable in planar posets of large dimension? We answer the question affirmatively by proving that every $d$-dimensional planar poset contains a standard example of order $\Omega(d)$. More generally, we prove that every poset from Kelly’s construction appears in every poset with a planar cover graph of sufficiently large dimension. joint work with Heather Smith Blake, Jędrzej Hodor, Piotr Micek, and William T. Trotter.

In this talk, we will introduce vector field method for the wave equation. The key step is to establish the Klainerman-Sobolev inequality developed in [1]. Using this inequality, we will provide dispersive estimates of the linear wave equation, and prove small-data global existence for some nonlinear wave equations. The main reference will be Chapter II in [2]. 참고문헌: [1]. Sergiu Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math. 38 (1985), no. 3, 321–332. MR 784477 [2]. Christopher D. Sogge, Lectures on Nonlinear Wave Equations, Second Edition

The standard theory of infectious diseases, tracing back to the work of Kermack and McKendrick nearly a century ago, has been a triumph of mathematical biology, a rare marriage of theory and application. Yet the limitations of its most simple representations, which has always been known, have been laid bare in dealing with COVID-19, sparking a spate of extensions of the basic theory to deal more effectively with aspects of viral evolution, asymptotic stages, heterogeneity of various kinds, the ambiguities of notions of herd immunity, the role of social behaviors and other features. This lecture will address some progress in addressing these, and open challenges in expanding the mathematical theory.

TBD

Since the proof of the graph minor structure theorem by Robertson and Seymour in 2004, its underlying ideas have found applications in a much broader range of settings than their original context. They have driven profound progress in areas such as vertex minors, pivot minors, matroids, directed graphs, and 2-dimensional simplicial complexes. In this talk, I will present three open problems related to this development, each requiring some background.

Abstract: We consider the initial-boundary value problem (IBVP) for the 1D isentropic Navier-Stokes equation (NS) in the half space. Unlike the whole space problem, a boundary layer may appear due to the influence of viscosity. In this talk, we first briefly study the asymptotic behavior for the initial value problem of NS in the whole space. Afterwards, we will present the characterization of the expected asymptotics for the IBVP of NS in the half space. Here, we focus only on the inflow problem, where the fluid velocity is positive on the boundary. Reference: Matsumura, Akitaka. 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.

In this talk, we explore some ordinary and partial differential equations (ODEs and PDEs) in a class of completely integrable systems. We begin by introducing Hamiltonian systems in classical mechanics and their integrability. We then discuss completely integrable ODEs and introduce the Lax pair formulation, a powerful framework for analyzing complete integrability. As a concrete example, we examine the classical Calogero-Moser system, a well-known completely integrable many-body system with remarkable mathematical properties. We then investigate the Calogero-Moser derivative nonlinear Schrödinger equation (CM-DNLS), which is a completely integrable PDE that arises as the continuum limit of the classical Calogero-Moser system. Finally, we present recent developments in the study of CM-DNLS, such as well-posedness and long-time dynamics.

We study stochastic motion of objects in micrometer-scale living systems: tracer particles in living cells, pathogens in mucus, and single cells foraging for food. We use stochastic models and state space models to track objects through time and infer properties of objects and their surroundings. For example, we can calculate the distribution of first passage times for a pathogen to cross a mucus barrier, or we can spatially resolve the fluid properties of the cytoplasm in a living cell. Recently developed computational tools, particularly in the area of Markov Chain Monte Carlo, are creating new opportunities to improve multiple object tracking. The primary remaining challenge, called the data association problem, involves mapping measurement data (e.g., positions of objects in a video) to objects through time. I will discuss new developments in the field and ongoing efforts in my lab to implement them. I will motivate these techniques with specific examples that include tracking salmonella in GI mucus, genetically expressed proteins in the cell cytoplasm, active transport of nuclei in multinucleate fungal cells, and raphid diatoms in seawater surface interfaces.

Molecular simulations serve as fundamental tools for understanding and predicting the system of interest at atomic level. It is significant for applications like drug and material discovery, but often cannot scale to real-world problems due to the computational bottleneck. In this seminar, I will briefly introduce this area and recent machine learning algorithms that have shown great promise in accelerating the molecular simulations. I will also introduce some of my recent research in this direction. First work is about structure prediction of metal-organic frameworks using geometric flow matching (or neural ODE on SO(3) manifolds) and (2) simulating chemical reactions / transition paths through RL-like training of diffusion models (or log-divergence minimization between path measures).

The concept of p-th variation of a real-valued continuous function along a general class of refining sequence of partitions is presented. We show that the finiteness of the p-th variation of a given function is closely related to the finiteness of ℓp-norm of the coefficients along a Schauder basis, similar to the fact that Hölder coefficient of the function is connected to ℓ∞-norm of the Schauder coefficients. This result provides an isomorphism between the space of α-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.

We present a full characterization of the unavoidable induced subgraphs of graphs with large pathwidth. This consists of two results. The first result says that for every forest H, every graph of sufficiently large pathwidth contains either a large complete subgraph, a large complete bipartite induced minor, or an induced minor isomorphic to H. The second result describes the unavoidable induced subgraphs of graphs with a large complete bipartite induced minor. We will also try to discuss the proof of the first result with as much detail as time permits. Based on joint work with Maria Chudnovsky and Sophie Spirkl.

A family $\mathcal{F}$ of graphs is said to satisfy the Erdős-Pósa property if there exists a function $f$ such that for every positive integer $k$, every graph $G$ contains either $k$ (vertex-)disjoint subgraphs in $\mathcal{F}$ or a set of at most $f(k)$ vertices intersecting every subgraph of $G$ in $\mathcal{F}$. We characterize the obstructions to the Erdős-Pósa property of $A$-paths in unoriented group-labelled graphs. As a result, we prove that for every finite abelian group $\Gamma$ and for every subset $\Lambda$ of $\Gamma$, the family of $\Gamma$-labelled $A$-paths whose lengths are in $\Lambda$ satisfies the half-integral relaxation of the Erdős-Pósa property. Moreover, we give a characterization of such $\Gamma$ and $\Lambda\subseteq\Gamma$ for which the same family of $A$-paths satisfies the full Erdős-Pósa property. This is joint work with Youngho Yoo.

An induced packing of cycles in a graph is a set of vertex-disjoint cycles such that the graph has no edge between distinct cycles of the set. The classic Erdős-Pósa theorem shows that for every positive integer $k$, every graph contains $k$ vertex-disjoint cycles or a set of $O(k\log k)$ vertices which intersects every cycle of $G$. We generalise this classic Erdős-Pósa theorem to induced packings of cycles of length at least $\ell$ for any integer $\ell$. We show that there exists a function $f(k,\ell)=O(\ell k\log k)$ such that for all positive integers $k$ and $\ell$ with $\ell\geq3$, every graph $G$ contains an induced packing of $k$ cycles of length at least $\ell$ or a set $X$ of at most $f(k,\ell)$ vertices such that the closed neighbourhood of $X$ intersects every cycle of $G$. Furthermore, we extend the result to long cycles containing prescribed vertices. For a graph $G$ and a set $S\subseteq V(G)$, an $S$-cycle in $G$ is a cycle containing a vertex in $S$. We show that for all positive integers $k$ and $\ell$ with $\ell\geq3$, every graph $G$, and every set $S\subseteq V(G)$, $G$ contains an induced packing of $k$ $S$-cycles of length at least $\ell$ or a set $X$ of at most $\ell k^{O(1)}$ vertices such that the closed neighbourhood of $X$ intersects every cycle of $G$. Our proofs are constructive and yield polynomial-time algorithms, for fixed $\ell$, finding either the induced packing of the constrained cycles or the set $X$. This is based on joint works with Pascal Gollin, Tony Huynh, and O-joung Kwon.

The group \( S_n \) of permutations on \([n]=\{1,2,\dots,n\} \) is generated by simple transpositions \( s_i = (i,i+1) \). The length \( \ell(\pi) \) of a permutation \( \pi \) is defined to be the minimum number of generators whose product is \( \pi \). It is well-known that the longest element in \( S_n \) has length \( n(n-1)/2 \). Let \( F_n \) be the semigroup of functions \( f:[n]\to[n] \), which are generated by the simple transpositions \( s_i \) and the function \( t:[n]\to[n] \) given by \( t(1) =t(2) = 1 \) and \( t(i) = i \) for \( i\ge3 \). The length \( \ell(f) \) of a function \( f\in F_n \) is defined to be the minimum number of these generators whose product is \( f \). In this talk, we study the length of longest elements in \( F_n \). We also find a connection with the Slater index of a tournament of the complete graph \( K_n \). This is joint work with Yasuhide Numata.