Every (finite) matroid consists of a (finite) set called the ground set, and a collection of distinguished subsets called the independent sets. A classic example arises when the ground set is a finite set of vectors from a vector space, and the independent subsets are exactly the subsets that are linearly independent. Any such matroid is said to be representable. We can think of a representable matroid as being a geometrical configuration where the points have been given coordinates from a field. Another important class arises when the points are given coordinates from a group. Such a class is said to be gain-graphic. Monadic second-order logic is a natural language for matroid applications. In this language we are able to quantify only over subsets of the ground set. The importance of monadic second-order logic comes from its connections to the theory of computation, as exemplified by Courcelle’s Theorem. This theorem provides polynomial-time algorithms for recognising properties defined in monadic second-order logic (as long as we impose a bound on the structural complexity of the input objects). It is natural to ask which classes of matroids can be defined by sentences in monadic second-order logic. When the class consists of the matroids that are coordinatized by a field we have a complete answer to this question. When the class is coordinatized by a group the problem becomes much harder. This talk will contain a brief introduction to matroids. Based on work with Sapir Ben-Shahar, Matt Conder, Daryl Funk, Angus Matthews, Mike Newman, and Gabriel Verret.

An introduction and overview will be given of extreme events in Wetropolis flood investigator and of extreme water-wave motion in a novel wave-energy device. I will give an overview of the mathematical models and simulations of the phenomena seen in the following movies as well as related phenomena. Subsequently, mathematical and numerical aspects of the novel wave-energy device will be highlighted. As well as a discussion on the Wetropolis World proposal. The related movies which explain Wetropolis: Wetropolis-II (temp link) 2024: https://www.youtube.com/watch?v=g8znktYpxvY, Wetropolis-I 2022: https://www.youtube.com/watch?v=rNgEqWdafKk, The wave-energy device was motived by the bore-soliton-splash (2010): https://www.youtube.com/watch?v=YSXsXNX4zW0&list=FL6mc7mUa6M4Bo2VkD970urw, as well as a first proof of principle (2013): https://www.youtube.com/watch?v=SZhe_SOxBWo

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The related movies which explain Wetropolis: Wetropolis-II (temp link) 2024: https://www.youtube.com/watch?v=g8znktYpxvY, Wetropolis-I 2022: https://www.youtube.com/watch?v=rNgEqWdafKk, The wave-energy device was motived by the bore-soliton-splash (2010): https://www.youtube.com/watch?v=YSXsXNX4zW0&list=FL6mc7mUa6M4Bo2VkD970urw, as well as a first proof of principle (2013): https://www.youtube.com/watch?v=SZhe_SOxBWo

In this talk, I will talk about isotopy problems of Seifert surfaces pushed in to the 4-ball. In particular, I will prove that every Seifert surface of a non-split alternating link become isotopic in the 4-ball. This is a joint work with Maggie Miller and Jaehoon Yoo.

In the first part of the talk, I will discuss the asymptotic expansions of the Euclidean Φ^4-measure in the low-temperature regime. Consequently, we derive limit theorems, specifically the law of large numbers and the central limit theorem for the Φ^4-measure in the low-temperature limit. In the second part of the talk, I will focus on the infinite volume limit of the focusing Φ^4-measure. Specifically, with appropriate scaling, the focusing Φ^4-measure exhibits Gaussian fluctuations around a scaled solitary wave, that is, the central limit theorem. This talk is based on joint works with Benjamin Gess, Pavlos Tsatsoulis, and Philippe Sosoe.

Let S be a simply-connected rational homology complex projective plane with quotient singularities. The algebraic Montgomery-Yang problem conjectures that the number of singular points of S is at most three. In this talk, we leverage results from the study of smooth 4-manifolds, such as the Donaldson diagonalization theorem, to establish additional conditions for S. As a result, we eliminate the possibility of a rational homology complex projective plane of specific types with four singularities. We also identify infinite families of singularities that satisfy properties in algebraic geometry, including the orbifold BMY inequality, but are obstructed from being a rational homology complex projective plane due to smooth conditions. Additionally, we discuss experimental results related to this problem. This is joint work with Jongil Park and Kyungbae Park.

In this talk, we first review nonlinear elliptic equations when the right-hand side is a finite measure. We discuss global gradient estimates of a solution for such measure data problems in bounded nonsmooth domains. We provide proper solutions and conditions which guarantee the regularity results. If time permits, we will consider parabolic problems with measure data.

As part of the Langlands conjecture, it is predicted that every $\ell$-adic Galois representation attached to an algebraic cuspidal automorphic representation of $\mathrm{GL}_n$ over a number field is irreducible. In this talk, we will prove that a type $A_1$ Galois representation attached to a regular algebraic (polarized) cuspidal automorphic representation of $\mathrm{GL}_n$ over a totally real field $K$ is irreducible for all $\ell$, subject to some mild conditions. We will also prove that the attached Galois representation is residually irreducible for almost all $\ell$. Moreover, if $K=\mathbb Q$, we will prove that the attached Galois representation can be constructed from two-dimensional modular Galois representations up to twist. This is a joint work with Professor Chun-Yin Hui.

For the past few years, I’ve been working on formalizing proofs in matroid theory using the Lean proof assistant. This has led me to many interesting and unexpected places. I’ll talk about what formalization looks like in practice from the perspective of a combinatorialist.

We consider the global dynamics of finite energy solutions to energy-critical equivariant harmonic map heat flow (HMHF). It is known that any finite energy equivariant solutions to (HMHF) decompose into finitely many harmonic maps (bubbles) separated by scales and a body map, as approaching to the maximal time of existence. Our main result for (HMHF) gives a complete classification of their dynamics for equivariance indices D≥3; (i) they exist globally in time, (ii) the number of bubbles and signs are determined by the energy class of the initial data, and (iii) the scales of bubbles are asymptotically given by a universal sequence of rates up to scaling symmetry. In parallel, we also obtain a complete classification of $\dot{H}^1$-bounded radial solutions to energy-critical heat equations in dimensions N≥7, building upon soliton resolution for such solutions. This is a joint work with Frank Merle (IHES and CY Cergy-Paris University).

Mr. Saqib Mushtaq Shah, a KAIX visiting graduate student from ISI Bangalore who will stay at KAIST for 8 weeks, is going to give a series of weekly talks on the Milnor K-theory from the beginning. It is part of his KAIX summer internship works.

We consider Calogero—Moser derivative NLS (CM-DNLS) equation which can be seen as a continuum version of completely integrable Calogero—Moser many-body systems in classical mechanics. Soliton resolution refers to the phenomenon where solutions asymptotically decompose into a sum of solitons and a dispersive radiation term as time progresses. Our work proves soliton resolution for both finite-time blow-up and global solutions without radial symmetry or size constraints. Although the equation exhibits integrability, our proof does not depend on this property, potentially providing insights applicable to other non-integrable models. This research is based on the joint work with Soonsik Kwon (KAIST).

For motivational purposes, we begin by explaining the classical Satake isomorphism from which we deduce the unramified local Langlands correspondence. Then we explain a geometric interpretation of the Satake isomorphism. More precisely, we explain how one can view Hecke operators as global functions on the moduli space of unramified L-parameters. This viewpoint arises from the categorical local Langlands correspondence. The main content of the talk is p-adic and mod p analogues of this interpretation, where the space of unramified L-parameters is replaced by certain loci in the moduli stack of p-adic Galois representations (so-called the Emerton-Gee stack). We will also discuss their relationship with the categorical p-adic local Langlands program.

Depth and width parameters of graphs, e.g., tree-width, path-width and tree-depth, play a crucial role in algorithmic and structural graph theory. These notions are of fundamental importance in the theory of graph minors, fixed parameter complexity and the theory of sparsity. In this talk, we will survey structural and algorithmic results that concern width and depth parameters of matroids. We will particularly focus on matroid depth parameters and discuss the relation of the presented concepts to discrete optimization. As an application, we will present matroid based algorithms that uncover a hidden Dantzig-Wolfe-like structure of an input instance (if such structure is present) and transform instances of integer programming to equivalent ones, which are amenable to the existing tools in integer programming. The most recent results presented in the talk are based on joint work with Marcin Briański, Jacob Cooper, Timothy F. N. Chan, Martin Koutecký, Ander Lamaison, Kristýna Pekárková and Felix Schröder.

Mr. Saqib Mushtaq Shah, a KAIX visiting graduate student from ISI Bangalore who will stay at KAIST for 8 weeks, is going to give a series of weekly talks on the Milnor K-theory from the beginning. It is part of his KAIX summer internship works.

Kahn-Sujatha's birational motive is a variant of Chow motive that synthesis the ideas of birational geometry and motives. We explain our result saying that the unramified cohomology is a universal invariant for torsion motives of surfaces. We also exhibit examples of complex varieties violating the integral Hodge conjecture. If time permits, we discuss a pathology in positive characteristic. (Joint work with Kanetomo Sato.)

This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. It will summarize about 70-80% of the book.

For each positive integer q del Pezzo q-secant varieties are subextremal objects, in a natural sense, among q-secant varieties to nondegenerate projective varieties. In this talk we review their definition, properties, and examples, together with those of extremal objects, namely q-secant varieties of minimal degree.

For a family F of graphs, the F-Deletion Problem asks to remove the minimum number of vertices from a given graph G to ensure that G belongs to F. One of the most common ways to obtain an interesting family F is to fix another family H of graphs and let F be the set of graphs that do not contain any graph H as some notion of a subgraph, including (standard) subgraph, induced subgraph, and minor. This framework captures numerous basic graph problems, including Vertex Cover, Feedback Vertex Set, and Treewidth Deletion, and provides an interesting forum where ideas from approximation and parameterized algorithms influence each other. In this talk, I will give a brief survey on the state of the art on the F-Deletion Problems for the above three notions of subgraphs, and talk about a recent result on Weighted Bond Deletion.

Mr. Saqib Mushtaq Shah, a KAIX visiting graduate student from ISI Bangalore who will stay at KAIST for 8 weeks, is going to give a series of weekly talks on the Milnor K-theory from the beginning. It is part of his KAIX summer internship works.

Knot homology theories revolutionized the study of knots and links, much like (simplicial or singular) homology theory revolutionized the study of topological spaces. One of the major knot homology theories, Khovanov homology, was introduced by M. Khovanov in 2000 as a "categorification of the Jones polynomial." One notable feature of Khovanov homology is its ability to detect the unknot, a feature not known to be possessed by the Jones polynomial. Recently, it has found notable applications in low-dimensional topology, including the detection of exotic surfaces in the 4-ball. Day 3: Recent developments of Khovanov homology and its applications to low-dimensional topology.

We introduce a version of Heegaard diagrams for 5-dimensional cobordisms with 2- and 3-handles, 5-dimensional 3-handlebodies, and closed 5-manifolds. We show that every such 5-manifold can be represented by a Heegaard diagram, and two Heegaard diagrams represent diffeomorphic 5-manifolds if and only if they are related by certain moves. As an application, we construct Heegaard diagrams for 5-dimensional cobordisms from the standard 4-sphere to the Gluck twists along knotted 2-spheres. This provides some equivalent statements regarding the Gluck twists being diffeomorphic to the standard 4-sphere.

Knot homology theories revolutionized the study of knots and links, much like (simplicial or singular) homology theory revolutionized the study of topological spaces. One of the major knot homology theories, Khovanov homology, was introduced by M. Khovanov in 2000 as a "categorification of the Jones polynomial." One notable feature of Khovanov homology is its ability to detect the unknot, a feature not known to be possessed by the Jones polynomial. Recently, it has found notable applications in low-dimensional topology, including the detection of exotic surfaces in the 4-ball. Day 2: Numerical invariants from Khovanov homology and their applications.

In this work, we investigate optimal control problems in heterogeneous porous media. Based on the partial differential equation constraint connecting the state and the control, we produce the associated control as a dependent quantity of the state. Then, we introduce the reduced optimal control problem which contains only the state variable. Here we employ $C^0$ interior penalty finite element methods for the spatial discretization to solve the reduced optimal control problem resulting in a fourth-order variational inequality. We provide a priori error estimates and stability analyses. Several numerical examples validate and illustrate the capabilities of the proposed algorithm.

Knot homology theories revolutionized the study of knots and links, much like (simplicial or singular) homology theory revolutionized the study of topological spaces. One of the major knot homology theories, Khovanov homology, was introduced by M. Khovanov in 2000 as a "categorification of the Jones polynomial." One notable feature of Khovanov homology is its ability to detect the unknot, a feature not known to be possessed by the Jones polynomial. Recently, it has found notable applications in low-dimensional topology, including the detection of exotic surfaces in the 4-ball. Day 1: Jones polynomial and its categorification.

In this lecture, we will discuss the formal/mathematical connection between the Boltzmann equation and the compressible Euler equation. For the mathematical justification we study the convergence of real analytic solutions of Boltzmann equations toward smooth solutions of the compressible Euler equation (before shock). This lecture will be accessible to graduate students.

Mr. Saqib Mushtaq Shah, a KAIX visiting graduate student from ISI Bangalore who will stay at KAIST for 8 weeks, is going to give a series of weekly talks on the Milnor K-theory from the beginning. It is part of his KAIX summer internship works.

Phenotypic selection occurs when genetically identical cells are subject to different reproductive abilities due to cellular noise. Such noise arises from fluctuations in reactions synthesizing proteins and plays a crucial role in how cells make decisions and respond to stress or drugs. We propose a general stochastic agent-based model for growing populations capturing the feedback between gene expression and cell division dynamics. We devise a finite state projection approach to analyze gene expression and division distributions and infer selection from single-cell data in mother machines and lineage trees. We use the theory to quantify selection in multi-stable gene expression networks and elucidate that the trade-off between phenotypic switching and selection enables robust decision-making essential for synthetic circuits and developmental lineage decisions. Using live-cell data, we demonstrate that combining theory and inference provides quantitative insights into bet-hedging–like response to DNA damage and adaptation during antibiotic exposure in Escherichia coli.

In general relativity, spacetime is described by a (1+3)-dimensional Lorentzian manifold satisfying the Einstein equations, and initial data sets (i.e., fixed-time configurations) correspond to embedded spacelike hypersurfaces. The initial data sets are required to satisfy underdetermined PDEs called constraint equations -- in the language of differential geometry, these are exactly the Gauss and Codazzi equations. The goal of my talk will be to elucidate the flexibility of these objects -- specific results to be presented include extension, gluing, asymptotics-prescription, and parametrization of asymptotically flat initial data sets, often with sharp assumptions. Basic to our approach is a novel way to construct solution operators for divergence-type equations with prescribed support properties, which should be of independent interest. This part is based on joint work with Phil Isett (Caltech), Yuchen Mao (UC Berkeley), and Zhongkai Tao (UC Berkeley).

In this talk, we will discuss cylindrical and hypoelliptic extensions of Hardy, Sobolev, Rellich, Caffarelli-Kohn-Nirenberg, and other related functional inequalities. We will then concentrate on discussing their best constants, ground states for higher-order hypoelliptic Schrödinger-type equations, and solutions to the corresponding variational problems.

This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. It will summarize about 70-80% of the book.

The classical nonlinear potential theory has recently been extended to nonlocal nonlinear potential theory, which studies harmonic functions associated with nonlocal nonlinear operators. In this talk, we focus on the harmonic functions solving the nonlocal Dirichlet problem. As in the study of classical Dirichlet problem, the nonlocal Dirichlet problem can be solved by using Sobolev and Perron solutions. We provide several properties of such solutions. This talk is based on joint works with Anders Björn, Jana Björn, Ki-Ahm Lee and Se-Chan Lee.

Stochastic models of gene expression are routinely used to explain large variability in measured mRNA levels between cells. These models typically predict the distribution of the total mRNA level per cell but ignore compartment-specific measurements which are becoming increasingly common. Here we construct a two-compartment model that describes promoter switching between active and inactive states, transcription of nuclear mRNA and its export to the cytoplasm where it decays. We obtain an analytical solution for the joint distribution of nuclear and cytoplasmic mRNA levels in steady-state conditions. Based on this solution, we build an efficient and accurate parameter inference method which is orders of magnitude faster than conventional methods. If you want to participate in the seminar, you need to enter IBS builiding (https://www.ibs.re.kr/bimag/visiting/). Please contact if you first come IBS to get permission to enter IBS building.

This talk presents the C^{1,\alpha}-regularity for viscosity solutions of degenerate/singular fully nonlinear parabolic equations. For this purpose, we develop a new type of Bernstein technique in view of the difference quotient to obtain a priori estimates of the regularized equations. Also, we establish the well-posedness and the uniform C^{1,\alpha}-estimates for the regularized Cauchy-Dirichlet problem.

In this talk, we prove the Aleksandrov--Bakelman--Pucci estimate for non-uniformly elliptic equations in non-divergence form. Moreover, we investigate local behaviors of solutions of such equations by developing local boundedness and weak Harnack inequality. Here we impose an integrability assumption on ellipticity representing degeneracy or singularity, instead of specifying the particular structure of ellipticity.

In the 1980's Casson and Gordon produced the first non slice knots which are trivial in Levine's algebraic concordance group, and in 2003 Cochran-Orr-Teichner produced the first no slice knots undetectable by Casson and Gordon's invariants. They do so by producing a filtration of the concordance group by subgroups a knot in the 1.5th term of this filtration has vanishing Casson-Gordon invariants. Since then this work has been central to the study of knot concordance. We will introduce this filtration and review just enough of the theory of L^2 homology to prove that the successive quotients of this filtration are nontrivial.

In the 1970's J. Levine produced a surjection from the knot concordance group to the so called algebraic concordance group. This captured the known features of the knot concordance group to that point and classifies high dimensional concordance. During this survey talk we will explore the construction of the algebraic concordance group and explain some of its consequences.

Mr. Saqib Mushtaq Shah, a KAIX visiting graduate student from ISI Bangalore who will stay at KAIST for 8 weeks, is going to give a series of weekly talks on the Milnor K-theory from the beginning. It is part of his KAIX summer internship works.

For a given hypergraph $H$ and a vertex $v\in V(H)$, consider a random matching $M$ chosen uniformly from the set of all matchings in $H.$ In $1995,$ Kahn conjectured that if $H$ is a $d$-regular linear $k$-uniform hypergraph, the probability that $M$ does not cover $v$ is $(1 + o_d(1))d^{-1/k}$ for all vertices $v\in V(H)$. This conjecture was proved for $k = 2$ by Kahn and Kim in 1998. In this paper, we disprove this conjecture for all $k \geq 3.$ For infinitely many values of $d,$ we construct $d$-regular linear $k$-uniform hypergraph $H$ containing two vertices $v_1$ and $v_2$ such that $\mathcal{P}(v_1 \notin M) = 1 – \frac{(1 + o_d(1))}{d^{k-2}}$ and $\mathcal{P}(v_2 \notin M) = \frac{(1 + o_d(1))}{d+1}.$ The gap between $\mathcal{P}(v_1 \notin M)$ and $\mathcal{P}(v_2 \notin M)$ in this $H$ is best possible. In the course of proving this, we also prove a hypergraph analog of Godsil’s result on matching polynomials and paths in graphs, which is of independent interest.

The analysis on the limiting behavior of solution is pivotal for equations in geometric analysis, mathematical physics and application in optimization. In 80s, Rene Thom conjectured that if an analytic gradient flow has a limit, then it approaches to the limit along a unique asymptotic direction. This represents a next-order question following the seminal works by Lojasiewicz and L. Simon. In 2000, Thom's conjecture was affirmatively proved by Kurdyka, Mostowski, and Parusinski for finite dimensional gradient flows. In this first part, we will discuss about the basics about theory of Lojasiewicz concerning the uniqueness of limits. Then we explore vast applications in PDEs which were initiated by Leon Simon.

Following the brief introduction to Lojasiewicz's theory in the first part, in the second part we discuss Thom's gradient conjecture and our recent joint work with Pei-Ken Hung where we generalized this conjecture to the class of PDEs. The result classifies the next-order asymptotics by revealing both the rate and the direction of convergence to the limit. Finally we talk about possible future applications and working directions.

This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. It will summarize about 70-80% of the book.

Tropical geometry replaces usual addition and multiplication with tropical addition (the min) and tropical multiplication (the sum), which offers a polyhedral interpretation of algebraic variety. This talk aims to pitch the usefulness of tropical geometry in understanding classical algebraic geometry. As an example, we introduce the tropicalization of the variety of symmetric rank 2 matrices. We discuss that this tropicalization has a simplicial complex structure as the space of symmetric bicolored trees. As a result, we show that this space is shellable and delve into its matroidal structure. It is based on the joint work with May Cai and Josephine Yu.