Consider the following hat guessing game: $n$ players are placed on $n$ vertices of a graph, each wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. Given a graph $G$, its hat guessing number $HG(G)$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. In 2008, Butler, Hajiaghayi, Kleinberg, and Leighton asked whether the hat guessing number of the complete bipartite graph $K_{n,n}$ is at least some fixed positive (fractional) power of $n$. We answer this question affirmatively, showing that for sufficiently large $n$, $HG(K_{n,n})\ge n^{0.5-o(1)}$. Our guessing strategy is based on some ideas from coding theory and probabilistic method. Based on a joint work with Noga Alon, Omri Ben-Eliezer, and Itzhak Tamo.

TBA

We introduce configurations of lines in the combinatorial and geometric setting. After a brief summary of the classical theory we will discuss results in the 4-dimensional setting. These include work of Ruberman and Starkston in the topological category and work in progress in the smooth category that is joint work with D. McCoy And J. Park.

We discuss an explicit formula for the structure of Bloch–Kato Selmer groups of the central critical twist of modular forms if the analytic rank is ≤ 1 or the Iwasawa main conjecture localized at the augmentation ideal holds. This formula reveals more refined arithmetic information than the p-part of the Tamagawa number conjecture for motives of modular forms and reduces the corresponding Beilinson–Bloch–Kato conjecture to a purely analytic statement. Our formula is insensitive to the local behavior at p.

A meandric system of size n is the set of loops formed from two arc diagrams (non-crossing perfect matchings) on {1,⋯,2n}, one drawn above the real line and the other below the real line. Equivalently, a meandric system is a coupled collection of meanders of total size n. I will discuss a conjecture which describes the large-scale geometry of a uniformly sampled meandric system of size n in terms of Liouville quantum gravity (LQG) decorated by certain Schramm-Loewner evolution (SLE) type curves. I will then present several rigorous results which are consistent with this conjecture. In particular, a uniform meandric system admits macroscopic loops; and the half-plane version of the meandric system has no infinite paths. Based on joint work with Jacopo Borga and Ewain Gwynne.

한국의 수학과 박사과정 학생이 해외로 포닥을 지원하는 방법, 그리고 2년간의 포닥 생활 후기에 대해 발표합니다. 포닥 지원 시기와 절차는 어떤지, 포닥 지원을 위한 CV, 추천서, research statement, teaching statement 등을 어떻게 준비하는 지 설명드립니다. 해외 정착, 기존 연구 마무리 및 포닥으로서의 새로운 연구 시작 방법에 대해 설명드립니다. 이 발표는 University of Wisconsin, Madison의 Van Vleck assistant professor 오퍼를 받은 홍혁표 학생과 University of Michigan, Ann Arbor에서 2년간 James Van Loo assistant professor로 근무한 김대욱 박사가 진행합니다. We will present on how a doctoral student in the field of mathematics in Korea can apply for a postdoctoral position abroad and share the experience of living as a postdoc for two years. We will explain the timing and procedures for applying for a postdoc position, as well as how to prepare a CV, recommendation letters, research statement, teaching statement, and other documents required for the application. Also, we will provide information on how to settle down in a foreign country, wrap up existing research, and start new research as a postdoc. This presentation will be conducted by Hyukpyo Hong, a student who recently received an offer for the Van Vleck Assistant Professor position at the University of Wisconsin, Madison, and Dr. Dae Wook Kim, who worked as a James Van Loo Assistant Professor at the University of Michigan, Ann Arbor for two years.

An archetype problem in extremal combinatorics is to study the structure of subgraphs appearing in different classes of (hyper)graphs. We will focus on such embedding problems in uniformly dense hypergraphs. In precise, we will mention the uniform Turan density of some hypergraphs.

In this talk, we explore a duality between federated learning and subspace correction, which are concepts from two very different fields. Federated learning is a paradigm of supervised machine learning in which data is decentralized into a number of clients and each client updates a local correction of a global model independently via the local data. Subspace correction is an abstraction of general iterative algorithms such as multigrid and domain decomposition methods for solving scientific problems numerically. Based on the duality between federated learning and subspace correction, we propose a novel federated learning algorithm called DualFL (Dualized Federated Learning). DualFL is the first federated learning algorithm that achieves communication acceleration, even when the cost function is either nonsmooth or non-strongly convex.

In astrophysical fluid dynamics, stars are considered as isolated fluid masses subject to self-gravity. A classical model of a self-gravitating Newtonian star is given by the gravitational Euler-Poisson system, while a relativistic star is modeled by the Einstein-Euler system. I will review some recent progress on the local and global dynamics of Newtonian stars, and discuss mathematical constructions of gravitational collapse that show the existence of smooth initial data leading to finite time collapse, characterized by the blow-up of the star density. For Newtonian stars, dust-like collapse and self-similar collapse will be presented, and the relativistic analogue and formation of naked singularities for the Einstein-Euler system will be discussed.

Tropicalizations of affine varieties give interesting ways to sketch and study affine varieties, whose tools are astonishingly elementary at the algebraic level. Not only that, studying algebraic dynamics on varieties may give interesting pictures under tropicalizations, as worked by Spalding and Veselov, or Filip. In this talk, we will introduce some basicmost ideas of tropicalizations, and play with the Markov cubic surfaces $$X^2+Y^2+Z^2+XYZ=AX+BY+CZ+D,$$ where A, B, C, D are parameters, as an example of tropical study of algebraic dynamics. It turns out that we obtain a $(\infty,\infty,\infty)$-triangle group action on the hyperbolic plane as a model of dynamics of interest. 언어: Korean (possibly English, depending on the audience)

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심사위원장 : 김재경 / 심사위원 : 김용정, 정연승, 김진수(POSTECH), 이승규(고려대학교)

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심사위원장: 엄상일, 심사위원 : 안드레아스 홈슨, 김재훈, 권오정(한양대학교), 오은진(POSTECH)

A graph class $\mathcal{G}$ has the strong Erdős-Hajnal property (SEH-property) if there is a constant $c=c(\mathcal{G}) > 0$ such that for every member $G$ of $\mathcal{G}$, either $G$ or its complement has $K_{m, m}$ as a subgraph where $m \geq \left\lfloor c|V(G)| \right\rfloor$. We prove that the class of chordal graphs satisfies SEH-property with constant $c = 2/9$. On the other hand, a strengthening of SEH-property which we call the colorful Erdős-Hajnal property was discussed in geometric settings by Alon et al.(2005) and by Fox et al.(2012). Inspired by their results, we show that for every pair $F_1, F_2$ of subtree families of the same size in a tree $T$ with $k$ leaves, there exist subfamilies $F'_1 \subseteq F_1$ and $F'_2 \subseteq F_2$ of size $\theta \left( \frac{\ln k}{k} \left| F_1 \right|\right)$ such that either every pair of representatives from distinct subfamilies intersect or every such pair do not intersect. Our results are asymptotically optimal. Joint work with Andreas Holmsen, Jinha Kim and Minki Kim.

TBA

TBD

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

Sequential decision making under uncertainty is a problem class with solid real-life foundation and application. We overview the concept of Knowledge Gradient (KG) from the perspective of multi-armed bandit (MAB) problem and reinforcement learning. Then we discuss the first KG algorithm with sublinear regret bounds for Gaussian MAB problems.

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(Online participation) Zoom Link: https://kaist.zoom.us/j/87516570701

To understand nonparametric regression, we should know first what the parametric model is. Simply speaking, the parametric regression model consists of many assumptions and the nonparametric regression model eases the assumptions. I will introduce what assumptions the parametric regression model has and how the nonparametric regression model relieves them. In addition, their pros and cons will be also presented.

A digital twin is a virtual representation of real-world physical objects. Through accurate and streamlined simulations, it effectively enhances our understanding of the real world, enabling us to predict complex and dynamic phenomena in a fraction of the time. In this talk, we will explore real-world applications of AI-based partial differential equation (PDE) solvers in various fields. Additionally, we will examine how such AI can be utilized to facilitate downstream tasks related to PDEs.

We introduce elliptic curves, their Mordell-Weil group structure, and isogenies over number fields. At the last of the talk, some results on the torsion subgroups of Mordell-Weil groups of elliptic curves defined over a number field will be given. The results are joint works with my advisor Bo-Hae Im.

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Discipline of talk : Number Theory / Advisor: 임보해(Bo-Hae Im)

The Erdős-Sós conjecture states that the maximum number of edges in an $n$-vertex graph without a given $k$-vertex tree is at most $\frac {n(k-2)}{2}$. Despite significant interest, the conjecture remains unsolved. Recently, Caro, Patkós, and Tuza considered this problem for host graphs that are connected. Settling a problem posed by them, for a $k$-vertex tree $T$, we construct $n$-vertex connected graphs that are $T$-free with at least $(1/4-o_k(1))nk$ edges, showing that the additional connectivity condition can reduce the maximum size by at most a factor of 2. Furthermore, we show that this is optimal: there is a family of $k$-vertex brooms $T$ such that the maximum size of an $n$-vertex connected $T$-free graph is at most $(1/4+o_k(1))nk$.

Despite much recent progress in analyzing algorithms in the linear MDPs and their variants, the understanding of more general transition models is still very restrictive. We study provably efficient RL algorithms for the MDP whose state transition is given by a multinomial logistic model. We establish the regret guarantees for the algorithms based on multinomial logistic function approximation. We also comprehensively evaluate our proposed algorithm numerically and show that it consistently outperforms the existing methods, hence achieving both provable efficiency and practical superior performance.

This talk presents new methods of solving machine learning problems using probability models. For classification problems, the classifier referred to as the class probability output network (CPON) which can provide accurate posterior probabilities for the soft classification decision, is proposed. In this model, the uncertainty of decision is defined using the accuracy of estimation. The deep structure of CPON is also presented to obtain the best classification performance for the given data. Applications of CPON models are also addressed.

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(Online participation) Zoom Link: https://kaist.zoom.us/j/87516570701

In this talk, we are going to consider the Fermi-Pasta-Ulam (FPU) system with innitely many oscil- lators. We particularly see that Harmonic analysis approaches allow us to observe dispersive properties of solutions to a reformulated FPU system, and with this observation, solutions to the FPU system can be approximated by counter-propagating waves governed by the Korteweg de-Vries (KdV) equation as the lattice spacing approaches zero. Additionally, we see dierent phenomena detected in the periodic FPU system.

We propose a kernel-based estimator to predict the mean response trajectory for sparse and irregularly measured longitudinal data. The kernel estimator is constructed by imposing weights based on the subject-wise similarity on L2 metric space between predictor trajectories, where we assume that an analogous fashion in predictor trajectories over time would result in a similar trend in the response trajectory among subjects. In order to deal with the curse of dimensionality caused by the multiple predictors, we propose an appealing multiplicative model with multivariate Gaussian kernels. This model is capable of achieving dimension reduction as well as selecting functional covariates with predictive significance. The asymptotic properties of the proposed nonparametric estimator are investigated under mild regularity conditions. We illustrate the robustness and flexibility of our proposed method via the simulation study and an application to Framingham Heart Study

While deep learning has many remarkable success stories, finding a satisfactory mathematical explanation on why it is so effective is still considered an open challenge. One recent promising direction for this challenge is to analyse the mathematical properties of neural networks in the limit where the widths of hidden layers of the networks go to infinity. Researchers were able to prove highly-nontrivial properties of such infinitely-wide neural networks, such as the gradient-based training achieving the zero training error (so that it finds a global optimum), and the typical random initialisation of those infinitely-wide networks making them so called Gaussian processes, which are well-studied random objects in machine learning, statistics, and probability theory. These theoretical findings also led to new algorithms based on so-called kernels, which sometimes outperform existing kernel-based algorithms. The purpose of this talk is to explain these recent theoretical results on infinitely wide neural networks. If time permits, I will briefly describe my work in this domain, which aims at developing a new neural-network architecture that has multiple nice theoretical properties in the infinite-width limit. This work is jointly pursued with Fadhel Ayed, Francois Caron, Paul Jung, Hoil Lee, and Juho Lee.

Stochasticity in gene expression is an important source of cell-to-cell variability (or noise) in clonal cell populations. So far, this phenomenon has been studied using the Gillespie Algorithm, or the Chemical Master Equation, which implicitly assumes that cells are independent and do neither grow nor divide. This talk will discuss recent developments in modelling populations of growing and dividing cells through agent-based approaches. I will show how the lineage structure affects gene expression noise over time, which leads to a straightforward interpretation of cell-to-cell variability in population snapshots. I will also illustrate how cell cycle variability shapes extrinsic noise across lineage trees. Finally, I outline how to construct effective chemical master equation models based on dilution reactions and extrinsic variability that provide surprisingly accurate approximations of the noise statistics across growing populations. The results highlight that it is crucial to consider cell growth and division when quantifying cellular noise.

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

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심사위원장: 임미경 / 심사위원: 김용정, 신연종, 권기운(동국대학교), 이은정(연세대학교)

Domain adaptation (DA) is a statistical learning problem that arises when the distribution of the source data used to train a model differs from that of the target data used to test the model. While many DA algorithms have demonstrated considerable empirical success, the unavailability of target labels in DA makes it challenging to determine their effectiveness in new datasets without a theoretical basis. Therefore, it is essential to clarify the assumptions required for successful DA algorithms and quantify the corresponding guarantees. In this work, we focus on the assumption that conditionally invariant components (CICs) useful for prediction exist across the source and target data. Under this assumption, we demonstrate that CICs found via conditional invariant penalty (CIP) play three essential roles in providing guarantees for DA algorithms. First, we introduce a new CIC-based algorithm called importance-weighted conditional invariant penalty (IW-CIP), which has target risk guarantees beyond simple settings like covariate shift and label shift. Second, we show that CICs can be used to identify large discrepancies between source and target risks of other DA algorithms. Finally, we demonstrate that incorporating CICs into the domain invariant projection (DIP) algorithm helps to address its known failure scenario caused by label-flipping features. We support our findings via numerical experiments on synthetic data, MNIST, CelebA, and Camelyon17 datasets.

The compressible Euler system (CE) is one of the oldest PDE models in fluid dynamics as a representative model that describes the flow of compressible fluids with singularities such as shock waves. But, CE is regarded as an ideal model for inviscid gas, and may be physically meaningful only as a limiting case of the corresponding Navier-Stokes system(NS) with small viscosity and heat conductivity that can be negligible. Therefore, any stable physical solutions of CE should be constructed by inviscid limit of solutions of NS. This is known as the most challenging open problem in mathematical fluid dynamics (even for incompressible case). In this talk, I will present my recent works that tackle the open problem, using new methods: the (so-called) weighted relative entropy method with shifts (for controlling shocks) and the viscous wave-front tracking method (for handling general solution with small total variation).

In this talk, we consider the problem of minimizing multi-modal loss functions with a large number of local optima. Since the local gradient points to the direction of the steepest slope in an infinitesimal neighborhood, an optimizer guided by the local gradient is often trapped in a local minimum. To address this issue, we develop a novel nonlocal gradient to skip small local minima by capturing major structures of the loss's landscape in black-box optimization. The nonlocal gradient is defined by a directional Gaussian smoothing (DGS) approach. The key idea is to conducts 1D long-range exploration with a large smoothing radius along orthogonal directions, each of which defines a nonlocal directional derivative as a 1D integral. Such long-range exploration enables the nonlocal gradient to skip small local minima. We use the Gauss-Hermite quadrature rule to approximate the d 1D integrals to obtain an accurate estimator. We also provide theoretical analysis on the convergence of the method on nonconvex landscape. In this work, we investigate the scenario where the objective function is composed of a convex function, perturbed by a highly oscillating, deterministic noise. We provide a convergence theory under which the iterates converge to a tightened neighborhood of the solution, whose size is characterized by the noise frequency. Furthermore, if the noise level decays to zero when approaching global minimum, we prove that the DGS optimization converges to the exact global minimum with linear rates, similarly to standard gradient-based method in optimizing convex functions. We complement our theoretical analysis with numerical experiments to illustrate the performance of this approach.

We introduce concepts of parameterized complexity, especially, kernelization. Kernelization is a polynomial-time preprocessing algorithm that converts a given instance for a problem to a smaller instance while keeping the answer to the problem. Delicate kernelization mostly boosts the speed of solving the problem. We explain standard techniques in kernelizations, for instance, the sunflower lemma. Most optimization problems can be reformulated in the Hitting Set problem format, and the sunflower lemma gives us a simple yet beautiful kernelization for the problem. We further introduce our recent work about the Hitting Set problem on sparse graph classes.

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Discipline of talk: Graph Theory, Complexity Theory / Advisor: 엄상일 (Sang-il Oum)

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심사위원장: 임보해, 심사위원 : 김완수, 백상훈, 최도훈(고려대학교), 선해상(UNIST)

We present how to construct a stochastic process on a finite interval with given roughness and finite joint moments of marginal distributions. Our construction method is based on Schauder representation along a general sequence of partitions and has two ramifications. The variation index of a process (the infimum value p such that the p-th variation is finite) may not be equal to the reciprocal of Hölder exponent. Moreover, we can construct a non-Gaussian family of stochastic processes mimicking (fractional) Brownian motions. Therefore, when observing a path of process in a financial market such as a price or volatility process, we should not measure its Hölder roughness by computing p-th variation and should not conclude that a given path is sampled from Brownian motion or fractional Brownian motion even though it exhibits the same properties of those Gaussian processes. This talk is based on joint work with Erhan Bayraktar and Purba Das.

We study the fundamental problem of finding small dense subgraphs in a given graph. For a real number $s>2$, we prove that every graph on $n$ vertices with average degree at least $d$ contains a subgraph of average degree at least $s$ on at most $nd^{-\frac{s}{s-2}}(\log d)^{O_s(1)}$ vertices. This is optimal up to the polylogarithmic factor, and resolves a conjecture of Feige and Wagner. In addition, we show that every graph with $n$ vertices and average degree at least $n^{1-\frac{2}{s}+\varepsilon}$ contains a subgraph of average degree at least $s$ on $O_{\varepsilon,s}(1)$ vertices, which is also optimal up to the constant hidden in the $O(.)$ notation, and resolves a conjecture of Verstraëte. Joint work with Benny Sudakov and Istvan Tomon.

In geometric variational problems and non-linear PDEs, challenges often reduce down to questions on the asymptotic behavior near singularity and infinity. In this talk, we discuss the rate and direction of convergence for slowly converging solutions. Previously, they were constructed under so called the Adams-Simon positivity condition on the limit. We conversely prove that every slowly converging solution necessarily satisfies such a condition and the condition dictates possible dynamics. The result can be placed as a generalization of Thom's gradient conjecture. This is a joint work with Pei-Ken Hung at Minnesota

Time-series data analysis is found in various applications that deal with sequential data over the given interval of, e.g. time. In this talk, we discuss time-series data analysis based on topological data analysis (TDA). The commonly used TDA method for time-series data analysis utilizes the embedding techniques such as sliding window embedding. With sliding window embedding the given data points are translated into the point cloud in the embedding space and the method of persistent homology is applied to the obtained point cloud. In this talk, we first show some examples of time-series data analysis with TDA. The first example is from music data for which the dynamic processes in time is summarized by low dimensional representation based on persistence homology. The second is the example of the gravitational wave detection problem and we will discuss how we concatenate the real signal and topological features. Then we will introduce our recent work of exact and fast multi-parameter persistent homology (EMPH) theory. The EMPH method is based on the Fourier transform of the data and the exact persistent barcodes. The EMPH is highly advantageous for time-series data analysis in that its computational complexity is as low as O(N log N) and it provides various topological inferences almost in no time. The presented works are in collaboration with Mai Lan Tran, Chris Bresten and Keunsu Kim.

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(Online participation) Zoom Link: https://kaist.zoom.us/j/87516570701

In this talk, we present a formula for the degree of the 3-secant variety of a nonsingular projective variety embedded by a 3-very ample line bundle. The formula is provided in terms of Segre classes of the tangent bundle of a given variety. We use the generalized version of double point formula to reduce the calculation into the case of the 2-secant variety. Due to the singularity of the 2-secant variety, we use secant bundle as a nonsingular birational model and compute multiplications of desired algebraic cycles.

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Discipline of talk: Algebraic geometry / Advisor: 이용남 교수님 ( Yong nam, Lee)

Tree decompositions are a powerful tool in both structural graph theory and graph algorithms. Many hard problems become tractable if the input graph is known to have a tree decomposition of bounded “width”. Exhibiting a particular kind of a tree decomposition is also a useful way to describe the structure of a graph. Tree decompositions have traditionally been used in the context of forbidden graph minors; bringing them into the realm of forbidden induced subgraphs has until recently remained out of reach. Over the last couple of years we have made significant progress in this direction, exploring both the classical notion of bounded tree-width, and concepts of more structural flavor. This talk will survey some of these ideas and results.

TBD

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

We obtain uniform in time L^\infty -bounds for the solutions to a class of thermo-diffusive systems. The nonlinearity is assumed to be at most sub-exponentially growing at infinity and have a linear behavior near zero. This is a joint work with Lenya Ryzhik and Jean-Michel Roquejoffre.

Recently, Letzter proved that any graph of order n contains a collection P of $O(n \log^*n)$ paths with the following property: for all distinct edges e and f there exists a path in P which contains e but not f. We improve this upper bound to 19n, thus answering a question of Katona and confirming a conjecture independently posed by Balogh, Csaba, Martin, and Pluhar and by Falgas-Ravry, Kittipassorn, Korandi, Letzter, and Narayanan. Our proof is elementary and self-contained.

We present a framework of predictive modeling of unknown system from measurement data. The method is designed to discover/approximate the unknown evolution operator, i.e., flow map, behind the data. Deep neural network (DNN) is employed to construct such an approximation. Once an accurate DNN model for the evolution operator is constructed, it serves as a predictive model for the unknown system and enables us to conduct system analysis. We demonstrate that flow map learning (FML) approach is applicable for modeling a wide class of problems, including dynamical systems, systems with missing variables and hidden parameters, as well as partial differential equations (PDEs).

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KAI-X Distinguished Lecture Series

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심사위원장: 이창옥, 심사위원:김동환, 신연종, 예종철(겸임교수), 신원용(연세대학교)

Collective cell movement is critical to the emergent properties of many multicellular systems including microbial self-organization in biofilms, wound healing, and cancer metastasis. However, even the best-studied systems lack a complete picture of how diverse physical and chemical cues act upon individual cells to ensure coordinated multicellular behavior. Myxococcus xanthus is a model bacteria famous for its coordinated multicellular behavior resulting in dynamic patterns formation. For example, when starving millions of cells coordinate their movement to organize into fruiting bodies – aggregates containing tens of thousands of bacteria. Relating these complex self-organization patterns to the behavior of individual cells is a complex-reverse engineering problem that cannot be solved solely by experimental research. In collaboration with experimental colleagues, we use a combination of quantitative microscopy, image processing, agent-based modeling, and kinetic theory PDEs to uncover the mechanisms of emergent collective behaviors.

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Professor of Bioengineering & BioSciences, Associate Chair of Bioengineering, Rice U

Compressible Euler system (CE) is a well-known PDE model that was formulated in the 19th century for dynamics of compressible fluid. The most important feature of CE is the finite-time breakdown of smooth solutions, especially, the formation of shock wave as severe singularity. Therefore, a fundamental question (since Riemann 1858) is on what happens after a shock occurs. This is the problem on well-posedness (that is, existence, uniqueness, stability) of CE in a suitable class of solutions. We will discuss on the well-posedness problem, and its generalization for applications to other PDE models arising in various contexts such as magnetohydrodynamics, tumor angiogenesis, vehicular traffic flow, etc.

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첫수융합포럼 The First Wednesday Multidisciplinary Forum, May 2023 with School of Business and Technology Management ZOOM Link: https://kaist.zoom.us/j/84028206160?pwd=VzNPRGxSR2hRcnJTNk4rMHQ4Z1hiQT09

The Ramsey number $R(k)$ is the minimum n such that every red-blue colouring of the edges of the complete graph on n vertices contains a monochromatic copy of $K_k$. It has been known since the work of Erdős and Szekeres in 1935, and Erdős in 1947, that $2^{k/2} < R(k) < 4^k$, but in the decades since the only improvements have been by lower order terms. In this talk I will sketch the proof of a very recent result, which improves the upper bound of Erdős and Szekeres by a (small) exponential factor. Based on joint work with Marcelo Campos, Simon Griffiths and Julian Sahasrabudhe.