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.

The well-known two-process model of sleep regulation makes accurate predictions of sleep timing and duration, as well as neurobehavioral performance, for a variety of acute sleep deprivation and nap sleep scenarios, but it fails to predict the effects of chronic sleep restriction on neurobehavioral performance. The two-process model belongs to a broader class of coupled, non-homogeneous, first-order, ordinary differential equations (ODEs), which can capture the effects of chronic sleep restriction. These equations exhibit a bifurcation, which appears to be an essential feature of performance impairment due to sleep loss. The equations implicate a biological system analogous to two connected compartments containing interacting compounds with time-varying concentrations, such as the adenosinergic neuromodulator/receptor system, as a key mechanism for the regulation of neurobehavioral functioning under conditions of sleep loss. The equations account for dynamic interaction with circadian rhythmicity, and also provide a new approach to dynamically tracking the magnitude of sleep inertia upon awakening from restricted sleep. This presentation will describe the development of the ODE system and its experimental calibration and validation, and will discuss some novel predictions.

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

In many different areas of mathematics (such as number theory, discrete geometry, and combinatorics), one is often presented with a large "unstructured" object, and asked to find a smaller "structured" object inside it. One of the earliest and most influential examples of this phenomenon was the theorem of Ramsey, proved in 1930, which states that if n = n(k) is large enough, then in any red-blue colouring of the edges of the complete graph on n vertices, there exists a monochromatic clique on k vertices. In this talk I will discuss some of the questions, ideas, and new techniques that were inspired by this theorem, and present some recent progress on one of the central problems in the area: bounding the so-called "diagonal" Ramsey numbers. Based on joint work with Marcelo Campos, Simon Griffiths and Julian Sahasrabudhe.

For a given graph $H$, we say that a graph $G$ has a perfect $H$-subdivision tiling if $G$ contains a collection of vertex-disjoint subdivisions of $H$ covering all vertices of $G.$ Let $\delta_{sub}(n, H)$ be the smallest integer $k$ such that any $n$-vertex graph $G$ with minimum degree at least $k$ has a perfect $H$-subdivision tiling. For every graph $H$, we asymptotically determined the value of $\delta_{sub}(n, H)$. More precisely, for every graph $H$ with at least one edge, there is a constant $1 < \xi^*(H)\leq 2$ such that $\delta_{sub}(n, H) = \left(1 - \frac{1}{\xi^*(H)} + o(1) \right)n$ if $H$ has a bipartite subdivision with two parts having different parities. Otherwise, the threshold depends on the parity of $n$.

[1] 인간의 질병 발생과 예방을 근본적으로 파악하려면 '인간에 대한 이해'가 필요합니다. 인간의 몸은 물질이며, 물질은 특성상 물리적 및 화학적 반복자극에 반드시 손상됩니다. 부모님께 몸을 받아 수십 년간 살다 보면 인간 내부의 태생적-구조적 요인과 외부의 환경적 요인에 의하여 부지불식중 가해지는 내부 및 외부 자극에 반복적으로 노출될 수밖에 없습니다. 이에 그와 같이 질병으로 진행될 수 밖에 없는 인간의 특성을 명화(그림)을 통하여 소개할 예정입니다. [2] 인간의 또 다른 이해로 과학적 혹은 수학적 평가로 가시화(객관화)하기 어려운 부문에 대한 내용을 역시 명화를 통하여 논의할 예정입니다.

Knowledge graphs represent human knowledge as a directed graph, representing each fact as a triplet consisting of a head entity, a relation, and a tail entity. Knowledge graph embedding is a representation learning technique that aims to convert the entities and relations into a set of low-dimensional embedding vectors while preserving the inherent structure of the given knowledge graph. Once the entities and relations in a knowledge graph are represented as a set of feature vectors, those vectors can be easily integrated into diverse downstream tasks. This talk introduces a new concept of knowledge graph called a bi-level knowledge graph, where the higher-level relationships between triplets can be represented. Learning representations on a bi-level knowledge graph, machines are allowed to solve problems requiring more advanced reasoning than simple link prediction. Also, as a practical example of knowledge graph embedding, how one can utilize the knowledge representations to operate a real robot is briefly explained. This talk discusses how knowledge graph embedding models effectively deliver human knowledge to machines, which is critical in many AI applications.

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(Online participation) Zoom Link: https://kaist.zoom.us/j/87516570701 ACMseminar mailing list registration: https://mathsci.kaist.ac.kr/mailman/listinfo/acmseminar

We study logarithmic spiraling solutions to the 2d incompressible Euler equations which solve a nonlinear transport system on the unit circle. We show that this system is locally well-posed for L^p data as well as for atomic measures, that is logarithmic spiral vortex sheets. We prove global well-posedness for almost bounded logarithmic spirals and give a complete characterization of the long time behavior of logarithmic spirals. This is due to the observation that the local circulation of the vorticity around the origin is a strictly monotone quantity of time. We are then able to show a dichotomy in the long time behavior, solutions either blow up (in finite or infinite time) or completely homogenize. In particular, bounded logarithmic spirals converge to constant steady states. For vortex logarithmic spiral sheets the dichotomy is shown to be even more drastic where only finite time blow up or complete homogenization of the fluid can and does occur.

We study logarithmic spiraling solutions to the 2d incompressible Euler equations which solve a nonlinear transport system on the unit circle. We show that this system is locally well-posed for L^p data as well as for atomic measures, that is logarithmic spiral vortex sheets. We prove global well-posedness for almost bounded logarithmic spirals and give a complete characterization of the long time behavior of logarithmic spirals. This is due to the observation that the local circulation of the vorticity around the origin is a strictly monotone quantity of time. We are then able to show a dichotomy in the long time behavior, solutions either blow up (in finite or infinite time) or completely homogenize. In particular, bounded logarithmic spirals converge to constant steady states. For vortex logarithmic spiral sheets the dichotomy is shown to be even more drastic where only finite time blow up or complete homogenization of the fluid can and does occur.

We first survey on nodal solutions for coupled elliptic equations, using results from nonlinear scalar field equations as motivations. Then we discuss work for constructing multiple nodal solutions using various variational methods. In particular we discuss in some details the results about solutions having componentwisely-shared nodal numbers of coupled elliptic systems. These works are done by further developing minimax type critical point theory with built-in flow invariance of the associated gradient or parabolic flows, which has been a useful tool to give locations of critical points via minimum methods, also revealing complex dynamic behavior of the flow.

Pivot-minors can be thought of as a dense analogue of graph minors. We shall discuss pivot-minors and two recent results for proper pivot-minor-closed classes of graphs. In particular, that for every graph H, the class of graphs containing no H-pivot-minor is 𝜒-bounded, and also satisfies the (strong) Erdős-Hajnal property.

다양한 소비재 중에서 유독 패션 카테고리는 그간 디지털 전환이 느린 분야였으며, 유통과 마케팅의 영역에서만 데이터를 주로 활용하는 양상을 보여왔다. 비정형적, 주관적인 의사 결정이 주를 이루는 '패션'에서 수학은 어떤 의미와 효용이 있는지 제조업을 운영하는 디자이너의 관점에서 실무 사례 위주로 이야기하려 한다.

The Harnack inequality plays a crucial role in elliptic and parabolic PDEs. In particular, one can characterize ancient positive solutions to parabolic PDEs by using the Harnack inequality. In this talk, we consider the mean curvature flow, a parabolic PDE of hypersurfaces. To study its stability, it is important to show the uniqueness of ancient flows staying in an one-side of self-similarly shrinking flows. After rescaling the ancient one-sided flow converges to the static self-similar solution, and so it is the graph of an evolving positive function defined over the self-similar solution. Then, the positive function is a solution to a parabolic PDE, and we can show the uniqueness by using the Harnack inequality.

We will present a new approach to develop a data-driven, learning-based framework for predicting outcomes of biophysical systems and for discovering hidden mechanisms and pathways from noisy data. We will introduce a deep learning approach based on neural networks (NNs) and on generative adversarial networks (GANs). Unlike other approaches that rely on big data, here we “learn” from small data by exploiting the information provided by the mathematical physics, e.g.., conservation laws, reaction kinetics, etc,. which are used to obtain informative priors or regularize the neural networks. We will demonstrate how we can train BINNs from multifidelity/multimodality data, and we will present several examples of inverse problems, e.g., in systems biology for diabetes and in biomechanics for non-invasive inference of thrombus material properties. We will also discuss how operator regression in the form of DeepOnet can be used to accelerate inference based on historical data and only a few new data, as well its generalization and transfer learning capacity.

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

In this talk, we provide an overview of the historical development of fast solution methods for partial differential equations, as well as their current status and potential for future advancements. We first begin with a historical survey and describe recent advances in efficient techniques, such as multigrid and domain decomposition methods. In addition, we will explore the potential of emerging methods in the realm of scientific machine learning.

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

In this seminar, we will talk about the chemotaxis model, which is a diffusion model for biological dispersion. Chemotaxis is the movement of biological organisms in response to chemical stimuli. The chemotaxis model has nonlinear diffusion with no reaction term and has been extensively studied in the sense of a diffusion model for heterogeneous media. The nonlinear diffusion alone makes it possible to allow us to observe various spatial patterns. We will see what kind of pattern formation the model provides and what mathematical problems this model can be applied to. Language : Korean but English if it is requested

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Discipline of talk: Analysis / Advisor: 김용정 (Yongjung Kim )

Configurations of axis-parallel boxes in $\mathbb{R}^d$ are extensively studied in combinatorial geometry. Despite their perceived simplicity, there are many problems involving their structure that are not well understood. I will talk about a construction that shows that their structure might be more complicated than people conjectured.

In this talk, we look at the results of various studies in which computational mathematics is used in medical imaging. Through the various scope of research from mathematical modeling to data-based methodology, we can think about the future direction by examining what we can do in data science can contribute and what contribution we can make to medical imaging.