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.

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.

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)

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

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.

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

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.

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

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.

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

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

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.

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