Tuesday, March 14, 2023

2023-03-20 / 11:00 ~ 12:00

by ()

The standard approach to problem-solving in physics consists of identifying state variables of the system, setting differential equations governing the state evolution, and solving the obtained. The behavior of the system for different values of parameters can be examined only as a fourth step. On the contrary, the modern approach to studying dynamical systems relies on Morphological/Topological analysis which alleviates the necessity for the explicit solution of differential equations. The stability analysis of the parabolic swing will demonstrate the merit of such an approach. It will be shown how to construct a qualitatively correct model of system dynamics that is surprisingly quantitatively correct as well. The sudden (catastrophic) change in the swing’s stability, caused by a slight change in the critical value of system parameters, will be linked to the drastic topological change of the corresponding phase-space portraits. It will be shown that for a system’s parameters close to critical ones, the system’s behavior is identical to a specific simple universal prototype given by catastrophe theory. A short survey of the simplest elementary catastrophes will be given that represents the basis for applying catastrophe theory in other fields of science.

2023-03-17 / 10:00 ~ 11:00

SAARC 세미나 - SAARC 세미나:

by 강남규(고등과학원)

Since conformal field theory (CFT) was introduced to relate various critical statistical physics models to Virasoro representation theory, it has been applied to string theory, condensed matter physics, vertex operator algebra, and probability theory. In this talk, I will explain how the boundary CFT of central charges c (less than or equal to 1) relates to the boundary CFT of 26 - c in terms of boundary conditions. (On the algebraic side, Feigin and Fuchs described the duality between the categories of Verma modules with central charges c and 26 - c to explain the appearance of the critical dimension 26 of the bosonic string theory.) I will present the connection between the boundary CFT and the theory of Schramm-Loewner evolution in various conformal types.

2023-03-21 / 10:30 ~ 11:30

SAARC 세미나 - SAARC 세미나: Determinantal expressions for eigenvalue statistics of random matrices

by 정성우 박사(MIT 수학과)

We present new (mostly determinantal) expressions for various eigenvalue statistics in random matrix theory. Whenever the eigenvalue $n$-point correlation function is given in terms of $n \times n$ determinants with some kernel, we propose a new kernel that gives the $n$-point correlation function of the eigenvalues conditioned on the event of some eigenvalues already existing at fixed positions. Using such new kernels we obtain determinantal expressions for the joint densities of the $k$ largest eigenvalues, probability density function of the $k$-th largest eigenvalue, density of the first eigenvalue spacing, and many more. Our formulae is highly amenable to numerical computation through the method proposed by Bornemann (2008).

2023-03-20 / 16:30 ~ 17:30

학과 세미나/콜로퀴엄 - 계산수학 세미나: Fifty three matrix factorizations, symmetric spaces and generalized Cartan decomposition

by 정성우 박사(MIT 수학과)

Symmetric spaces from Lie theory and differential geometry are often represented by special set of structured matrices. The Cartan decomposition and its generalization of symmetric spaces and classical Lie groups recover many of the known matrix factorizations in numerical linear algebra, such as the singular value decomposition, CS decomposition, generalized SVD and many more. We discuss a blueprint for generating fifty-three matrix factorizations from the generalized Cartan decomposition, most of which appear to be new. The underlying mathematics may be traced back to Cartan (1927), Harish-Chandra (1956), and Flensted-Jensen (1978). This is joint work with Alan Edelman.

2023-03-17 / 11:00 ~ 12:00

학과 세미나/콜로퀴엄 - 응용 및 계산수학 세미나:

by 이재용(고등과학원)

Many differential equations and partial differential equations (PDEs) are being studied to model physical phenomena in nature with mathematical expressions. Recently, new numerical approaches using machine learning and deep learning have been actively studied. There are two mainstream deep learning approaches to approximate solutions to the PDEs, i.e., using neural networks directly to parametrize the solution to the PDE and learning operators from the parameters of the PDEs to their solutions. As the first direction, Physics-Informed Neural Network was introduced in (Raissi, Perdikaris, and Karniadakis 2019), which learns the neural network parameters to minimize the PDE residuals in the least-squares sense. On the other side, operator learning using neural networks has been studied to approximate a PDE solution operator, which is nonlinear and complex in general. In this talk, I will introduce these two ways to approximate the solution of PDE and my research related to them.

2023-03-17 / 14:00 ~ 16:00

IBS-KAIST 세미나 - 수리생물학:

by ()

The seasonal outbreaks of influenza infection cause globally respiratory illness, or even death in all age groups. Given early-warning signals preceding the influenza outbreak, timely intervention such as vaccination and isolation management effectively decrease the morbidity. However, it is usually a difficult task to achieve the real-time prediction of influenza outbreak due to its complexity intertwining both biological systems and social systems. By exploring rich dynamical and high-dimensional information, our dynamic network marker/biomarker (DNM/DNB) method opens a new way to identify the tipping point prior to the catastrophic transition into an influenza pandemics. In order to detect the early-warning signals before the influenza outbreak by applying DNM method, the historical information of clinic hospitalization caused by influenza infection between years 2009 and 2016 were extracted and assembled from public records of Tokyo and Hokkaido, Japan. The early-warning signal, with an average of 4-week window lead prior to each seasonal outbreak of influenza, was provided by DNM-based on the hospitalization records, providing an opportunity to apply proactive strategies to prevent or delay the onset of influenza outbreak. Moreover, the study on the dynamical changes of hospitalization in local district networks unveils the influenza transmission dynamics or landscape in network level.

2023-03-15 / 16:00 ~ 17:00

IBS-KAIST 세미나 - 수리생물학:

by ()

Multi-omics technologies, and in particular those with single-cell and spatial resolution, provide unique opportunities to study the deregulation of intra- and inter-cellular signaling processes in disease. I will present recent methods and applications from our group toward this aim, focusing on computational approaches that combine data with biological knowledge within statistical and machine learning methods. This combination allows us to increase both the statistical power of our analyses and the mechanistic interpretability of the results. These approaches allow us to identify key processes, that can be in turn studied in detailed with dynamic mechanistic models. I will then present how cell-specific logic models, trained with measurements upon perturbations, can provides new biomarkers and treatment opportunities. Finally, I will show how, using novel microfluidics-based technologies, this approach can also be applied directly to biopsies, allowing to build mechanistic models for individual cancer patients, and use these models to prose new therapies.

2023-03-16 / 11:50 ~ 12:40

대학원생 세미나 - 대학원생 세미나:

by 장원용(카이스트)

For complex number $\alpha$, let $ A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 \\ \alpha & 1 \end{bmatrix} $ be two parabolic matrices and let $G_\alpha$ be the group generated by these two matrices. Determining whether $G_\alpha$ is the free group rank 2 or not, is one of the important problems. In this talk, I will introduce a geometric aspect of the group $G_\alpha$ and give previous results of the problem. Next, I will introduce my work joint with KyeongRo Kim.

2023-03-14 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Recent progress on the Union-closed conjecture and related

by Stijn Cambie(IBS 극단조합및확률그룹)

We give a summary on the work of the last months related to Frankl's Union-Closed conjecture and its offsprings. The initial conjecture is stated as a theorem in extremal set theory; when a family F is union-closed (the union of sets of F is itself a set of $\mathcal F$), then $\mathcal F$ contains an (abundant) element that belongs to at least half of the sets. Meanwhile, there are many related versions of the conjecture due to Frankl. For example, graph theorists may prefer the equivalent statement that every bipartite graph has a vertex that belongs to no more than half of the maximal independent sets. While even proving the ε-Union-Closed Sets Conjecture was out of reach, Poonen and Cui & Hu conjectured already stronger forms. At the end of last year, progress was made on many of these conjectures. Gilmer proved the ε-Union-Closed Sets Conjecture using an elegant entropy-based method which was sharpened by many others. Despite a sharp approximate form of the union-closed conjecture as stated by Chase and Lovett, a further improvement was possible. In a different direction, Kabela, Polak and Teska constructed union-closed family sets with large sets and few abundant elements. This talk will keep the audience up-to-date and give them insight in the main ideas. People who would like more details, can join the Ecopro-reading session on the 7th of March (10 o'clock, B332) as well. Here we go deeper in the core of the proofs and discuss possible directions for the full resolution.

Events for the 취소된 행사 포함