5月演講

Applied Mathematics Seminar, Prof. Moody T. Chu, North Carolina State University Wednesday, May 1, 15:10—16:00 數學系館3樓會議室 Title: On the Dynamics of the Maximin Flow Abstract: In a complex system, such as the molecular dynamics, chemical kinetics, nucleation mechanism, or even the Lagrangian of a constrained convex programming problem, the presence of a saddle point often represents that a transition of events has occurred. Determining the locations of saddle points in the configuration space and the way they affect the transition provide critical information about the underlying complex system. This paper proposes a dynamical system approach to explore this problem. In addition to being capable of finding saddle points, the flow exhibits some intriguing behavior nearby a saddle point, which is demonstrated by graphic examples in various settings. Maximin flows also arise naturally from complex-valued differential equations over analytic vector fields due to the Cauchy-Riemann equations. The maximin flow can be cast as a gradient flow in the Krein space under indefinite inner product, whence the Lojasiewicz gradient inequality can be generalized. It is proved that a solution trajectory has finite arc length and, hence, converges to a singleton saddle point.

Colloquium, 日本東京大學數學系 川又雄二郎教授 Thursday, May 2, 16:10—17:00 化學系館 1F36104教室 Title: Derived McKay Correspondence and Non-Commutative Deformations Abstract:The derived MacKay correspondence conjecture states that there is an equivalence of triangulated categories between commutative and non-commutative crepant resolutions of singularities. It is proved in dimensions at most 3 or for toric singularities. We investigate how the derived McKay correspondence extends under non-commutative deformationso f commutative and non-commutative crepant resolutions in the example of toric surface singularities.

升等演講, 國立成功大學數學系 賴青瑞教授 Thursday, May 9, 15:10—16:00 化學系館 1F36104教室 Title: On Movable Cones of Some Calabi-Yau Threefolds of Picard Rank Two Abstract:The birational geometry of Calabi-Yau threefolds is intricate even with many advances in modern higher dimension geometry. For example, Reid's fantasy and the Morrison-Kawamata cone conjecture are only partially solved. In this talk, we report a recent joint work with Atsushi Ito (Okayama) and Sz-Sheng Wang (Academic Sinica at Taiwan), in which we describe explicitly the movable cones of a class of Calabi-Yau threefolds of Picard rank two. These examples may shed some light on the above-mentioned conjectures.

Colloquium, 陽明交通大學應用數學系 吳昌鴻教授 Thursday, May 9, 16:10—17:00 化學系館 1F36104教室 Title: Spreading Fronts Arising from the Singular Limit of Reaction-Diffusion Systems Abstract:In this presentation, we will focus on the singular limit of reaction-diffusion systems to gain insight into the formation of spreading fronts of invasive species. We will derive several free boundary problems and provide interpretations for spreading fronts from a modeling perspective. Additionally, numerical examples will be presented to facilitate discussion on invasion speed. The talk is based on joint works with Hirofumi Izuhara and Harunori Monobe.

Colloquium, 中山大學應用數學系 黃毅青教授 Thursday, May 16, 15:10—16:00 數學系3174教室 Title: On Linear Preservers Between Matrices Over an Arbitrary Field Abstract:演講摘要

Colloquium, 中山大學應用數學系 卓建宏教授 Thursday, May 16, 16:10—17:00 數學系3174教室 Title: On the Convergence of the Numerical Blow-Up Time for a Rescaling Algorithm Abstract:Berger and Kohn (1988) proposed an algorithm to compute approximate blow-up times for those evolution equations whose solutions blow up in a finite time and possess a scaling invariance property. Later, Anada et al. (2018) used this algorithm to compute the blow-up rates of various blow-up problems which turned out to be very effective. However, the convergence analysis for this algorithm seemed to be not well-studied. In this talk, we analyze the numerical implementation for this algorithm via a nonlinear ODE blow-up problem. The convergence order of the numerical blow-up time is also verified.

Analysis Seminar, Prof. Wu Hau-Tieng, Department of Mathematics Courant Institute of Mathematical Sciences, New York University, USA. Tuesday, May 21, 9:30—11:30 數學系3樓會議室 Title: An introduction of applying unsupervised manifold learning algorithms to analyze time series. Unsupervised manifold learning algorithms will be introduced with existing theoretical support. Its application to time series analysis will be demonstrated by exploring real world biomedical signal processing challenges.

Colloquium, Prof. Kazuo Aoki, National Cheng Kung University, Tainan, Taiwan and Kyoto University, Kyoto, Japan Tuesday, May 28, 16:10—17:00 數學系3樓會議室 Title: Models of Boundary Conditions for the Boltzmann Equation Based on a Kinetic Model of Gas-Surface Interaction Abstract:Boundary conditions for the Boltzmann equation are investigated on the basis of a kinetic model of gas-surface interaction. The model takes into account the interaction of gas molecules with the molecules forming the solid. The layer of interaction is assumed to be thinner than the mean free path of the gas molecules, and the solid molecules are in equilibrium. The asymptotic kinetic equation for the interaction layer (physisorbate layer), which forms a steady half-space problem, is derived and used to investigate boundary conditions for the Boltzmann equation that is valid outside the physisorbate layer. To be more specific, new models of the boundary condition are proposed on the basis of iterative solutions of the half-space problem and are assessed by the direct numerical analysis of the problem. In addition, some rigorous mathematical results for the half-space problem are presented. This is a joint work with Vincent Giovangigli, Fran?ois Golse (Ecole Polytechnique) and Shingo Kosuge (Kyoto University).

Analysis Seminar, Prof. Wu Hau-Tieng, Department of Mathematics Courant Institute of Mathematical Sciences, New York University, USA. Thursday, May 30, 10:00—12:00 數學系3樓會議室 Title: Some Topics in Time-Frequency Analysis and their Applications Abstract:We will discuss some time-frequency analysis tools for nonstationary time series analysis. A comparison of their properties will be provided. Some applications to biomedical signals will be shown.

Colloquium, 陽明交通大學應用數學系 蘇承芳教授 Thursday, May 30, 16:10—17:00 數學系3174 Title: Introduction to Quantum Monte Carlo Methods and Related Issues Abstract:The challenge of option pricing is pivotal for financial experts globally, as it aims to predict the price distributions of assets to facilitate informed investment decisions. This talk primarily introduces the significance of the call option pricing problem, discusses the complexity of solving the high-dimensional Black-Scholes model, and then considers Quantum Monte Carlo simulations, explaining their fundamental structure and advantages. We hope to study how this methodology provides financial professionals with new tools for more accurate assessment and management of investment risks, thereby opening new avenues in quantitative finance.

Analysis Seminar, Prof. Wu Hau-Tieng, Department of Mathematics Courant Institute of Mathematical Sciences, New York University, USA. Friday, May 31, 10:00—12:00 數學系3樓會議室 Title: Analyzing nonstationary time series with manifold learning algorithms Abstract:Compared with snapshot health information, long-term and high-frequency physiological time series provides health information from the other dimension. I will discuss recently developed graph-Laplacian based manifold learning algorithms for such time series. From the clinical aspect, its application to estimating and forecasting airflow signal from thoracic and abdominal respiratory efforts for sleep apnea application will be discussed. From the theoretical aspect, we will discuss some topics toward statistical inference, like L^\infty spectral convergence and local law and rigidity of eigenvalue distribution of graph Laplacian. The current efforts toward including longitudinal data analysis will also be discussed if time permits.