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12月演講

Colloquium, 北海道大學理學院 中村玄教授
Wednesday, December 5, 10:10—13:00 數學系3樓會議室
Title: Linear Sampling Method and Related Method (1)(2)

Colloquium, 中正大學數學系 王琪仁教授
Thursday, December 6, 16:00—17:00 數學系3174
Title: Critical Droplet Solutions for Non-Equilibrium Phase Transitions in Crystal Lattice Systems
Abstract: Discontinuous phase transitions are common in the steady states of diverse non-equilibrium systems describing catalytic reaction-diffusion processes, biological transport, spatial epidemics, etc. These transitions are usually associated with equistability of two stable states, as can be determined by stationarity of a planar interface separating these states. For equilibrium systems, this criterion is equivalent to the Maxwell construction determining coexistence of two states at a unique equistability point. Analyses of nucleation phenomena near such transitions aims in part to characterize critical droplets of the more stable state embedded in the less stable metastable state, where these droplets correspond to stationary curved interfaces between the two states. There is a range of critical droplets and their critical sizes are expected to diverge when approaching the transition. The critical curvature which arrests propagation should vanish linearly approaching the transition. However, the analysis of discontinuous transitions in spatially discrete non-equilibrium systems also reveals an interface propagation failure. As a non-equilibrium counterpart to the classic Ising model, we consider stochastic lattice-versions of Schloegl’s 2nd model involving spontaneous annihilation X→ and autocatalytic creation +2X3X.

Colloquium, 北海道大學理學院 中村玄教授
Wednesday, December 12, 10:10—13:00 數學系3樓會議室
Title: No Response Test and Related Methods (1)(2)

Colloquium, 台灣大學醫學工程學系 林發暄教授
Thursday, December 13, 16:00—17:00 數學系3174
Title: Spatial Encoding and Decoding in Magnetic Resonance Imaging: The Forward and the Inverse Problems
Abstract:Magnetic resonance imaging (MRI) is a powerful diagnostic imaging tool in both clinical medicine and neuroscience because of its non-invasiveness, versatile contrasts, and high spatial resolution. In this talk, we will first briefly introduce the spatial encoding and decoding formalism of MRI and the associated imaging hardware. Then we will summarize recent technological advances in magnet, gradient coils, and radio-frequency coils. Preliminary results enabled by these hardware advances and the associated challenges and opportunities will also be presented. Lastly, we will discuss open questions related to how to further advance these technologies and how to optimally use these tools in order to achieve MRI with high spatiotemporal resolution.

PDE Seminar, 中央研究院 陸冰瑩博士
Thursday, December 20, 15:10—16:00 數學系3174
Title: The Universality of the Semi-Classical Sine-Gordon Equation at the Gradient Catastrophe
Abstract: We study the semi-classical sine-Gordon equation with pure impulse initial data below the threshold of rotation: , u(x,0) \equiv 0, \epsilon u_t(x,0)=G(x)\leq 0, and |G(0)|<2. A dispersively-regularized shock forms in finite time. We found, in accordance with a conjecture made by Dubrovin et. al., that the asymptotics near a certain gradient catastrophe is universally (insensitive to initial condition) described by the tritronqu\’ee solution to the Painlev\’e-I equation. Furthermore, we are able to universally characterize the shapes of the spike-like local structures (similar to rogue wave on periodic background for the focusing nonlinear Schr\"odinger equation) on top of the poles of the tritronqu\’ee solution. Our technique is the Deift-Zhou steepest descent analysis of the Riemann-Hilbert problem associated with the sine-Gordon equation. Our approach is inspired by a study of universality for the focusing nonlinear Schr\”odinger equation by Bertola-Tovbis. (joint work with Peter Miller)

Colloquium, 俄亥俄州立大學 Prof. Yuji Kodama
Thursday, December 20, 16:10—17:00 數學系3174
Title: Mathematics for Web-Like Patterns of Solitary Waves in Shallow Water
Abstract: We often observe web-like patterns of waves on the surface of shallow water. They are examples of nonlinear waves, and these patterns are generated by nonlinear interactions among several obliquely propagating solitary waves. The aim of the talk is to explain these wave patterns based on a two-dimensional nonlinear dispersive wave equation called the KP equation invented by Kadomtsev and Petviashvili in 1970. Recently a large variety of the exact solutions of the KP equation, referred to as the KP solitons, has been found and classified by using modern mathematical tools from several mathematical areas including algebraic geometry, algebraic combinatorics and representation theory. In this talk, I will give a brief summary of the theory and, in particular, discuss an application of the KP solitons to the Mach reflection problem in shallow water, which has an important implication to tsunami amplification along the shore. The problem describes the resonant interaction of solitary waves appearing in the reflection of an obliquely incident wave onto a vertical wall. The talk will be elementary and include many figures of the wave-patterns from real ocean.

Algebraic Geometry Seminar, 東京大學數學系 中村勇哉教授
Friday, December 21, 11:10—12:00 數學系3樓會議室
Title: A Rational Point Problem on Fano Varieties
Abstract: Fano varieties defined over a certain field are believed to have a k-rational point. For instance, the classically known Chevalley--Warning theorem states that homogeneous polynomials f_1,\ldots, f_l \in \mathbb{F} _q [x_0, \ldots, x_n] with (n+1)-many variables over a finite field \mathbb{F} _q have a non-trivial common solution if \sum _{1 \le i \le l} \deg f_i \le n holds. More precisely, it states that the number of common solution is divisible by the characteristic p of the field \mathbb{F} _q. Geometrically, it can be interpreted as that the number of the \mathbb{F} _q-rational points on a complete intersection Fano variety is one modulo p. This kind of problem was answered in the smooth case by Esnault. She proved that smooth Fano varieties defined over a finite field have a rational point. It is natural to ask whether we can generalize this results to singular Fano varieties or not. In this talk, we study the rational point problem on singular Fano varieties and the relation to the minimal model program.

Algebraic Geometry Seminar, 東京大學數學系 中村勇哉教授
Friday, December 21, 14:10—15:00 數學系3樓會議室
Title: Dual Complex of Fano Carieties and an Application to Vanishing of Witt Vector Cohomology
Abstract: I will explain that the dual complex of a log Fano pair is simply connected. As a corollary, we can get a vanishing theorem of Witt vector cohomology for log canonical Fano threefolds. This is an analogy of the Ambro-Fujino vanishing theorem in characteristic zero.

NCTS/NCKU Algebraic Geometry Seminar, 香港科技大學 張懷良教授
Tuesday, December 25, 13:10—15:00 數學系3樓會議室
Title: Feynman Structures in Gromov Witten Theory
Abstract: For compact CY 3fold, such as the quintic 3fold, the math study of higher genus GW potentials Fg has been open for over two decades. Recently, we prove quintic's Fg's are analytic functions, satisfy Yamaguchi-Yau finite generation conjecture(2004), and BCOV Feynman graph conjecture(1993). This determines the infinite series Fg up to 3g-3 unknowns. For example we recover F1 and obtain F2 completely. Our approach is packaging ``N-Mixed-Spin-P fields" (NMSP) moduli for large N. This is a joint work with Shuai Guo and Jun Li.

   
 


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國立成功大學數學系
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