|Colloquium, 國立政治大學應用數學系 黃建豪博士因疫情取消|
Thursday, May 5, 16:10—17:00 數學系3174
Title: The Statistical Mechanics of Wiener Sausages
Abstract:We first discuss one basic idea in mathematical statistical mechanics: large deviations. Secondly, we review the materials in the theory of random walks. Physicists and mathematicians use those concepts to discuss phase transitions in polymer models. In this talk, the directed polymer model is the statistical mechanics model with the hamiltonian described by the d-dimensional random walk in space-time random environments. The particles gain energy whenever they visit the potential sites. The analogous continuum model, namely, the model with noise formally defined on R^d, which is the stochastic heat equation with space-time white noise, is not well-defined. Recently many discussions on it. We propose a new model that the particles only gain energy at their first visit with the noise being independent of time. In the continuum and weak disorder regime, the partition function of our model as a random variable converges weakly to a Wiener Chaos expansion, however, it describes a different phenomenon.
|Colloquium, 中央研究院數學所 尤釋賢特聘研究員|
Friday, May 6, 10:10—12:00 數學系3174
Title: A Computable Approach to Wave and Heat Equation
Abstract:In this talk we will give an elementary method to study wave and heat equation including various physical condition imposed.
|Colloquium, 國立中央大學數學系 曾國師教授|
線上專題演講(會議號:2511 759 8689，會議密碼：238172)
Thursday, May 12, 16:10—17:00 數學系3174
Title: 3D Map Exploration using Topological Fourier Sparse Set
Abstract:3D map exploration is one of key technologies in robotics. However, finding an optimal exploration path is a challenge due to unknown environments. This research proposed the Topological Fourier Sparse Set (TFSS) algorithm to enable an unmanned aerial vehicle (UAV) to explore 3D environments with theoretical guarantees. The algorithm combines the Rips complex with Fourier sparse set representation to take the advantages of topological and submodular approaches. More specifically, the Rips complex is used for expanding the exploration subgoals, while the Fourier sparse set encodes a learned representation of the subgoal selection problem in the form of a submodular optimization problem. Since the objective function of spatial exploration is reformulated as a maximizing submodular function with path constraints, greedy algorithms can achieve 0.5(1 - e^(-1)) of the optimum. Experiments conducted with this algorithm demonstrates that the TFSS explores unknown environments 25% to 127% more than the NBV algorithm does. The TFSS exploration performance is close to the SFSS but it is 50 times faster than the SFSS.
|理學論壇暨數學系專題演講, 國立陽明交通大學應用數學系 林文偉 教授 (國家講座)|
線上專題演講(會議號:2517 757 6638，會議密碼：n8UGpnGC4J3)
Thursday, May 19, 15:10—17:00 理學院36102
Title: Computational Conformal Geometry with Optimal Mass Transportations and its Application on 3D Brain Tumor Segmentations
Abstract:In this talk, we would like to introduce the computational conformal geometry with optimal mass transportation techniques and its applications on 3D medical image detections and segmentations. The well-known uniformization theorem shows that a closed surface of genus-zero is equivalently conformal to a unit sphere. However, the numerical method and its convergence should be addressed. We will propose efficient algorithms on conformal energy minimization (CEM), stretch energy minimization (SEM) and volume stretch energy minimization (VSEM) for finding the conformal (angle-preserving) and equiareal (area-preserving) parametrizations, respectively, between a simply connected closed surface and a sphere, as well as, the volume-preserving parametrization between a 3-manifold with a genus-zero boundary and a unit ball. Based on the SEM and VSEM algorithms we further develop the reliable and robust algorithms for solving the optimal mass transportation (OMT) between an irregular 3D domain and a unit ball, while minimizing the deformation cost, and keeping the minimal distortion and the local mass ratios unchanged. Combining the proposed OMT with the U-net machine learning algorithm, we develop a novel two-phase OMT algorithm successfully applying for the detection and segmentation of 3D brain tumors with high training and validation Dice scores. For training, good Dice scores: 0.9538 for the WT (whole tumor), 0.9546 for the TC (tumor core) and 0.9093 for the ET (enhanced tumor) can be obtained. For validation, the Dice scores of WT, TC and ET with mesh refinement and ensemble voting postprocessing can reach 0.9371, 0.9062 and 0.8747. A significant accuracy improvement in brain tumor detection and segmentation is achieved. Furthermore, It takes within 200 seconds to complete the whole brain tumor segmentation process for each new brain sample.
|Analysis Seminar, Duke University 吳浩榳教授|
Tuesday, May 24, 11:10—12:00 數學系3F會議室
Title: A Novel Spatiotemporal Analysis Based on Kernel Fusion and Multiresolution Analysis
Abstract:I will discuss a new spatiotemporal analysis algorithm based on kernel fusion and multiresolution analysis. The key ingredient is the kernel based sensor fusion algorithm that captures the "common" and "different" information shared by two consecutive time points. Some preliminary theoretical analyses will be provided, and some numerical results will be discussed. If time permits, its application to body area networks datasets will be demonstrated.
|Colloquium, 國立陽明交通大學應用數學系 吳昌鴻教授|
線上專題演講(會議號:2515 508 1445，會議密碼：i2ReJKEYy33)
Thursday, May 26, 16:10—17:00 線上演講
Title: Some Spreading Properties of Two-Species Lotka-Volterra Competition-Diffusion Systems
Abstract:The Lotka-Volterra competition-diffusion system is a classical model to describe the interaction between competing species. For the two species case, it is well known that traveling waves solutions exist and these waves can be used to determine the spreading behaviors of species. In this talk, we shall present some recent progress on the spreading properties of this system with two different settings: the first one is the Cauchy problem with two different situations: either one species is an invasive one, and the other is a native one, or both are invasive species; the second is the free boundary problem, with both being invasive species.