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 4¤ëºtÁ¿ Colloquium, °ê¥ß¶§©ú¥æ³q¤j¾Ç²z¾Ç°|°|ªø ¿à©úªv±Ð±Â Thursday, April 14, 16:10¡X17:00 ¼Æ¾Ç¨t3174 Title: Solving PDEs Using Machine Learning: How and Why? Abstract:In this talk, we will present how to use machine learning methods to solve partial differential equations, especially when PDEs become difficult to handle with traditional numerical methods (such as finite difference, finite element or spectral methods). We start by introducing some fundamental mathematical backgrounds for neural network representations. Then we introduce our developments for machine learning methodologies by considering elliptic interface problems where the solutions are usually not smooth. We have successfully developed two different types of completely shallow neural network to solve above elliptic PDEs. To the best of our knowledge, this is the first work in literature that uses only one hidden layer with moderate number of neurons to solve the problems even in higher dimensions. Colloquium, °ê¥ß»OÆW®v½d¤j¾Ç¼Æ¾Ç¨t ªL«T¦N±Ð±Â Friday, April 15, 11:10¡X12:00 ¼Æ¾Ç¨t3173 Title: An Approach of Geometric Flows for the Path-planning Problem --- Motivated by Geometric Control Theory and Quantum Circuit Design Abstract:Given a set of points $\{p_0,\cdots,p_q\}$ in a (sub-)Riemannian manifold, how to find a curve connecting these points in the given order and fulfilling with certain smoothness and constraints is called the path-planning problem. Such a problem is motivated by several disciplines including geometric control theory and quantum circuit design. In this talk, I would like to present the setting of problems, the connection with various disciplines, and the approach of geometric flows. Colloquium, °ê¥ß²MµØ¤j¾Ç¼Æ¾Ç¨t ÃQºÖ§ø±Ð±Â Thursday, April 21, 16:10¡X17:00 ¼Æ¾Ç¨t3174 Title: CM Periods and Special Gamma Values Abstract:The gamma function is one commonly used extension of the factorial function to complex numbers. It shows up in various research areas, and the special values of which are closely related to ¡§CM periods¡¨. The celebrated Chowla-Selberg formula expresses the periods of elliptic curves with ¡§complex multiplications¡¨ in terms of a very beautiful monomial combinations of gamma values at fractions. In fact, the recipe comes precisely from the ¡§Stickelberger¡¨ evaluators of Dirichlet L-values. This ¡§Chowla-Selberg phenomenon¡¨ is regarded as a geometric aspect of the Kronecker-Weber theorem for abelian extensions of the rational number field, which is studied extensively for CM periods in the higher dimensional cases based on the work of Weil, Shimura, Deligne, Gross, Anderson, Colmez, etc. In this talk, we shall start from basic properties of the gamma function, and then discuss algebraic relations among special gamma values and their ¡§Chowla-Selberg¡¨ connections with CM periods. A brief summary of my current progress on a ¡§positive-characteristic¡¨ analogue will be mentioned if time permits.

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 °ê¥ß¦¨¥\¤j¾Ç¼Æ¾Ç¨t 70101 ¥x«n¥«¤j¾Ç¸ô¤@¸¹ ¹q¸Ü¡J(06) 2757575 Âà 65100   ¶Ç¯u¡J(06) 2743191 em65100[at]email.ncku.edu.tw