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Registration:
https://goo.gl/4PYZay
10:00-15:30, July 25 - August 12, 2016
Room 101, Astronomy-Mathematics Building, NTU
Speaker:
Yi-Fan Yang ( National Chiao Tung University )
Wu-yen Chuang ( National Taiwan University )
Chin-Yu Hsiao ( Academia Sinica )
Organizers:
Jungkai Chen ( National Taiwan University & National Center for Theoretical Sciences )
Week 1. 7/25-7/29
Title:
Introductory and Arithmetic aspects
Lecturer:
Prof. Yi-Fan Yang (NCTU)
Abtract:
The first week serves as an introductory to elliptic curves together with some basic number theoretical discussion. It is consists of the following four parts.
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Elliptic curves over C: elliptic functions, Weierstrass equation, Weierstrass P-function, addition law, modular curve as the moduli space of elliptic curves, and classification of endomorphism rings of elliptic curves over C.
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Elliptic curves over Q: points of finite order on elliptic curves (the Nagell-Lutz theorem), and the Mordell theorem.
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Elliptic curves over finite fields: Hasse's theorem, application to integer factorization (Lenstra's algorithm) and cryptography.
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L-functions of elliptic curves over Q.
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Week 2. 8/1-8/5
Title:
Algebraic aspects : Dualities of Elliptic Curves
Lecturer:
Prof. Wu-Yen Chuang (NTU/NCTS)
Abstract:
We will first introduce the line bundles on complex tori, Poincare line bundle, cohomology of the line bundles, and theta functions. After introducing the basics, I will talk about the homological mirror symmetry for elliptic curves to show how the basics are put into use.
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Week 3. 8/8-8/12
Title:
An introduction to complex analytic geometry: Bergman kernel and Demailly's Morse inequalities
Lecturer:
Prof. Chin-Yu Hsiao (AS/NCTS)
Abstract:
An elliptic curve over C is a Riemann surface and is isomorphic as a complex torus. Theta functions are holomorphic functions on C g , quasi-periodic with respect to a lattice. From modern point of view, elliptic curve over C is an 1-dimensional complex manifold and Theta functions are sections of line bundles on complex torus. Complex geometry is the study of complex manifolds (Elliptic curve over C) and the behavior of holomorphic sections (Theta functions). Demailly’s Morse inequalities and Bergman kernel are powerful tool to study the behavior of holomorphic sections. In this course, I will give a Bergman kernel proof of Demailly’s Morse inequalities. In the proof, we only need undergraduate P.D.E and basic complex analysis. We will not assume any background knowledge of complex geometry.
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The concept of complex manifolds, describe line bundles on complex torus by using theta functions.
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holomorphic vector bundle, connection, curvature.
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Dolbeault cohomology, Elliptic P.D.E on compact complex manifold, Hodge theory.
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Bergman kernel function and Proof of Demailly's Morse inequalities.
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