2016清华大学华罗庚讲座
报告人：Don Zagier，Max Planck Institute for Mathematics
时间：16:30-17:30, 4月6日，8日，11日，13日
地点：清华大学近春园西楼三层报告厅
报告人简介
Don Bernard Zagier是一位美国数学家，他的主要研究领域是数论。目前，他是德国马克斯•普朗克数学研究所的主任之一。他曾与2006至2014年在巴黎法兰西学院任教授。自2014年十月起，他同时也是国际理论物理中心的杰出科学家。
Zagier在麻省理工学院学习三年，完成他的学士学位和硕士学位，后在牛津大学获得博士学位。1976年，年仅24岁的Zagier成为波恩大学教授，是德国当时最年轻的教授。
Zagier和Hirzebruch合作进行希尔伯特模曲面的研究。他们合著的《Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus》一书证明了一个希尔伯特模曲面上的代数闭链的相交数可表为一个模型式的Fourier系数。
Zagier的一个重要研究成果是与Benedict Gross合作证明的Gross–Zagier公式。此公式将一个椭圆曲线的复L函数在1处的一阶导数与某个Heegner点的高度联系起来。这个定理有很多应用，包括Birch--Swinnerton-Dyer猜想的一些情况，还有作为Dorian Goldfeld解决理想类数问题的关键一步。作为他们工作的一部分， Gross和Zagier发现一种奇异模量的差异规范（范数？）公式。后来，Zagier又发现一个奇异模的迹的公式，将其表为一个权为3/2的模型式的Fourier系数。
Zagier与John Harer合作计算代数曲线模空间的orbifold欧拉示性数，将它们与黎曼Zeta函数的特殊值联系起来。
Zagier通过研究三维算术双曲流形，发现了一个任一数域的Dedekind zeta 函数在s=2处的取值的公式，用dilogarithm 函数表达。后来他又提出了一个将Dedekind zeta函数的特殊值用多对数函数表达的一般猜想。他还发现了对费马关于两个平方之和的定理的一个简短而初等的证明。
Zagier教授获得了很多荣誉，包括1987年的科尔数论奖，2001年的 von Staudt奖和2007年的德国数学学会的高斯讲座奖。他于1997年当选为荷兰皇家艺术学院的外籍院士。
2016 Tsinghua University Loo-Keng Hua Distinguished Lecture
Speaker: Don Zagier [Max Planck Institute for Mathematics]
Time: 16:30-17:30, April 6 (Wed.), April 8 (Fri.), April 11 (Mon.), April 13 (Wed.)
Place: Lecture Hall, Floor 3, Jin Chun Yuan West Building
Title: Partitions, modular forms, and applications to surfaces
Abstract:
Partitions and modular forms, both of which go back to Euler, are among the most important and beautiful objects studied in mathematics, and are closely intertwined: computing partitions led Euler to discover their generating function, which turns out to be the first example of a modular form, and then the modularity of this function led to proofs of the main properties of partitions like their asymptotic behavior and the congruences they satisfy. The first two lectures will give an introduction and survey of these two topics and their interrelationships, while the third and fourth will discuss more recent developments: an application of modular forms to counting the number of coverings of a torus by a surface of genus g ("mirror symmetry in dimension one"), a wonderful generalization of this result by Bloch and Okounkov, with a new and very simple proof, and recent joint work with Dawei Chen and Martin Moeller that generalizes and applies this theorem to questions concerning moduli spaces of Riemann surfaces.
All of the lectures will be aimed at non-experts, and are intended to be accessible to enthusiastic senior undergraduate students as well as to graduate students or researchers from other fields. At least the first talk should be understandable even for first or second year undergraduates.
TALK 1: 16:30-17:30, 2016-4-6
In how many ways can you divide a set of 4 apples into smaller subsets?
We assume that the apples are identical, so only the size of the subsets counts. Then the answer is "5 ways", with the sizes of the subsets being (1,1,1,1), (1,1,2), (1,3), (2,2), or (4). We say that 4 has 5 partitions, or p(4)=5. Partitions, first studied by Euler, are a basic object of combinatorics, and have many wonderful properties. In this lecture I will describe some of these, e.g.: how the number p(200) = 3972999029388 could be computed by hand, long before there were electronic calculators; how the study of partitions led Euler to the wonderful idea of "generating functions"; the Hardy-Ramanujan asymptotic formula for the size of p(n) when n is large; and the famous Ramanujan congruences like the divisibility of p(11n+6) by 11.
TALK 2: 16:30-17:30, 2016-4-8
Modular forms have been studied for about 150 years and are among the basic tools of modern number theory. They have a dual nature: one the one hand they are functions of a complex variable with a very large non-commutative symmetry group, and on the other hand they can be seen as power series whose coefficients include many interesting arithmetical functions like partitions, sums of powers of divisors, or number of representations of integers as sums of squares. Because of this they have many applications in all parts of number theory, e.g., they played the key role in the proof of Fermat's Last Theorem. I will give the main concepts and lots of examples, and discuss how modular forms can be used to prove some of the properties of partitions discussed in the first lecture.
TALKS 3 and 4: 16:30-17:30, 2016-4-11/13
In "mirror symmetry", which plays a central role in string theory as well as in several parts of modern algebraic geometry, one is interested in particular in counting the number of maps of a Riemann surface of given genus g to an n-dimensional complex manifold of a special type called Calabi-Yau varieties. For n=1 this leads to the question of counting the ramified covers of a torus (= surface of genus one = one-dimensional Calabi-Yau manifold) by a surface of genus g. The answer turns out to be given by a (nearly) modular form, as discovered by Dijkgraaf and other string theorists and proved by Kaneko and myself. I will describe this theorem and then a huge generalization of it found by Bloch and Okounkov, together with a new and very simple proof of that theorem. Finally, I will describe recent joint work with Moeller and Chen that uses this connection with modular forms to solve various problems in the moduli space of surfaces, including a proof of the Eskin-Zorich conjecture on the large genus asymptotics of Masur-Veech volumes and the calculation of the so-called "Siegel-Veech constants" in the theory of flat surfaces.
Introduction of Speaker:
Don Bernard Zagier is an American mathematician whose main area of work is number theory. He is currently one of the directors of the Max Planck Institute for Mathematics in Bonn, Germany. He has been a professor at the Collège de France in Paris, France from 2006 to 2014. Since October 2014, he is also a Distinguished Staff Associate at ICTP.
Zagier studied for three years at MIT, completing his bachelor's and master's degrees and then receiving his PhD at Oxford University. He received his Habilitation at University of Bonn. In 1976, aged only 24, he became Germany's youngest professor.
Zagier collaborated with Hirzebruch in work on Hilbert modular surfaces. Hirzebruch and Zagier coauthored Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, where they proved that intersection numbers of algebraic cycles on a Hilbert modular surface occur as Fourier coefficients of a modular form.
One of Zagier’s results is a joint work with Benedict Gross (the so-called Gross–Zagier formula). This formula relates the first derivative of the complex L-series of an elliptic curve evaluated at 1 to the height of a certain Heegner point. This theorem has some applications including implying cases of the Birch and Swinnerton-Dyer conjecture along with being an ingredient to Dorian Goldfeld's solution of the class number problem. As a part of their work, Gross and Zagier found a formula for norms of differences of singular moduli. Zagier later found a formula for traces of singular moduli as Fourier coefficients of a weight 3/2 modular form.
Zagier collaborated with John Harer to calculate the orbifold Euler characteristics of moduli spaces of algebraic curves, relating them to special values of the Riemann zeta function.
Zagier found a formula for the value of the Dedekind zeta function of an arbitrary number field at s = 2 in terms of the dilogarithm function, by studying arithmetic hyperbolic 3-manifolds. He later formulated a general conjecture giving formulas for special values of Dedekind zeta functions in terms of polylogarithm functions. He discovered a short and elementary proof of Fermat's theorem on sums of two squares.
Zagier won the Cole Prize in Number Theory in 1987, the von Staudt Prize in 2001 and the Gauss Lectureship of the German Mathematical Society in 2007. He became a foreign member of the Royal Netherlands Academy of Arts and Sciences in 1997.
Source: https://en.wikipedia.org/wiki/Don_Zagier
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