这是闭幕式前的最后一个 invited talk. 张大师习惯手写, 当场演算.

Yitang Zhang stepped onto the main stage of mathematics last year with the announced of his achievement that is hailed as “a landmark theorem in the distribution of prime numbers”.

Aug 212014

这是闭幕式前的最后一个 invited talk. 张大师习惯手写, 当场演算.

Yitang Zhang stepped onto the main stage of mathematics last year with the announced of his achievement that is hailed as “a landmark theorem in the distribution of prime numbers”.

Aug 152014

每 4 年举行一次世界数学家大会从 13 日到 21 日在首尔会展中心(COEX)举行.

Martin Groetschel, Secretary-General of IMU, 在开幕式上的讲话说, IMU 有一些倡议. 这些打算之一是 adopt-a-graduate-student: IMU 会扶持发达国家的数学家, 这些数学家愿意给不那么发达国家的工作在相近领域的数学博士提供指导(mentorship).

今年的 Chern Medal 的得主 Phillip Griffiths 选择了 African Mathematics Millennium Science Initiative(AMMSI) 来接受 $250,000.

Donaldson, Tao, Kontsevich, Lurie and Taylor, winners of the Breakthrough Prizes in mathematics, 每人给了 $100,000 给一个目的是支持发展中国家的博士的$500,000 基金会. 具体采用何种方式来实施帮助不得而知, 但已经使用了 “breakout graduate fellowships” 这样的措词.

Martin Groetschel 还指出, 韩国的数学出版物的数量当前是世界第 11 位, 但韩国数学家 1981 年发表在国际期刊上的论文仅仅只有 3 篇. 韩国从几乎一无所有, 建立了现在的数学传统, 仅仅过了一代人的时间.

1981 年成为国际数学联盟的最低等级第1军成员国的韩国时隔 33 年从援助受惠国成为供应国, 这将成为向全世界宣传韩国数学的契机.

韩国总统朴槿惠出席了当天的开幕式, 她强调了数学给我们的生活带来的影响, 向帮助韩国数学升至世界水平的世界数学界表达了谢意.

ICM 开幕式的一个小插曲是, 戴着面具的舞蹈演员走上舞台时, Maryam Mirzakhani 的不到三岁的女儿 Anahita 发出恐怖的尖叫, 许久才平静下来. Timothy Gowers 有一个 6 岁的儿子.

The Fields Medal Committee for 2014 consisted of Daubechies, Ambrosio, Eisenbud, Fukaya, Ghys, Dick Gross, Kirwan, Kollar, Kontsevich, Struwe, Zeitouni and Günter Ziegler.

The program committee consisted of Carlos Kenig (chair), Bolthausen, Alice Chang, de Melo, Esnault, me, Kannan, Jong Hae Keum, Le Bris, Lubotsky, Nesetril and Okounkov.

Kyoto University professor Shigefumi Mori has been elected president of the International Mathematical Union(IMU), becoming the first head of the group from Asia.

The ICM executive committee for the next four years will be Shigefumi Mori (president), Helge Holden (secretary), Alicia Dickenstein (VP), Vaughan Jones (VP), Dick Gross, Hyungju Park, Christiane Rousseau, Vasudevan Srinivas, John Toland and Wendelin Werner.

好像, 国际数学联盟已经讨论过, 在大会开幕很久之前公布大奖得主的名字, 是否可行.

下次全世界数学家的聚会, 2018 年的八月在巴西. The General Assembly of the IMU in Gyeongju announced on Aug. 11 that Rio de Janeiro would be the site of ICM 2018.

- Timothy Gowers, ICM2014 — opening ceremony, August 13, 2014
- ICM 2014

Aug 132014

Artur Avila

Manjul Bhargava

Martin Hairer

Maryam Mirzakhani

At the opening ceremony of the International Congress of Mathematicians 2014 on August 13, 2014, the Fields Medals (started in 1936), the Nevanlinna Prize (started in 1982), the Gauss Prize (started in 2006), and the Chern Medal Award (started in 2010) were awarded. In addition, the winner of the Leelavati Prize (started in 2010) and the speaker of the ICM Emmy Noether Lecture (started in 1994) were announced.

Subhash Khot

Stanley Osher

Phillip Griffiths

Adrián Paenza

The 2014 ICM Emmy Noether lecturer is Georgia Benkart.

Stanford University, USA

[Maryam Mirzakhani is awarded the Fields Medal]

for her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces.

- Maryam Mirzakhani has made stunning advances in the theory of Riemann surfaces and their moduli spaces, and led the way to new frontiers in this area. Her insights have integrated methods from diverse fields, such as algebraic geometry, topology and probability theory.
- In hyperbolic geometry, Mirzakhani established asymptotic formulas and statistics for the number of simple closed geodesics on a Riemann surface of genus g. She next used these results to give a new and completely unexpected proof of Witten’s conjecture, a formula for characteristic classes for the moduli spaces of Riemann surfaces with marked points.
- In dynamics, she found a remarkable new construction that bridges the holomorphic and symplectic aspects of moduli space, and used it to show that Thurston’s earthquake flow is ergodic and mixing.
- Most recently, in the complex realm, Mirzakhani and her coworkers produced the long sought-after proof of the conjecture that – while the closure of a real geodesic in moduli space can be a fractal cobweb, defying classification – the closure of a complex geodesic is always an algebraic subvariety.
- Her work has revealed that the rigidity theory of homogeneous spaces (developed by Margulis, Ratner and others) has a definite resonance in the highly inhomogeneous, but equally fundamental realm of moduli spaces, where many developments are still unfolding

CNRS, France & IMPA, Brazil

[Artur Avila is awarded a Fields Medal] for his profound contributions to dynamical systems theory have changed the face of the field, using the powerful idea of renormalization as a unifying principle.

- Avila leads and shapes the field of dynamical systems. With his collaborators, he has made essential progress in many areas, including real and complex one-dimensional dynamics, spectral theory of the one-frequency Schródinger operator, flat billiards and partially hyperbolic dynamics.
- Avila’s work on real one-dimensional dynamics brought completion to the subject, with full understanding of the probabilistic point of view, accompanied by a complete renormalization theory. His work in complex dynamics led to a thorough understanding of the fractal geometry of Feigenbaum Julia sets.
- In the spectral theory of one-frequency difference Schródinger operators, Avila came up with a global de- scription of the phase transitions between discrete and absolutely continuous spectra, establishing surprising stratified analyticity of the Lyapunov exponent.
- In the theory of flat billiards, Avila proved several long-standing conjectures on the ergodic behavior of interval-exchange maps. He made deep advances in our understanding of the stable ergodicity of typical partially hyperbolic systems.
- Avila’s collaborative approach is an inspiration for a new generation of mathematicians.

Princeton University, USA

[Manjul Bhargava is awarded a Fields Medal]

for developing powerful new methods in the geometry of numbers and applied them to count rings of small rank and to bound the average rank of elliptic curves.

- Bhargava’s thesis provided a reformulation of Gauss’s law for the composition of two binary quadratic forms. He showed that the orbits of the group \(SL(2, \Bbb Z)3\) on the tensor product of three copies of the standard integral representation correspond to quadratic rings (rings of rank \(2\) over \(\Bbb Z\)) together with three ideal classes whose product is trivial. This recovers Gauss’s composition law in an original and computationally effective manner. He then studied orbits in more complicated integral representations, which correspond to cubic, quartic, and quintic rings, and counted the number of such rings with bounded discriminant.
- Bhargava next turned to the study of representations with a polynomial ring of invariants. The simplest such representation is given by the action of \(PGL(2, \Bbb Z)\) on the space of binary quartic forms. This has two independent invariants, which are related to the moduli of elliptic curves. Together with his student Arul Shankar, Bhargava used delicate estimates on the number of integral orbits of bounded height to bound the average rank of elliptic curves. Generalizing these methods to curves of higher genus, he recently showed that most hyperelliptic curves of genus at least two have no rational points.
- Bhargava’s work is based both on a deep understanding of the representations of arithmetic groups and a unique blend of algebraic and analytic expertise.

University of Warwick, UK

[Martin Hairer is awarded a Fields Medal]

for his outstanding contributions to the theory of stochastic partial differential equations, and in particular created a theory of regularity structures for such equations.

- A mathematical problem that is important throughout science is to understand the influence of noise on differential equations, and on the long time behavior of the solutions. This problem was solved for ordinary differential equations by Itó in the 1940s. For partial differential equations, a comprehensive theory has proved to be more elusive, and only particular cases (linear equations, tame nonlinearities, etc.) had been treated satisfactorily.
- Hairer’s work addresses two central aspects of the theory. Together with Mattingly he employed the Malliavin calculus along with new methods to establish the ergodicity of the two-dimensional stochastic Navier-Stokes equation.
- Building on the rough-path approach of Lyons for stochastic ordinary differential equations, Hairer then created an abstract theory of regularity structures for stochastic partial differential equations (SPDEs). This allows Taylor-like expansions around any point in space and time. The new theory allowed him to construct systematically solutions to singular non-linear SPDEs as fixed points of a renormalization procedure.
- Hairer was thus able to give, for the first time, a rigorous intrinsic meaning to many SPDEs arising in physics.

New York University, USA

[Subhash Khot is awarded the Nevanlinna Prize]

for his prescient definition of the “Unique Games” problem, and his leadership in the effort to understand its complexity and its pivotal role in the study of efficient approximation of optimization problems, have produced breakthroughs in algorithmic design and approximation hardness, and new exciting interactions between computational complexity, analysis and geometry.

- Subhash Khot defined the “Unique Games” in 2002 , and subsequently led the effort to understand its complexity and its pivotal role in the study of optimization problems. Khot and his collaborators demonstrated that the hardness of Unique Games implies a precise characterization of the best approximation factors achievable for a variety of NP-hard optimization problems. This discovery turned the Unique Games problem into a major open problem of the theory of computation.
- The ongoing quest to study its complexity has had unexpected benefits. First, the reductions used in the above results identified new problems in analysis and geometry, invigorating analysis of Boolean functions, a field at the interface of mathematics and computer science. This led to new central limit theorems, invariance principles, isoperimetric inequalities, and inverse theorems, impacting research in computational complexity, pseudorandomness, learning and combinatorics. Second, Khot and his collaborators used intuitions stemming from their study of Unique Games to yield new lower bounds on the distortion incurred when embedding one metric space into another, as well as constructions of hard families of instances for common linear and semi- definite programming algorithms. This has inspired new work in algorithm design extending these methods, greatly enriching the theory of algorithms and its applications.

Institute for Advanced Study, USA

[Phillip Griths is awarded the 2014 Chern Medal]

for his groundbreaking and transformative development of transcendental methods in complex geometry, particularly his seminal work in Hodge theory and periods of algebraic varieties.

- Phillip Griffiths’s ongoing work in algebraic geometry, differential geometry, and differential equations has stimulated a wide range of advances in mathematics over the past 50 years and continues to influence and inspire an enormous body of research activity today.
- He has brought to bear both classical techniques and strikingly original ideas on a variety of problems in real and complex geometry and laid out a program of applications to period mappings and domains, algebraic cycles, Nevanlinna theory, Brill-Noether theory, and topology of K¨ahler manifolds.
- A characteristic of Griffithss work is that, while it often has a specific problem in view, it has served in multiple instances to open up an entire area to research.
- Early on, he made connections between deformation theory and Hodge theory through infinitesimal methods, which led to his discovery of what are now known as the Griffiths infinitesimal period relations. These methods provided the motivation for the Griffiths intermediate Jacobian, which solved the problem of showing algebraic equivalence and homological equivalence of algebraic cycles are distinct. His work with C.H. Clemens on the non-rationality of the cubic threefold became a model for many further applications of transcendental methods to the study of algebraic varieties.
- His wide-ranging investigations brought many new techniques to bear on these problems and led to insights and progress in many other areas of geometry that, at first glance, seem far removed from complex geometry. His related investigations into overdetermined systems of differential equations led a revitalization of this subject in the 1980s in the form of exterior differential systems, and he applied this to deep problems in modern differential geometry: Rigidity of isometric embeddings in the overdetermined case and local existence of smooth solutions in the determined case in dimension \(3\), drawing on deep results in hyperbolic PDEs(in collaborations with Berger, Bryant and Yang), as well as geometric formulations of integrability in the calculus of variations and in the geometry of Lax pairs and treatises on the geometry of conservation laws and variational problems in elliptic, hyperbolic and parabolic PDEs and exterior differential systems.
- All of these areas, and many others in algebraic geometry, including web geometry, integrable systems, and
- Riemann surfaces, are currently seeing important developments that were stimulated by his work.
- His teaching career and research leadership has inspired an astounding number of mathematicians who have gone on to stellar careers, both in mathematics and other disciplines. He has been generous with his time, writing many classic expository papers and books, such as “Principles of Algebraic Geometry”, with Joseph Harris, that have inspired students of the subject since the 1960s.
- Griffiths has also extensively supported mathematics at the level of research and education through service on and chairmanship of numerous national and international committees and boards committees and boards. In addition to his research career, he served 8 years as Duke’s Provost and 12 years as the Director of the Institute for Advanced Study, and he currently chairs the Science Initiative Group, which assists the development of mathematical training centers in the developing world.
- His legacy of research and service to both the mathematics community and the wider scientific world continues to be an inspiration to mathematicians world-wide, enriching our subject and advancing the discipline in manifold ways.

Aug 122014

明天上午(韩国时间, 东九区时间)九点, ICM 2014 会准时在韩国首尔(Coex , Seoul , Korea)开幕. 依惯例, 开幕式上会为引人注目的 Fields Medal 获得者颁发奖章. 坊间流传的一个(如若发生)会载入历史的传奇是:

本届 ICM 会有一个女性数学家获得 Fields Medal!

这真是一个令人振奋的消息! 全世界的数学工作者屏住呼吸兴奋的等待着亲眼目睹见证这个激动人心的时刻!

Maryam Mirzakhani(born May 1977) is an Iranian mathematician, Professor of Mathematics (since September 1, 2008) at Stanford University.

Maryam Mirzakhani 是今年非常有力的竞争者, 和任何候选人站在一起都是那样出众引人注目. 她还是 IMO 满分.

1995 年, 今年中国领队姚一隽去加拿大参加 IMO 的时候, 这一年也是张筑生第一次做领队, 一共有 14 个满分, 其中有两个女生: 咱们中国的朱晨畅, 现在在德国 Gottingen University; 来自伊朗的 Maryam Mirzakhani, 8 月 16 日上午作 ICM 一小时报告.

很多人看好法国的 Sophie Morel. Sophie Morel 专长数论. 不过, 恐怕 Sophie Morel 今年拿不到奖章, 但她四年后还有一次机会.

此外, Laure Saint-Raymond 和 Marianna Csörnyei 也是相当给力的人选.

一个不好的消息是: 国际数学家大会召开在即, 韩劝阻埃博拉(Ebola virus)疫区数学家不要与会.

Aug 052014

ICM 2014 Program

这届国际数学界大会(International Congress of Mathematicians, ICM)的安排, 已经明确无误的说明:

4 个数学家将获得本届大会的 Fields Medal.

张大师将于 8 月 21 日作 ICM 闭幕式之前的压轴报告, 这是只有今年的 Fields Medalist, Gauss Prize, Chern Medal 得主才有的殊荣

张益唐 7 月1 日在北大本科生毕业典礼有一个讲话

这个暑假, 张大师在中国科学院晨兴数学中心和他的母校北京大学做了好几次讲座.

1. A Transition Formula for Mean Values of Dirichlet Polynomials

2014,6.23./6.25. 9:30-11:30

晨兴 110

主持人: 王元

2. 关于 Siegel 零点

2014.7.2.9:30-11:30

晨兴 110

3. Distribution of Prime Numbers and the Riemann Zeta Function

July 8, 10, 2014 16:00-17:00, 镜春园82号甲乙丙楼的中心报告厅

July 15, 16:30-17:30 镜春园 78 号院的 77201 室.

主持人: 刘若川

4. 关于 Siegel 零点(2)

014.7.16./7.30./8.4./8.6. 9:30-11:30

N820

Jul 242014

There exist elliptic curve groups \(E(\Bbb Q)\) of arbitrarily large rank.

用 \(r\) 表示 \(\Bbb Q\) 上的椭圆曲线 \(E\) 的秩—the rank of the Mordell–Weil group \(E(\Bbb Q)\).

一个悬而未决的著名难题是: \(r\) 是否可以任意大?

Martin-McMillen 2000 年有一个 \(r\geq24\) 的例子:

\begin{equation*}\begin{split}y^2+xy+y&=x^3-120039822036992245303534619191166796374x\\&+ 504224992484910670010801799168082726759443756222911415116\end{split}\end{equation*}

Hasse-Weil \(L\)-function \(L(s, E)\) 在 \(s=1\) 处的零点的阶数 \(r_a\) 称为 \(E\) 的 analytic rank(解析秩).

Manjul Bhargava, Christopher Skinner, Wei Zhang(张伟) 7 月 7 日在 arXiv 上传的论文 “A majority of elliptic curves over \(Q\) satisfy the Birch and Swinnerton-Dyer conjecture“, 宣布了取得的进展:

- \(\Bbb Q\) 上的椭圆曲线, when ordered by height(同构类以高排序), 至少有 \(66.48\%\) 满足 BSD conjecture;
- \(\Bbb Q\) 上的椭圆曲线, when ordered by height, 至少有 \(66.48\%\) 有有限 Tate–Shafarevich group;
- \(\Bbb Q\) 上的椭圆曲线, when ordered by height, 至少有 \(16.50\%\) 满足 \(r=r_a=0\), 至少有 \(20.68\%\) 满足 \(r=r_a=1\).

谁将在 8 月 13 日的 ICM 2014 开幕式上获得 Fields medal？坊间向来不缺传闻. 数论大牛 Manjul Bhargava 无疑是最耀眼的明星.