Artur Avila 是第一个不在美国和欧洲得博士的拿到 fields medal 的数学家. Artur Avila 21 岁在世界上著名的动力系统研究中心 Instituto Nacional de Matemática(IMPA ) 得到博士. 他接受的包括博士在内的教育, 全部在巴西完成.

## Artur Avila’s work

Etienne Ghys 的报告讲述了 Artur Avila 的工作

## IMPA

IMPA是国际一流数学研究所, 有不少世界一流的专家. 比如, 2014年韩国世界数学家大会, IMPA有四位教授受邀请做大会报告, 有两位教授参选2014 年菲尔兹奖, 他们是 Artur Avila 和 Fernando Codá Marques.

IMPA的博士加硕士学生总数 100 名左右. IMPA入学以后, 有一定淘汰率. 被退学的学生, 很多在里约市其他大学找到了机会继续学习.

IMPA 的学生主要来自以葡萄牙语和西班牙语为母语的国家, 授课百分之九十使用葡语, 但是考试可以使用英语作答. 偶尔会有教授使用英文授课, 但是这种情况很少, 而且往往不是常规课程. 过去的三五年期间, IMPA每个学期都会挑选3, 5门课程录影并提供免费下载. 这些录影多数也有被上传到 YouTube.

### References

1. Thomas Lin,  Erica Klarreich, A Brazilian Wunderkind Who Calms Chaos, August 12, 2014
2. Timothy Gowers, ICM2014 — Avila laudatio, August 14, 2014
3. 刘小川, IMPA简介, November 19, 2013

Maryam Mirzakhani, 37 岁, 刚刚在第二十七届国际数学家大会(ICM 2014)开幕式上, 从韩国女总统朴槿惠手中接过菲尔兹奖章. Mirzakhani 是首位荣获 fields medal 的女性.

Maryam Mirzakhani (R) was given the top mathematics award by South Korean president Park Geun-Hye (L)

Mirzakhani grew up in Iran and was at first more interested in reading and writing fiction than doing mathematics

Mirzakhani 在 IMO 表现杰出.

### References

1. Erica Klarreich, A Tenacious Explorer of Abstract Surfaces, August 12, 2014

Maryam Mirzakhani

## Fields Medals 2014

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.

Artur Avila

Manjul Bhargava

Martin Hairer

Phillip Griffiths Chern Medalist 2014

ICM 2014

Subhash Khot

Stanley Osher

## Chern Medal Award 2014

Phillip Griffiths

## ICM Emmy Noether Lecture 2014

The 2014 ICM Emmy Noether lecturer is  Georgia Benkart.

### Maryam Mirzakhani

Mirzakhani was given the top mathematics award by South Korean president Park Geun-Hye

The Work of Maryam Mirzakhani

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

### Artur Avila

Avila was given the top mathematics award by South Korean president Park Geun-Hye

The Work of Artur Avila

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.

### Manjul Bhargava

Bhargava was given the top mathematics award by South Korean president Park Geun-Hye

The Work of Manjul Bhargava

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.

### Martin Hairer

Hairer was given the top mathematics award by South Korean president Park Geun-Hye

The Work of Martin Hairer

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.

### Subhash Khot

Khot was given the Rolf Nevanlinna Prize by South Korean president Park Geun-Hye

Subhash Khot

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.

### Phillip Griffiths

Phillip Griffiths was given the Chern Medal by South Korean president Park Geun-Hye

[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.

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.

## Conjecture

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

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:

1. $$\Bbb Q$$ 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;
2. $$\Bbb Q$$ 上的椭圆曲线, when ordered by height, 至少有 $$66.48\%$$ 有有限 Tate–Shafarevich group;
3. $$\Bbb Q$$ 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 $$r=r_a=1$$.