The Shaw Prize in Mathematical Sciences 2013 is awarded to David L Donoho

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May 292013
 

The Shaw Prize in Mathematical Sciences 2013 is awarded to David L Donoho for his profound contributions to modern mathematical statistics and in particular the development of optimal algorithms for statistical estimation in the presence of noise and of efficient techniques for sparse representation and recovery in large data-sets.

The Shaw Prize in Mathematical Sciences for 2013 is awarded to David L Donoho, Anne T and Robert M Bass Professor of the Humanities and Sciences, and Professor of Statistics at Stanford University, USA for his profound contributions to modern mathematical statistics and in particular the development of optimal algorithms for statistical estimation in the presence of noise and of efficient techniques for sparse representation and recovery in large data-sets.

The dramatic developments in technology in the last half century present fundamental new challenges in theoretical and applied mathematical statistics. David Donoho has played a major role in developing new mathematical and statistical tools to deal with such problems ranging from large data-sets in high dimensions to contamination with noise. His work provides fast, efficient and often optimal algorithms which are founded on rigorous mathematical analysis.

Key themes introduced in his works, and which today are standard features of the theory, include the exploitation of sparseness of representation of complex objects, related adaptive nonlinear thresholding techniques and the deep relation between sparseness and certain penalty functions that are being minimized (specifically \(L^1\) norms).

Many of these emerge from his development of algorithms for statistical estimators in the presence of noise. These are remarkable in that they overcome the difficulties associated with noise, with very little loss of efficiency or reliability. Along the way, he demonstrated the power of the mathematical theory of wavelets in dealing with such problems in statistics. The Donoho–Johnstone soft-thresholding algorithm has been widely used in statistical and signal processing applications.

During the last 15 years Donoho has developed a theory of sparse and multi-scale representations of signals and data-sets using nonlinear \(L^1\) optimization methods. These combine very well with techniques of unstructured and redundant dictionaries of functions and provide a fundamental approach to lower the dimensionality of complex problems. Along with Candes and Tao, he made fundamental contributions to the development of “compressed sensing”. In terms of sparseness and recovery, this method which “compresses while sensing the data”, using dramatically fewer data points while retaining the ability to recover the correct signal, yields strikingly efficient and even optimal algorithms for compressing and decompressing complex signals (e.g. images). This area remains a very active area of research especially in view of its wide applications.

David L Donoho was born in 1957 in Los Angeles, USA and is currently Anne T and Robert M Bass Professor of the Humanities and Sciences, and Professor of Statistics at Stanford University, USA. He graduated from Princeton University in 1978 and received his PhD from Harvard University in 1983. From 1984 to 1990, he was on the faculty of the University of California, Berkeley before moving to Stanford. He is a fellow of the American Academy of Arts and Sciences, a SIAM Fellow, a foreign associate of the French Academy of Sciences, and a member of the US National Academy of Sciences.

Debate about presenting the winner of Abel Prize

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Mar 262013
 

今年的 Abel Prize 授予 Pierre Deligne.  Timothy Gowers, 1998 年的 Field Medal 得主, 写了一篇文章来介绍 Pierre Deligne 的工作. 这已经是 Timothy Gowers 连续第三次做这件事情了: 他 2012年介绍了 Endre Szemerédi 的工作, 2011年为 John Milnor 写了长文.

有一个数学家用 sowa 这个 ID 在 T. Gowers 的博客, 宣布今年的 Abel Prize 得主是 Pierre Deligne 的文章, 质疑 Abel Prize:

  1. 应该解释为什么选择 T. Gowers 连续三年来写得奖人工作介绍. T. Gowers 的数学知识是否可以承担这项光荣的职责?
  2. (有人向 sowa 透漏)Norwegian Academy 希望把奖授予那些会接受奖项, 会亲临现场并与国王交谈的候选人. 于是有些数学家就被排除了, 比如 H. Cartan, A. Grothendick. H. Cartan 因为年事太高, 不能亲临领奖(当然, H. Cartan 现在已经过世); A. Grothendick 肯定也不会亲临, 甚至会拒绝这个奖.
  3. 去年应该把奖颁给 W. Thurston. W. Thurston 是历史上最好的数学家之一. 去年, 他的病已经很严重了. 把 Abel Prize 授予他, “to do something good for a dying person”.
  4. 没有授奖给 H. Cartan, I.M. Gelfand 和  W. Thurston 这样伟大的数学家是不可原谅的.
  5. 2012 年的得主 Endre Szemerédi 是否能与 Milnor和 Deligne 相提并论?
  6. 组合数学的重要性是否与代数几何一样?

这个论战正在进行.  参与讨论的人包括 T. Gowers 和 Terry Tao.

George Mostow and Michael Artin win the 2013 Wolf Prize in Mathematics

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Jan 092013
 

Prof. George Mostow, Yale, USA:For his fundamental and pioneering contribution to geometry and Lie group theory.

Prof. Michael Artin, M.I.T, USA: For his fundamental contributions to algebraic geometry, both commutative and non-commutative.

George D. Mostow made a fundamental and pioneering contribution to geometry and Lie group theory. His most celebrated accomplishment in this fields is the discovery of the completely new rigidity phenomenon in geometry, the Strong Rigidity Theorems. These theorems are some of the greatest achievements in mathematics in the second half of the 20th century. This established a deep connection between continuous and discrete groups, or equivalently, a remarkable connection between topology and geometry. Mostow’s rigidity methods and techniques opened a floodgate of investigations and results in many related areas of mathematics. Mostow’s emphasis on the “action at infinity” has been developed by many mathematicians in a variety of directions. It had a huge impact in geometric group theory, in the study of Kleinian groups and of low dimensional topology , in work connecting ergodic theory and Lie groups. Mostow’s contribution to mathematics is not limited to strong rigidity theorems. His work on Lie groups and their discrete subgroups which was done during 1948-1965 was very influential. Mostow’s work on examples of nonarithmetic lattices in two and three dimensional complex hyperbolic spaces (partially in collaboration with P. Delinge) is brilliant and lead to many important developments in mathematics. In Mostow’s work one finds a stunning display of a variety of mathematical disciplines. Few mathematicians can compete with the breadth, depth, and originality of his works.

Michael Artin is one of the main architects of modern algebraic geometry. His fundamental contributions encompass a bewildering number of areas in this field.

To begin with, the theory of étale cohomology was introduced by Michael Artin jointly with Alexander Grothendieck. Their vision resulted in the creation of one of the essential tools of modern algebraic geometry. Using étale cohomology Artin showed that the finiteness of the Brauer group of a surface fibered by curves is equivalent to the Birch and Swinerton-Dyer conjecture for the Jacobian of a general fiber. In a very original paper Artin and Swinerton-Dyer proved the conjecture for an elliptic \(K_3\) surface.

He also collaborated with Barry Mazur to define étale homotopy- another important tool in algebraic geometry- and more generally to apply ideas from algebraic geometry to the study of diffeomorphisms of compact manifold.

We owe to Michael Artin, in large part, also the introduction of algebraic spaces and algebraic stacks. These objects form the correct category in which to perform most algebro-geometrical constructions, and this category is ubiquitous in the theory of moduli and in modern intersection theory. Artin discovered a simple set of conditions for a functor to be represented by an algebraic space. His ”Approximation Theorem” and his ”Existence Theorem” are the starting points of the modern study of moduli problems Artin’s contributions to the theory of surface singularities are of fundamental importance. In this theory he introduced several concepts that immediately became seminal to the field, such as the concepts of rational singularity and of fundamental cycle.

In yet another example of the sheer originality of his thinking, Artin broadened his reach to lay rigorous foundations to deformation theory. This is one of the main tools of classical algebraic geometry, which is the basis of the local theory of moduli of algebraic varieties.

Finally, his contribution to non-commutative algebra has been enormous. The entire subject changed after Artin’s introduction of algebro-geometrical methods in this field. His characterization of Azumaya algebras in terms of polynomial identities, which is the content of the Artin-Procesi theorem, is one of the cornerstones in non-commutative algebra. The Artin-Stafford theorem stating that every integral projective curve is commutative is one of the most important achievements in non-commutative algebraic geometry.

Artin’s mathematical accomplishments are astonishing for their depth and their scope . He is one of the great geometers of the 20th century.

Ian Agol and Daniel Wise receive 2013 AMS Veblen Prize

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Dec 262012
 

Providence, RI—Ian Agol of the University of California, Berkeley, is receiving the 2013 AMS Oswald Veblen Prize in Geometry. The Veblen Prize is given every three years for an outstanding publication in geometry or topology that has appeared in the preceding six years. The prize will be awarded on Thursday, January 10, 2013, at the Joint Mathematics Meetings in San Diego.

Agol is honored for “his many fundamental contributions to hyperbolic geometry, 3-manifold topology, and geometric group theory,” the prize citation says. The citation points in particular to the following papers:

  • I. Agol, P. Storm, and W. P. Thurston, “Lower bounds on volumes of hyperbolic Haken 3-manifolds” with an appendix by Nathan Dunfield, Journal of the AMS, 20 (2007), no. 4, 1053-1077;
  • I. Agol, “Criteria for virtual fibering,” Journal of Topology, 1 (2008), no. 2, 269-284; and
  • I. Agol, D. Groves, and J. F. Manning, “Residual finiteness, QCERF and fillings of hyperbolic groups,”Geometry and Topology, 13 (2009), no. 2, 1043-1073.

Providence, RI—Daniel Wise of McGill University is receiving the 2013 AMS Oswald Veblen Prize in Geometry. The Veblen Prize is given every three years for an outstanding publication in geometry or topology that has appeared in the preceding six years. The prize will be awarded on Thursday, January 10, 2013, at the Joint Mathematics Meetings in San Diego.

Wise is honored for “his deep work establishing subgroup separability (LERF) for a wide class of groups and for introducing and developing with Frederic Haglund the theory of special cube complexes which are of fundamental importance for the topology of three-dimensional manifolds,” the prize citation says. The citation mentions in particular the following papers:

  • D. T. Wise, “Subgroup separability of graphs of free groups with cyclic edge groups,” Quarterly Journal of Mathematics, 51 (2000), no. 1, 107-129;
  • D. T. Wise, “Residual finiteness of negatively curved polygons of finite groups,” Inventiones Mathematicae, 149 (2002), no. 3, 579-617;
  • F. Haglund and D. T. Wise, “Special cube complexes,” Geometric and Functional Analysis, 17 (2008), no. 5, 1551-1620; and
  • F. Haglund and D. T. Wise, “A combination theorem for special cube complexes,” Annals of Mathematics, 176 (2012), no. 3, 1427-1482.
 Posted by at 12:53 pm

Maxim Kontsevich wins the 2012 Shaw Prize in Mathematical Sciences

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Sep 212012
 

The Shaw Prize in Mathematical Sciences 2012 is awarded to Maxim Kontsevich for his pioneering works in algebra, geometry and mathematical physics and in particular deformation quantization, motivic integration and mirror symmetry.

2012 年的邵逸夫奖, 数学奖颁予法国高等科学研究所的教授马克西姆·康采维奇, 以表彰他在代数, 几何和数学物理上的开创性工作.

Maxim Kontsevich 也是 1998 年的菲尔兹奖(Fields Medal)得主.

颁奖典礼已经于 9 月 17 日进行.

邵逸夫奖的官网是 shawprize, 在这里可以找到获奖得主及其工作的简单介绍, 此外, 还有颁奖视频.

on the Prize in Mathematical Sciences 2012

Traditionally the interaction between mathematics and theoretical physics has been concerned with topics ranging from dynamical systems and partial differential equations to differential geometry to probability theory. For the last two decades, modern algebra and algebraic geometry (which is the study of the solutions of systems of polynomial equations in several variables via algebraic methods) have taken a central position in this interaction.  Physical insights and intuition, especially from string theory, have led to a number of unexpected and striking predictions in both classical and modern algebraic geometry.  Thanks to the efforts of many mathematicians new techniques and theories have been developed and some of these conjectures have been proven.

Maxim Kontsevich has led the way in a number of these developments.  Among his many achievements is his early work on Witten’s conjecture concerning the topology and geometry of the moduli (that is parameter) spaces of all algebraic curves of a given genus, his solution of the problem of deformation quantization, his work in mirror symmetry and in a different direction the theory of motivic integration.

Quantization is the process of passing from classical to quantum mechanics and it has been realized by different mathematical theories.  One of these is the algebraic theory of deformation quantization.  This takes place on a Poisson manifold (that is a manifold with a Poisson bracket on functions) for which there are two natural algebras, the classical observables which are the functions under point-wise multiplication and the Poisson algebra where the multiplication comes from the Poisson structure.  The problem is to give a formal deformation in powers of a parameter h, in which the zeroth order term is the classical algebra of observables and the next order term is the given Poisson algebra.  The construction of such a deformation was carried out in special cases (Weyl, Moyal, Fedosov…) but the general case proved formidable. It was resolved brilliantly by Kontsevich using ideas from quantum field theory.

The discovery by physicists of mirror pairs of Calabi–Yau manifolds has led to a rich and evolving mathematical theory of mirror symmetry.  The physics predicts that there is a relation between the symplectic geometry (that is a geometry coming from classical mechanics) on such a manifold and the algebraic/complex geometry of the mirror manifold.  When carried out in certain examples for which explicit computations can be made, this led to some remarkable predictions in classical enumerative geometry, concerning the counting of curves in higher dimensional spaces.  Some of these predictions have since been proven. Kontsevich introduced homological mirror symmetry which predicts that further refined objects associated with the symplectic geometry of the manifold are related to ones associated with the complex geometry of its mirror.  These conjectures and their generalizations have been proven in significant special cases.  From the beginning Kontsevich has played a leading role in the development of the mathematical theory of mirror symmetry.  He continues to revisit the original formulation and to provide clearer conceptual answers to the mathematical question:   “What is mirror symmetry?”

Motivic integration is another invention of Kontsevich.  It is an integration theory which applies in the setting of algebraic geometry.  Unlike the usual integral from calculus whose value is a number, the motivic integral has its values in a large ring which is built out of the collection of all varieties (the zero sets of polynomial equations).  It satisfies many properties similar to the usual integral and while appearing to be quite abstract, when computed and compared in different settings it yields some far reaching information about algebraic varieties as well as their singularities.  It has been used to resolve some basic questions about invariants of Calabi–Yau varieties and it is also central to many recent developments concerning the uniform structure of counting points on varieties over finite fields and rings.

Through his technical brilliance in resolving central problems, his conceptual insights and very original ideas, Kontsevich has played a substantial role in shaping modern algebra, algebraic geometry and mathematical physics and especially the connections between them.

Mathematical Sciences Selection Committee
The Shaw Prize

17 September 2012, Hong Kong

Autobiography of Maxim Kontsevich

I was born in 1964 in a suburb of Moscow, close to a big forest. My father is a well-known specialist in Korean language and history, my mоther was an engineer (she is retired now), and my elder brother is a specialist in computer vision.  The apartment where I grew up was very small and full of books – about half of them in Korean or Chinese.

I became interested in mathematics at age 10 – 11, mainly because of the influence of my brother. Several books at popular level were very inspiring. Also, my brother was subscribed to the famous monthly “Kvant” magazine containing many wonderful articles on mathematics and physics addressed to high-school kids, sometimes explaining even new results or unresolved problems.  Also, I used to take part in Olympiads at various levels and was very successful.

In the Soviet Union, some schools had special classes for gifted children, with an additional four hours per week devoted to extra-curricular education (usually in mathematics or physics) taught by university students who had passed through the same system themselves. At age 13 – 15 I was attending such a school in Moscow, and from 1980 till 1985 was studying mathematics at Moscow State University. Because of my previous training in High School, I never attended regular courses, but instead went to several graduate and research-level seminars where I learned a huge amount of material. My tutor was Israel Gelfand, one of the greatest mathematicians of the 20th Century. His weekly seminar, on Mondays, was completely unpredictable, and covered the whole spectrum of mathematics. Outstanding mathematicians, both Soviet and visitors from abroad, gave lectures. In a sense, I grew up in these seminars, and also had the great luck to witness the birth of conformal field theory and string theory in the mid-80s. The interaction with theoretical physics remains vitally important for me even now. After graduating from university, I became a researcher at the Institute for Information Transmission Problems. Simultaneously, I began to learn to play the cello and for several years enjoyed the good company of my musician friends with whom I played some obscure pieces of baroque and renaissance music.

In 1988, I went abroad for the first time, to Poland and France. Also in 1988, I wrote a short article concerning two different approaches to string theory, and maybe because of this result, was invited to visit the Max Planck Institute for Mathematics in Bonn for three months in 1990. At the end of my stay there was an annual informal meeting of mostly European mathematicians, called Arbeitstatgung, where the latest hot results were presented. The opening lecture by Michael Atiyah was about a new surprising conjecture of Witten concerning matrix models and the topology of moduli spaces of algebraic curves. In two days I came up with an idea of how to relate moduli spaces but with a completely new type of matrix model, and explained it to Atiyah. People at MPIM were very impressed and invited me to come back the following year. During the next 3 – 4 years I was visiting mostly Bonn, and also IAS in Princeton and Harvard. My then future wife Ekaterina, whom I met in Moscow, accompanied me, and in 1993 we were married. In Bonn I finished several works which became very well-known: one on Vassiliev invariants, and another on quantum cohomology (with Yu Manin, whose seminar I had attended back in Moscow).  Scientifically, a very important moment for me was Spring 1993 when I came to the idea of homological mirror symmetry, which was an opening of a grand new perspective. In 1994, I accepted an offer from Berkeley, but one year later I moved to IHES in France, where I continue to work. In 1999 my wife and I were granted French citizenship (keeping our Russian citizenship as well), and in 2001 our son was born.

For a few years I visited simultaneously Rutgers University, where my teacher Gelfand moved to after the perestroika, and IAS in Princeton. During the last six years I have regularly visited the University of Miami.

In my work I often change subjects, moving from Feynman graphs to abstract algebra, differential geometry, dynamical systems, finite fields. Still, mirror symmetry remains the major line. The interaction during the last two decades between mathematics and theoretical physics has been an amazing chain of breakthroughs. I am very happy to be a participant in this dialogue, not only absorbing mathematical ideas from string theory, but also giving something back, like a recent wall-crossing formula which I discovered with my long-term collaborator Yan Soibelman, and which became a very important tool in the hands of physicists, simultaneously answering questions concerning supersymmetric particles, and solving the classical problem about asymptotics for equations depending on small parameter.

17 September 2012

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