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发帖数: 905 | 1 The Shaw Prize in Mathematical Sciences 2015 is awarded to Gerd Faltings,
Managing Director at Max Planck Institute for Mathematics in Bonn, Germany,
and Henryk Iwaniec, New Jersey Professor of Mathematics at Rutgers
University, USA, for their introduction and development of fundamental tools
in number theory, allowing them as well as others to resolve some
longstanding classical problems.
Number theory concerns whole numbers, prime numbers, and polynomial
equations involving them. The central problems are often easy to state but
extraordinarily difficult to resolve. Success, when it is achieved, relies
on tools from many fields of mathematics. This is no coincidence since some
of these fields were introduced in attempts to resolve classical problems in
number theory. Faltings and Iwaniec have developed many of the most
powerful modern tools in algebra, analysis, algebraic and arithmetic
geometry, automorphic forms, and the theory of zeta functions. They and
others have used these tools to resolve longstanding problems in number
theory.
Gerd Faltings
A polynomial equation of degree n in one variable with coefficients which
are rational numbers has just n complex numbers as solutions. Such an
equation has a symmetry group, its Galois group, that describes how these
complex solutions are related to each other.
A polynomial equation in two variables with rational coefficients has
infinitely many complex solutions, forming an algebraic curve. In most cases
(that is, when the curve has genus 2 or more) only finitely many of these
solutions are pairs of rational numbers. This well-known conjecture of
Mordell had defied resolution for sixty years before Faltings proved it. His
unexpected proof provided fundamental new tools in Arakelov and arithmetic
geometry, as well as a proof of another fundamental finiteness theorem —
the Shaferavich and Tate Conjecture — concerning polynomial equations in
many variables. Later, developing a quite different method of Vojta,
Faltings established a far-reaching higher dimensional finiteness theorem
for rational solutions to systems of equations on Abelian Varieties (the
Lang Conjectures). In order to study rational solutions of polynomial
equations by geometry, one needs arithmetic versions of the tools of complex
geometry. One such tool is Hodge theory. Faltings’ foundational
contributions to Hodge theory over the p-adic numbers, as well as his
introduction of other related novel and powerful techniques, are at the core
of some of the recent advances connecting Galois groups (from polynomial
equations in one or more variables) and the modern theory of automorphic
forms (a vast generalization of the theory of periodic functions). The
recent striking work of Peter Scholze concerning Galois representations is a
good example of the power of these techniques.
Henryk Iwaniec
Iwaniec’s work concerns the analytic side of diophantine analysis, where
the goal is usually to prove that equations do have integral or prime
solutions, and ideally to estimate how many there are up to a given size.
One of the oldest techniques for finding primes is sieve theory, originating
in Erastosthenes’ description of how to list the prime numbers. Iwaniec’s
foundational works and breakthroughs in sieve theory and its applications
form a large part of this active area of mathematics. His proof (with John
Friedlander) that there are infinitely many primes of the form X2+Y4 is one
of the most striking results about prime numbers known; the techniques
introduced to prove it are the basis of many further works. The theory of
Riemann’s zeta function — and more generally of L-functions associated
with automorphic forms — plays a central role in the study of prime numbers
and diophantine equations. Iwaniec invented many of the powerful techniques
for studying L-functions of automorphic forms, which are used widely today.
Specifically, his techniques to estimate the Fourier coefficients of
modular forms of half-integral weight and for estimating L-functions on
their critical lines (the latter jointly with William Duke and John
Friedlander) have led to the solution of a number of longstanding problems
in number theory, including one of Hilbert’s problems: that quadratic
equations in integers (in three or more variables) can always be solved
unless there is an “obvious” reason that they cannot.
In a series of papers remarkable both in terms of its concept and novel
techniques, Iwaniec together with different authors (Ètienne Fouvry and
then Enrico Bombieri and John Friedlander) established results about the
distribution of primes in arithmetic progressions which go beyond the
notorious Riemann Hypothesis. This opened the door to some potentially very
striking applications. Yitang Zhang’s much celebrated recent result on
bounded gaps between primes relies heavily on the works of Iwaniec et al.
Iwaniec’s work mentioned above, together with his many other technically
brilliant works, have a central position in modern analytic number theory.
http://www.shawprize.org/en/shaw.php?tmp=3&twoid=98&threeid=245 | M****o
发帖数: 4860 | 2 老张呢？
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【在 x********i 的大作中提到】
: The Shaw Prize in Mathematical Sciences 2015 is awarded to Gerd Faltings,
: Managing Director at Max Planck Institute for Mathematics in Bonn, Germany,
: and Henryk Iwaniec, New Jersey Professor of Mathematics at Rutgers
: University, USA, for their introduction and development of fundamental tools
: in number theory, allowing them as well as others to resolve some
: longstanding classical problems.
: Number theory concerns whole numbers, prime numbers, and polynomial
: equations involving them. The central problems are often easy to state but
: extraordinarily difficult to resolve. Success, when it is achieved, relies
: on tools from many fields of mathematics. This is no coincidence since some
| x********i
发帖数: 905 | 3 按照最后一段，呃。。。貌似老张不够？？
In a series of papers remarkable both in terms of its concept and novel
techniques, Iwaniec together with different authors (Ètienne Fouvry and
then Enrico Bombieri and John Friedlander) established results about the
distribution of primes in arithmetic progressions which go beyond the
notorious Riemann Hypothesis. This opened the door to some potentially very
striking applications. Yitang Zhang’s much celebrated recent result on
bounded gaps between primes relies heavily on the works of Iwaniec et al.
Iwaniec’s work mentioned above, together with his many other technically
brilliant works, have a central position in modern analytic number theory.
【在 M****o 的大作中提到】
: 老张呢？
:
: ,
: tools
: some
: in
: cases
: His
: arithmetic
: complex
| s*****V
发帖数: 21731 | 4 Shaw prize 只发给这种WELL-ESTABLISHED是没前途的，人家根本不CARE这个 |
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