Contemporary Progress in Mathematics

Contemporary Progress in Mathematics

Soundararajan很可能就是下届的Fields奖。另两个大热人选是Hacon和Ngo.

In a seminar co-organized by Stanford University and the American Institute
of Mathematics, Soundararajan announced that he and Roman Holowinsky have pr
oven a significant version of the quantum unique ergodicity (QUE) conjecture
.

"This is one of the best theorems of the year," said Peter Sarnak, a mathema
tician from Princeton who along with Zeev Rudnick from the University of Tel
Aviv formulated the conjecture fifteen years ago in an effort to understand
the connections between classical and quantum physics.

"I was aware that Soundararajan and Holowinsky were both attacking QUE using
different techniques and was astounded to find that their methods miraculou
sly combined to completely solve the problem," said Sarnak. Both approaches
come from number theory, an area of pure mathematics which recently has been
found to have surprising connections to physics.

The motivation behind the problem is to understand how waves are influenced
by the geometry of their enclosure. Imagine sound waves in a concert hall. I
n a well-designed concert hall you can hear every note from every seat. The
sound waves spread out uniformly and evenly. At the opposite extreme are "wh
ispering galleries" where sound concentrates in a small area.

The mathematical world is populated by all kinds of shapes, some of which ar
e easy to picture, like spheres and donuts, and others which are constructed
from abstract mathematics. All of these shapes have waves associated with t
hem. Soundararajan and Holowinsky showed that for certain shapes that come f
rom number theory, the waves always spread out evenly. For these shapes ther
e are no "whispering galleries."

Quantum chaos

The quantum unique ergodicity conjecture (QUE) comes from the area of physic
s known as "quantum chaos." The goal of quantum chaos is to understand the r
elationship between classical physics--the rules that govern the motion of m
acroscopic objects like people and planets when their motion is chaotic, wit
h quantum physics--the rules that govern the microscopic world.

"The work of Holowinsky and Soundararajan is brilliant," said physicist Jens
Marklof of Bristol University, "and tells us about the behaviour of a parti
cle trapped on the modular surface in a strong magnetic field."

The problems of quantum chaos can be understood in terms of billiards. On
a standard rectangular billiard table the motion of the balls is predictable
and easy to describe. Things get more interesting if the table has curved e
dges, known as a "stadium." Then it turns out most paths are chaotic and ove
r time fill out the billiard table, a result proven by the mathematical phys
icist Leonid Bunimovich.

In their QUE conjecture, Rudnick and Sarnak hypothesized that for a large cl
ass of systems, unlike the stadium there are no scars or bouncing ball state
s and in fact all states become evenly distributed. Holowinsky and Soundarar
ajan's work shows that the conjecture is true in the number theoretic settin
g.

Highly excited states

The conjecture of Rudnick and Sarnak deals with certain kinds of shapes call
ed manifolds, or more technically, manifolds of negative curvature, some of
which come from problems in higher arithmetic. The corresponding waves are a
nalogous to highly excited states in quantum mechanics.

Soundararajan and Holowinsky each developed new techniques to solve a partic
ular case of QUE. The "waves" in this setting are known as holomorphic Hecke
eigenforms. The approaches of both researchers work individually most of th
e time and miraculously when combined they completely solve the problem. "Th
eir work is a lovely blend of the ideas of physics and abstract mathematics,
" said Brian Conrey, Director of the American Institute of Mathematics.

According to Lev Kaplan, a physicist at Tulane University, "This is a good e
xample of mathematical work inspired by an interesting physical problem, and
it has relevance to our understanding of quantum behavior in classically ch
aotic dynamical systems."