Professor Manjul Bhargava will speak in Golden Jubilee Colloqium series in Mathematics
on March 10, 2008.
Speaker : Professor Manjul Bhargava
Date : Mar 10, 2008
Venue : Ramanujan Hall (Room 214)
Time : 2:30 pm
Title : Gauss Composition Laws and their applications
In 1801 Gauss laid down a remarkable law of composition on
integral binary quadratic forms. This discovery, known as Gauss
composition, not only had a profound influence on elementary number
theory but also laid the foundations for ideal theory and modern
algebraic number theory. Even today, Gauss composition remains one of
the best ways of understanding ideal class groups of quadratic fields.
The question arises as to whether there might exist similar laws of com- position on other spaces of forms that could shed light on the structure of other algebraic number rings and fields. In this talk we describe several such higher analogues of Gauss composition, and we discuss some of their recent applications.
| About the Speaker:
Prof. Manjul Bhargava received Ph. D. from Princeton under the
Andrew Wiles in 2001. He is now Professor of Mathematics at Princeton
University, one of the youngest ever to receive that rank in
history at the age of 28.
He has received numerous awards for his outstanding research in Number Theory. He received the Cole Prize in Number Theory from the American Mathematical Society in 2008. In 2005 he received the Sastra Ramanujan Prize, the Clay Research Award, Leonard M. and Eleanor B. Blumenthal Award for the Advancement of Research in Pure Mathematics. He was awarded the Medal of the College de France in 2004 and the Mathematical Association of America's Merten M. Hasse Prize for Exposition in 2003.
He was named as one of Popular Science Magazine's "Brilliant 10" in 2002. He also received the first Five-Year Long-Term Prize Fellow of the Clay Mathematics Institute in 2000 and AMS-MAA-SIAM Frank and Brennie Morgan Prize for Outstanding Undergraduate Research in Mathematics in 1997.
The Cole Prize citation noted Bhargava's "revolutionary work on higher composition laws", which introduced completely new and unexpected ideas into a subject that began with work of Brahmagupta in 628 C.E. and Carl Friedrich Gauss in 1801.
"[Bhargava's] techniques and insights ... are dazzling; even in the case considered by Gauss, they lead to a new and clearer presentation of that theory," the prize citation says.
"If Bhargava had stopped with this discovery, his work would already be quite remarkable. But Bhargava has gone on to use his composition laws to solve a new case of one of the fundamental questions of number theory, that of asymptotic enumeration of number fields of given degree, as the discriminant grows.
Bhargava used his new composition laws to solve the degree 4 case, brilliantly overcoming very serious analytic problems that had completely blocked all previous work on the problem."