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Geometry and Dynamics IGERT Awarded

August 26, 2011 By

Brandeis has just been awarded an NSF Integrative Graduate Education and Research Traineeship (IGERT) grant in the mathematical sciences.  The grant, titled Geometry and Dynamics: integrated education in the mathematical sciences, is designed to foster interdisciplinary research and education by and for graduate students across the mathematical and theoretical sciences, including chemistry, economics, mathematics, neuroscience, and physics.  It is structured around a number of themes common to these disciplines: complex dynamical systems, stochastic processes, quantum and statistical field theory; and geometry and topology. We believe that it is the first IGERT awarded for the theoretical (as opposed to laboratory) sciences, and are very excited about what we believe to be a highly novel program which will cement existing interdepartmental relationships and encourage exciting new collaborations in the mathematical sciences, including collaborations between the natural sciences and the International Business School (IBS).

The resolution of a singularity that develops along Ricci flow, understood mathematically by Grigori Perelman.  If the red manifold represents the target space of a string, it is conjectured that the corresponding two-dimensonal field theory describing the string undergoes confinement and develops a mass gap for the degrees of freedom corresponding to the singular regime.

The award, for $2,867,668 spread out over five years, provides funds for graduate student stipends, travel, seminar speakers, and interdisciplinary course development.  It contains activities and research opportunities in partnership with the New England Complex Systems Institute (NECSI) in Cambridge, MA.  It also provides opportunities for research internships at the International Center for the Theoretical Sciences in Bangalore.

The PIs on the grant are: Bulbul Chakraborty (Physics); Albion Lawrence (Physics: lead PI); Blake LeBaron (IBS); Paul Miller (Neuroscience); and Daniel Ruberman (Mathematics).  There are 11 additional affiliated Brandeis faculty across biology, chemistry, mathematics, neuroscience, physics, and psychology.  Contact Albion Lawrence (albion@brandeis.edu) for more information about the program.

Arrays of repulsively coupled Kuramoto oscillators on a triangular lattice organize into domains with opposite helicities in which phases of any three neighboring oscillators either increase or decrease in a given direction. Fig. (a) illustrates these two helicities in which cyan, ma- genta and blue vary in opposite directions. In Fig. (b), white and green regions represent domains of opposite helicities. The red regions indicate the frequency entrained oscillators, which are predominantly seen in the interior of the domains.

Admission to the program is handled through the Ph.D programs in the various disciplines:

Filed Under: grad students, Mathematics, Neuroscience, Physics Tagged With: , , , , , , ,

Helfgott ’98 wins Adams Prize in mathematics

March 10, 2011 By

Harald Helfgott ’98 has been awarded the Adams Prize by the University of Cambridge (UK), one of its oldest and most prestigious prizes. The prize, awarded jointly to Helfgott and to Dr. Tom Sanders (University of Cambridge), honors young UK-based mathematicians  doing “first class international research in mathematical sciences”. Helfgott, currently a Reader at Univ. of Bristol and researcher at the CNRS/ENS (Paris), has been the recipient of additional prestigious prizes. In 2010 he was awarded the Whitehead Prize by the London Mathematical Society for his contributions to number theory and in 2008 he was awarded the Leverhulme Mathematics Prize for his work on number theory, diophantine geometry, and group theory.

Helfgott was a double major in Mathematics and Computer Science while at Brandeis, graduating summa cum laude with highest honors in both disciplines. Professors from both departments recall Harald as a top student, extremely well prepared, outspoken, and as one who truly loves to learn and  exchange ideas. He took full advantage of the opportunities for independent research in both departments, resulting in several conference papers and publications. In Computer Science, working with James Storer completed significant research projects on genetic algorithms for lossless image compression, Lempel-Ziv methods for two dimensional lossless compression, predictive coding, and maximal parsings. He formulated an approach to two dimensional coding that equaled one of the best methods in the literature at the time and had a number of computational advantages. According to Storer “He had an impact on nearly every research group in the Computer Science Department at that time.”

Regarding Helfgott’s work in the Math department, Ira Gessel remembers:

Although I never had him for a course, I did write a paper with him when he was an undergraduate here (the only paper I’ve ever written with an undergraduate).  Harald was involved in an undergraduate  research program with Jim Propp on tilings, and he had made some progress on solving some open problems on counting certain types of tilings. He was having trouble evaluating some determinants, and I helped him with that technical aspect of his work. But the main ideas of the paper were all Harald’s.

On graduation, Helfgott chose to focus on mathematics, doing his Ph.D. at Princeton and post-doctoral stints at Yale and at Concordia University before moving to his current position at Bristol. In addition to his current active research career, Helfgott also has been “strongly committed to the free sharing of information in all areas of intellectual activity“, giving lecture series to students and young researchers in the Third World, including lecture series in India, Cuba, Bolivia, and his native Peru.

According to Gessel:

It’s difficult to give a nontechnical account of most of Harald’s work, but here’s one of his results that’s not too hard to state.  He proved a difficult conjecture of Paul Erdős that if f(x) is a cubic polynomial with integer coefficients (satisfying some additional obvious necessary conditions that I’ll omit) then there are infinitely many primes p such that f(p) is not divisible by a square.

Filed Under: awards, Computer Science, Mathematics Tagged With: , , ,

Chiral Equivariant Cohomology

December 30, 2010 By

Prof. Bong Lian from Math writes:

In the 1950’s, French mathematicians Henri Cartan and Armand Borel defined a new topological invariant that was capable of distinguishing symmetries of certain geometric spaces known as G-manifolds. Cartan and Borel called their invariant the Equivariant Cohomology of a G-manifold. It was new in that it was able to capture essential aspects of geometric operations, called Lie group actions (after Sophus Lie), on manifolds that ordinary cohomology theory was unable to detect. Hence it provides a new conceptual framework for studying symmetries of spaces on the one hand, and offers a powerful tool for computing ordinary cohomology of these spaces, on the other.

In the late 80’s, physicists invented String Theory in their attempt to construct a grand unified field theory. They found that certain solutions to String Theory are essentially governed by an algebraic structure called a Chiral Algebra. This turns out to be a new structure that generalizes many fundamental algebraic constructs in mathematics, including commutative algebras and Lie algebras. A question was then raised as to whether there exists a natural theory that integrates both the Cartan-Borel invariant and Vertex Algebras. This hypothetical theory, which I learned about as a graduate student at Yale University, was dubbed the stringy analogue of the Equivariant Cohomology theory.

In 2004, Andrew Linshaw, a Brandeis PhD student (now Research Fellow, U. Darmstadt), and I constructed such a theory, which we coined the Chiral Equivariant Cohomology (CEC) of a G-manifold. In our latest paper, joint with another Brandeis PhD student, Bailin Song (now Assoc. Prof., Univ. of Science and Technology of China), we showed that not only does the CEC subsumes the Cartan-Borel theory, it goes well beyond that. For example, we have found an infinite family of Lie group actions on spheres that the Cartan-Borel theory is too weak to distinguish, but have non-isomorphic CEC. This proves that the CEC theory is strictly stronger as a topological invariant than the Cartan-Borel invariant. The paper appears in the December 2010 issue of the American Journal of Mathematics (Volume 132, Number 6).

Filed Under: grad students, labs, Mathematics Tagged With: , , , , ,

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