Olivier Bernardi to Join Math Faculty

Dr. Olivier Bernardi will be joining the mathematics department in Fall, 2012 as a tenure-track assistant professor. Bernardi’s research interests lie in combinatorics and probability. He has worked on problems arising from mathematical physics (statistical mechanics and  quantum gravity), computer science (algorithms and graph theory), and algebra (representation theory of the symmetric group). His Ph.D. thesis was on bijective approaches to the numeration of planar maps.

Bernardi received his Ph. D. in computer science in 2006 at the University of Bordeaux, under the direction of Mireille Bousquet-Mélou, and has worked as a postdoctoral researcher at the Center of Mathematical Research, Barcelona, Spain, and as a CNRS researcher in the Mathematics Department at Université Paris-Sud, in Orsay, France. He is currently an instructor in applied mathematics at MIT.

Six scientists secure fellowships

One current undergraduate, and five alumni, from the Brandeis Sciences were honored with offers of National Science Foundation Graduate Research Fellowships in 2012. The fellowships, which are awarded based on a national competition, provide three full years of support for Ph.D. research and are highly valued by students and institutions. These students are:

  • Samuel McCandlish ’12 (Physics) , a current student who did research with Michael Hagan and Aparna Baskaran, resulting in a paper “Spontaneous segregation of self-propelled particles with different motilities” in Soft Matter (as a junior). He then switched to work with Albion Lawrence for his senior thesis research. Sam will speak about “Bending and Breaking Time Contours: a World Line Approach to Quantum Field Theory” at the Berko Symposium on May 14.  Sam has been offered a couple of other fellowships as well, so he’ll have a nice choice to make. Sam will be heading to Stanford in the fall to continue his studies in theoretical physics.
  • Briana Abrahms ’08 (Physics). After graduating from Brandeis, Briana followed her interests in ecological and conversation issues, and  in Africa as a research assistant with the Botswana Predator Conservation Trust, Briana previously described some of her experiences here in “Three Leopards and a Shower“. Briana plans to pursue as Ph.D. in Ecology at UC Davis.
  • Sarah Robinson ’07 (Chemistry). Sarah did undergraduate research with Irving Epstein on “Pattern formation in a coupled layer reaction-diffusion system”. After graduating, Sarah spent time with the Peace Corps in Tanzania, returning to study Neurosciene at UCSF.
  • Si Hui Pan ’10 (Physics) participated in a summer REU program at Harvard, and continued doing her honors thesis in collaboration with the labs at Harvard. Her award is to study condensed matter physics at MIT.
  • Elizabeth Setren ’10 was a Mathematics and Economics double major who worked together with Donald Shepard (Heller School) on the cost of hunger in the US. She has worked as an Assistant Economist at the Federal Reserve Bank of New York and her award is to study Economics at Harvard.
  • Michael Ari Cohen ’01 (Psychology) worked as a technology specialist for several years before returning to academia as  PhD student in the Energy and Resources Group at UC Berkeley.

Congratulations to all the winners!

Eisenbud Lectures: “The Mathematics of Dynamic Random Networks”

This year’s Eisenbud Lectures in Mathematics and Physics will be given by Dr. Jennifer Chayes, Distinguished Scientist and Managing Director of Microsoft Research New England. Dr. Chayes is well known for her work on the phase transitions in combinatorial and computer science problems; she is a world expert on the study of random, dynamically growing graphs, which can be used to model real-world social and technological networks.

Dr. Chayes received her PhD in mathematical physics from Princeton.  After postdoctoral fellowships at Harvard and Cornell, she was on the faculty at UC Los Angeles before co-founding the theory group at Microsoft Research in Redmond, Washington.  In 2008 she co-founded Microsoft Research New England. She is a fellow of the American Association for the Advancement of Science, the Fields Institute, and the Association for Computing Machinery; she is also a National Associate of the National Academies.

The Eisenbud Lectures are the result of a generous donation by Leonard and Ruth-Jean Eisenbud, intended for a yearly set of lectures by an eminent physicist or mathematician working close to the interface of the two subjects. Dr. Chayes’ distinguished career working on fundamental issues in mathematics, physics, and computer science makes her an ideal speaker for this series.

The lectures will take place at 4 PM on Tuesday Nov. 29 and at 4:30 PM on Thursday Dec. 1. both in Abelson 131.  A full description of the lectures can be found below. Driving directions, maps, links to the MBTA, and so forth can be found at: http://www.brandeis.edu/about/visiting/directions.html.  If you need parking, please contact Catherine Broderick at cbroderi@brandeis.edu.  A reception will be held after the first lecture on Tuesday November 29th from 5pm – 7pm in the Faculty Club Lounge at Brandeis.  All are welcome.

Everybody should come out to hear this year’s lectures!  They promise to be a lot of fun.

THE MATHEMATICS OF DYNAMIC RANDOM NETWORKS
During the past decade, dynamic random networks have become increasingly important in communication and information technology.  Vast, self-engineered networks, like the Internet, the World Wide Web, and online social networks, have facilitated the flow of information, and served as media for social and economic interaction.  I will discuss both the mathematical challenges and opportunities that exist in describing these networks:  How do we model these networks – taking into account both observed features and incentives?  What processes occur on these networks, again motivated by strategic interactions and incentives, and how can we influence or control these processes?  What algorithms can we construct on these networks to make them more valuable to the participants?  In this talk, I will review the general classes of mathematical problems which arise on these networks, and present a few results which take into account mathematical, computer science and economic considerations.  I will also present a general theory of limits of sequences of networks, and discuss what this theory may tell us about dynamically growing networks.

LECTURE 1:  Models and Behavior of the Internet,  the World Wide Web and Online Social Networks
Although the Internet, the World Wide Web and online social networks have many distinct features, all have a self-organized structure, rather than the engineered architecture of previous networks, such as phone or transportation systems.  As a consequence of this self-organization, these networks have a host of properties which differ from those encountered in engineered structures:  a broad “power-law” distribution of connections (so-called “scale-invariance”), short paths between two given points (so-called “small world phenomena” like “six degrees of separation”), strong clustering (leading to so-called “communities and subcultures”), robustness to random errors, but vulnerability to malicious attack, etc.    During this lecture, I will first review some of the distinguishing observed features of these networks, and then discuss some of the models which have been devised to explain these features.  I will also discuss processes and algorithms on these networks, focusing on a few particular examples.

LECTURE 2:  Convergent Sequences of Networks
In the second lecture of this series, I will abstract some of the lessons of the first lecture.  Inspired by dynamically growing networks, I will ask how we can characterize general sequences of graphs in which the number of nodes grows without bound.   In particular, I will define various natural notions of convergence for a sequence of graphs, and show that, in the case of dense graphs and even some sparse graphs, many of these notions are equivalent.  I will also give a construction for a function representing the limit of a sequence of graphs.  I’ll review examples of some simple growing network models, and illustrate the corresponding limit functions.  I will also discuss the relationship between these convergent sequences and some notions from mathematical statistical physics.

Geometry and Dynamics IGERT Awarded

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:

Helfgott ’98 wins Adams Prize in mathematics

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.

Chiral Equivariant Cohomology

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).

Protected by Akismet
Blog with WordPress

Welcome Guest | Login (Brandeis Members Only)