The SPAMS Schedule - Spring 2009

Abstract: In the spirit of taking the start of the semester nice and easy, this talk will primarily consist of direct instruction in the construction of a simple origami unit (the Sonobe unit). This unit can (and will!) be used to construct beautiful polyhedra that arguably have mathematical interest. Origami paper will be provided.

Abstract: I will begin by reviewing the classical mathematics of how and why the earth orbits the sun. Then I'll move on to consider the case of a quantum earth.

It is known by many people that the Lie algebra so(3) plays an important role in quantum mechanics (as the algebra of angular momentum operators), and that the representation theory of so(3) is used to deduce the quantisation of angular momentum. Not quite so many people know that a clever trick can be used to deduce the possible energies of a quantum mechanical earth orbiting the sun... using a non-obvious so(4) symmetry inherent in the system. An extension of this material can be used to determine the wavefunctions of the earth using algebra as well.

Finally I will describe something relatively few people seem to aware of: That there is an unrelated method (involving S.G.A. and representations of sl(2, R)) that can be employed to determine the spectrum of the relativistic, quantum earth. "Look, Ma! No spinors or anything!"

Abstract: Evolutionary dynamics are strongly affected by population structure. The outcome of an evolutionary process in a well-mixed population can be very different from that in a structured population. There have been many attempts to study the effect of population structure on evolutionary and ecological dynamics. These approaches include spatial models in ecology, viscous populations, spatial games and games on graphs. In this talk, I will present a new way to think about population structure. Unlike previous structures, the one I introduce now is dynamical: the population structure itself is a consequence of evolutionary dynamics. I will present a general mathematical approach for studying any evolutionary game in this structure. As a particular example, I will discuss the evolution of cooperation and derive precise conditions for cooperators to be selected over defectors.

Finally, I will use the same mathematical tools to derive a general condition for strategy A to be favored over strategy B in a large variety of structured populations.

Abstract: Given some signal (for example, an image), how many Fourier coefficients do we need in order to recover it? If the signal is sparse in some sense, it turns out that surprisingly few Fourier coefficients suffice. Additionally, these coefficients can be chosen at random. Such phenomena lie at the heart of compressed sensing, which has recently become a very active research area. We will give a brief introduction to the field and survey some of its main results. There will be lots of pretty pictures.

In this talk, I will discuss Indian popular cinema, and exhibit its many connections to mathematics. In particular, I will demonstrate how all areas of mathematics could be taught more effectively using audio-visual aids from Hindi or Tamil films. This revolutionary technique can be used across a range of applications, from the earliest childhood education, through geometry, calculus, algebra, differential equations, combinatorics, and computer science. No knowledge of Hindi, Tamil, or mathematics will be assumed

Spoiler alert: Talk may include references to Alai Payuthey, Biwi No. 1, Devdas, Dil To Pagal Hai, Hum Aapke Hain Kaun, Hum Kisise Kum Nahin, Jaane Tu Ya Jaane Na, Jeans, Jo Jeeta Wohi Sikander, Kabhi Kabhie, Kal Ho Na Ho, Kandukondain Kandukondain, Khalnayak, Kisna, Koi Mil Gaya, Mughal-E-Azam, Om Shanti Om, Rang De Basanti, Roja, Silsila, Swades, Tezaab, and Umrao Jaan.

Abstract: Until last year, I was cache oblivious -- an unfortunate state to be in while programming. In this talk I will explain the basics of the cache hierarchy of modern computers and its implications (illustrated with a few possibly surprising examples), which in practice can change our understanding of the runtime of algorithms. I will also describe a few cache oblivious algorithms, a nice meeting place of theory and practice. The goals of this talk are to educate the novice programmer about an important practical consideration often left unmentioned in introductory classes, and to educate the novice computer consumer about what all that gobbledegook about L2, FSB, etc. means (and why you should care).

Abstract: Breaking a graph into two or more pieces without cutting too many edges is a long-studied optimization problem, with applications in scientific computing, machine vision, and VLSI design. The state of the art takes inspiration from Fourier analysis on manifolds to derive spectral methods, which employ eigenvectors of a graph version of the Laplacian operator.

We present the classical theory of spectral graph partitioning, in which a central concern is showing bounds for planar and other well-behaved graphs on the eigenvalues which govern the algorithms' performance. Recent work extends these bounds from the first positive eigenvalue to all eigenvalues of the Laplacian, and provides a shorter, combinatorial proof of Korevaar's corresponding result in differential geometry.

Joint work with Jonathan Kelner, James R. Lee, and Shang-Hua Teng.