The SPAMS Schedule - Spring 2011

Sparse representations have been studied in the past decades (and more actively in the past few years) for various engineering applications. The idea is that signals can be recovered from far lesser measurements than the Nyquist-Shannon sampling limit would suggest. What is meant by sparsity? Why do people like sparsity? What are the limits on the recovery of the signals? What algorithms have been suggested and what are some of the applications? We will briefly touch on these and other interesting questions in this talk.

Abstract: We propose a method which we call "Isotropic Entanglement" (IE), that takes inspiration from Free Probability Theory and other ideas in Random Matrix Theory to predict the eigenvalue distribution of quantum many body (spin) systems with generic interactions. At the heart of this method is a sliding scale, "the Slider", which interpolates the quantum problem between two extrema by matching fourth moments. The first extreme treats the non-commuting terms in the Hamiltonian classically and the second treats them isotropically. By isotropic we mean that the non-commuting subsets have eigenvectors that are in generic positions. We prove The Matching Three Moments and Slider theorems and further prove that the interpolation of the quantum problem in terms of the classical and isotropic is universal, i.e., independent of the choice of local terms. Examples show that IE provides an accurate picture of the spectra well beyond what one expects from the first four moments alone.

Abstract: This talk explains Sparse Grid Quadrature, a technique for numerically integrating a function of many variables. I compare it to the two "extremes" of tensor-product integration (which is dimensionally cursed) and pure Monte-Carlo integration (which is slow-converging), and give theoretical intuition into why it operates in the middle ground. Performance analysis is done on a few test integrands. Additionally, beautiful pictures of quadrature grids are shown.

Once upon a March SPAMS table, while I pondered how much cable I would need to tie some netting tightly 'round my unit orb, While the sphere I thus tried trapping, suddenly a bound came, capping How much string is needed wrapping, wrapping round my precious orb. Merely three pi is required to accomplish such a chore! Only this? Well, just \eps more.

Ah, distinctly I remember, how I then wished to dismember, Enough of the links that emb(e)raced that rare and radiant orb. Seeking to dislodge my prize, I quickly saw, to my surprise, by Choosing k strands to incise, my sphere from net could not be torn! How much - how much length is needed? - tell me - tell me, I implore! (2k+3)pi, and more.

Abstract: We have a list of all knots with up to 16 crossings. How would you program a computer to make a list of knots for you? If you do it a stupid way, the computer will laugh in your face. I'll show how to do it a smart way (there will be no code). I'll also present an unpublished list of knotted graphs, including the 'Mickey Mouse', 'Popeye', 'Oscar Wilde', 'The Hangover', 'Reclining Nude', and 'The Knuckle Sandwich'.

Abstract: If PI day was not enough and you can't wait until June 28, then you should attend this talk. Elkies in 2003 reviewed a new proof by Calabi of the result by Euler that sum_{k=-infty}^infty (4k+1)^-n is a rational multiple of PI^n and interpreted the result as the trace of a power of a self-adjoint operator in L^2, and as the number of alternating permutations on n+1 letters. First, we will talk about these results by Elkies focusing on the combinatorics. Second, since most talks of combinatorics should mention the Catalan numbers (which can be approximated by 4^n/n^(3/2)sqrt(PI)), we will talk about a polytope whose volume is the product of consecutive Catalan numbers and how this polytope is related to the Greek phrase ????? ??? (panta rei, everything flows).

Abstract: With growing interest in finding coherent structures in oceanic systems, it has become apparent that an efficient means of identifying transport boundaries is of great importance. Using tools from topology, in particular braid theory, I will present a method for finding good approximations for the location of transport boundaries from trajectory data by identifying isolated bundles within trajectory braids.

Initial concepts were developed by Thiffealt

http://arxiv.org/abs/0906.3647

Abstract: In many dispersive wave systems, the presence of a focusing nonlinearity can result in soliton and breather solutions, or approximate solutions, which remain spatially localized for very long times. For example, a large number of nonlinear field theories support such localized solutions. These "oscillons", which develop somewhat spontaneously, exist as slowly radiating, oscillating waves. With an eye towards their cosmological implications, I will discuss oscillons in an expanding universe. First, I'll provide an analytic solution for the resulting one-dimensional oscillons and discuss their generalization to three dimensions. Second, I'll outline the oscillons stability to long wavelength perturbations using ideas by Vakhitov and Kolokolov. The result is a family of flat-topped oscillons which are robust against collapse instabilities.

Abstract: We aim to show that homotopy theory is the appropriate tool for modeling questions of identity. In Heideggerian terms, it is the mathematical equivalent of Dasein.

The Schrodinger equation is a PDE that describes the evolution of a wave function \psi(x,t) in quantum mechanics. We're all comfortable with PDEs, but the strange thing is that \psi has a probabilistic interpretation. In fact, one of the postulates of quantum mechanics is that |\psi|^2 is the probability density function of the particle! This makes many people uncomfortable, and people have been searching for alternate theories for over 80 years.

One of these theories is stochastic mechanics, pioneered by mathematician/physicist Ed Nelson. In this talk, I will give a brief description of stochastic mechanics, and perhaps I will say something about its various drawbacks. *Please note:* No background in quantum mechanics or stochastic mechanics is assumed, necessary, or desired; I will develop everything we need in the talk.