The SPAMS Schedule - Spring 2008

Did you ever, at dawn, with eyes bloodshot from a night of doing LaTeX, promise yourself to quit the habit? Did you ever, frustrated by an untraceable error, curse and beat your computer, and then, full of shame, beg its broken case for forgiveness? If so, you may wish to consult with MIT Medical Mental Health --- this seminar will probably not help you.

For those of you in earlier stages of a LaTeX addiction, I will give a brief tour of LyX. It's a program designed to make producing LaTeX documents more intuitive, quick, and practically error-free. Instead of typing backslashes and arcane commands in a text editor, you use a graphical "what you see is what you mean" environment. You don't have to worry about closing curly braces. Your formulas don't take up half a page of jumbled text -- they appear as printed. However, you don't sacrifice flexibility; you can still insert straight LaTeX whenever you need it.

LyX isn't perfect, and it doesn't read your mind. So, I will cover a few common tricks and workarounds that would otherwise take a while to discover.

What kinds of surfaces can one form from a sheet of paper? If only bending is allowed -- i.e., the paper must remain smooth -- the answer is "developable surfaces." Such surfaces are well-studied in differential geometry and not hard to characterize. On the other hand, if arbitrary creasing is allowed, much more general behavior can occur; hence, little can be said about these "applicable surfaces." To convince yourself of this, find a paper you are tired of reading and crumple it into a ball. (For added stress-relief, throw it in the trash or at your office-mate.)

In between these two extremes lies the world of piecewise-developable surfaces, which will be the main topic of this talk. We will begin by reviewing the basic geometry of intrinsically flat surfaces which arises in studying developables. Then, applying our tools, we will explore some elementary but surprising results concerning the geometry of straight and curved creases. Finally, we will view some curved origami eye-candy that illustrates these effects in "real life."

Paper will be provided free of charge for a hands-on experience, though attendees assume the risk of paper cuts and all complications thereof.

In perturbative Quantum mechanics it is often required to compute the matrix elements of operators in some preferred basis. Knowing certain properties of an operator can reduce the work required in computing them. The Wigner-Eckart theorem is one such example where the class of 'tensor operators' have a certain transformation property, making the calculation of its matrix elements easier. A short proof of the theorem using representations of SO(3) will be presented and its applications to the Zeeman and Stark effects explored.

To say that a permutation w contains a pattern p is to say that w has a subsequence whose elements appear in the same relative order (or equivalently have the same relative sizes) as those of p. If we ask in addition that certain elements of this subsequence be adjacent, we are speaking of a generalized pattern. We will show (via numerous, hopefully interesting examples) that enumeration of patterns in permutations can do nearly everything the budding combinatorialist could desire: recover interesting sets of permutations, recover various classical and other permutation statistics, and generate positive-integer sequences of note.

In this talk, I will discuss four currently used algorithms for testing primality. These algorithms illustrate various ideas and techniques which are common in number theory, such as elliptic curves, cyclotomic fields, non-determinism, and polynomial arithmetic. I will conclude by presenting some practical applications of primality-testing algorithms.

Abstract: It is very well known in combinatorics that given an exponential generating function for labeled objects, its logarithm is the generating function for the connected ones. This simple idea is used in other fields: in basic quantum field theory the partition function Z of a classical action gives a sum of Feynman diagrams, and its logarithm gives the series of the connected diagrams. Interestingly, the story does not stop there. When calculating the effective action via the Legendre transform, such connected diagrams are further dissected into their 2-edge connected (1-particle irreducible) components. This involves trees which in turn are the simplest connected graphs. If we bring this last step back to combinatorics what does it mean? The talk will explore this and another loosely related question.

Our world is full of many different kinds of dense materials, fluids, solids, and inbetween. Granular materials like sand are in this inbetween category of amorphous matter. Depending on how compressed it is, a chunk of sand can flow like a fluid or resist deformation like a solid. In this talk we delve into the mathematical theory of continuum mechanics and apply it to the strange case of granular materials.