Thur, Sep 4 ALEXANDRE KIRILLOV (University of Pennsylvania) WHY SYMPLECTIC AND CONTACT GEOMETRIES ARE SO IMPORTANT ABSTRACT: It is known that symplectic geometry and its odd dimensional analog, contact geometry, occur in many different parts of mathematics. There is a nice explanation of this fact. It is based on the notion of a local Lie algebra structure in the space of all smooth functions on a given manifold. This notion unites in a natural way analysis, algebra and topology. Namely, we require that the commutator [ , ] should be a local operation: if one of two functions f, g vanishes in a neighborhood of a point x, then the commutator [f,g] (x) also vanishes. It turns out that the description of all local Lie algebra structures can be done in terms of symplectic and contact geometry.
Thur, Sep 18 JOHN ROE (Jesus College, Oxford, UK) MOUNTAINEERING AND QUANTUM MECHANICS ABSTRACT: One of the guiding ideas of topology is to try to understand the constraint that global structure imposes on local data. For instance, Euler's famous polyhedron formula V - E + F = 2 can be understood in this way: there is no local restriction on the way we triangulate a polyhedron, but all triangulations have to satisfy this same global constraint. A similar formula relating to the critical points of a function was discovered by Marston Morse in the thirties. In 1982, Ed Witten published a proof of this formula based on ideas from quantum mechanics; the numbers appearing in the formula are interpreted as the number of zero-energy modes of certain fictitious particles. By varying a parameter in the problem, one cools down the particles until they are moving slowly enough to count. In this talk, I will aim to explain this argument, which is a prototype for a number of discussions relating analysis to topology.
Thur, Sep 25 DOUG ARNOLD (Penn State) CONNECTING THE DOTS: THE THEORY AND PRACTICE OF INTERPOLATION ABSTRACT: If you know the value of a function at only a handful of points, what is the best way to guess to the function's value elsewhere? In other words: given a few dots on a graph, how should you connect them? This seemingly simple question inspired the rich subject known as interpolation theory. In this talk, which will be extensively illustrated with computer examples, I will survey some of the lovely, and often deep, mathematical results of this theory. We will mostly tour the classical world of polynomial interpolation, but will end with an excursion to the more modern land of piecewise polynomial interpolation and finite elements, and glimpse an application to the simulation of colliding black holes. THE COMPUTER RELATED MATERIALS for the lecture are available here
Thur, Oct 9 ADRIAN OCNEANU (Penn State) PLATONIC SOLIDS, THEIR SYMMETRY GROUPS AND KLEIN'S INVARIANT THEORY ABSTRACT: The finite subgroups of 3-dimensional rotation group arise as symmetrics of solids in R^3. We discuss Klein's Invariant polynomials and their relations to the Platonic solids.
Thur, Oct 16 SERGEI IVANOV (Steklow Institute for Math, Russia) PROBABILISTIC STRATEGIES IN GAMES ABSTRACT: We will discuss games with incomplete information. It turns out that a player who mixes several different strategies (using probabilistic choices) has better chances than one who follows a purely deterministic strategy. (This is just why bluff may be helpful in poker!) We will prove a theorem due to von Neumann about existence of optimal mixed strategies. The proof is a nice application of the theory of convex sets.
Thur, Nov 6 RICHARD L. BISHOP (University of Illinois at Urbana-Champaign) INVARIANTS OF CURVES BY EVOLUTION OF FRAMES ABSTRACT: The goal of a lot of excellent mathematics is to distinguish inequivalent objects by means of invariants. If the invariants devised are rich enough, then the distinction is complete, that is, two objects are equivalent exactly when the invariants are the same. For example, in algebra, linear operators on a vector space are distinguished by their eigenvalues, for equivalence under similarity. For the subclass of operators with distinct eigenvalues, the distinction is complete. For all operators, however, the richer invariant of the Jordan canonical form is needed. The situation is analogous for classes of curves in Euclidean spaces for equivalence under rigid motions. The classical invariants are the curvature and torsion, in the case of 3-dimensional space, and a natural generalization to a sequence of n-1 curvatures for the case of n-dimensional space; these are the Frenet invariants. They are complete for the class of curves with n-1 linearly independent derivative vectors (i.e., velocity, acceleration, etc.). For classes of curves defined by fewer regularity conditions, however, the Frenet invariants are undefined, but there are alternative systems of invariants, defined in the same way, in terms of the evolution of frames of vectors moving along the curve and adapted in various ways to the derivative vectors. In particular, the Bishop frames require only that there be a differentiable unit-length tangent field. The description of spherical and planar curves is much simpler in terms of the Bishop invariants than in terms of Frenet invariants. The classification and properties of all reasonable alternative moving-frame invariants has been carried out by an undergraduate, Matthew Rodriguez, at UIUC.
Thur, Nov 20 PAUL BAUM (Penn State) WINDING NUMBERS AND A PROOF OF THE FUNDAMENTAL THEOREM OF ALGEBRA The Mass students were invited to join the Undergraduate Mathematical Physics Reading Group and attend Professor Baum's talk