MASS Colloquia 2005

Thursday, September 15 Professor Mark Levi (Penn State)
2:30 p.m. Gravity-defying phenomena
ABSTRACT Spinning top is the best known of a group of devices which seem to defy gravity. These include the electro-magnetic particle traps and their striking mechanical analogues: the levitron (a spinning magnet capable of hovering in mid-air) and the inverted pendulum made stable in its upside-down position by rapid vibration of the hinge (there is no feedback—the hinge's motion is prescribed ahead of time). A short demonstration will be followed by a mathematical explanation of stability of the pendulum via topology of the group of matrices of determinant one.
Thursday, September 22 Professor Don Zagier (MPI-Bonn & College de France)
2:30 p.m. How old was Diophantus's son?
ABSTRACT One of the most beautiful and most accessible parts of number theory, or perhaps of all of mathematics, is the theory of Diophantine equations, named after the Greek mathematician Diophantus of Alexandria, who lived some time around the 2nd century AD. The lecture will try to describe some of the pearls of this field, both classical and recent. If you want to know the meaning of the mysterious title, you have to come.
Thursday, September 29 Professor James Propp (University of Wisconsin, Madison)
2:30 p.m. Bugs, Blobs and Roto-Routers
ABSTRACT A common fallacy among gamblers is that if you're observing a random process and the outcome you're waiting for hasn't occurred in a long time, it must occur soon. This belief stops being a fallacy, and becomes an important and useful fact of life, in the strange world of quasirandomness. With a few fun puzzles and lots of colorful graphics (see, for instance), I'll take you on a quick tour of this world, and show you how some quasirandom machines, built out of simple components called rotor-routers, can give surprisingly good estimates for quantities like the golden ratio, the square root of two, and pi.
Thursday, October 6 Professor Chengbo Yue (Academy of Mathematical and System Sciences, Chinese Academy of Sciences)
2:30 p.m. Schwarzian derivative, projectivity and some applications to rigidity
ABSTRACT On the real line, constant functions are characterized by their first derivative being equal to zero; affine functions are characterized by their second derivative being equal to zero; the projective transformations—the fractional linear functions—are characterized, not by their third derivative, but by their Schwarzian derivative, being equal to zero. In this talk, I'll introduce the notion of the Schwarzian derivative and prove a necessary and sufficient condition for a diffeomorphism on the circle to be smoothly conjugate to a projective transformation: its Schwarzian cocycle must be cohomologous to the zero cocycle. As application, I'll introduce a smooth rigidity result for a Fuchsian group action on the circle.
Thursday, October 20 Professor Adrian Ocneanu (Penn State)
4:00 p.m. Mathematics of symmetry in 4 dimensions - a sculpture
ABSTRACT We discuss several mathematical topics brought together by the sculpture in our lobby, among others
  • regular solids in higher dimensions - why the dimension 4 is special.
  • packing spheres in higher dimensions.
  • soap bubbles and conformal mappings.
  • root systems for Lie algebras—the sculpture illustrates D4, B4, C4 and F4—their Coxeter elements and the way to read the geometry from the eigenvalues of the adjacency matrix of the graph. Picturing Weyl groups with the sculpture.
  • finite subgroups of SU(2) and SO(3) illustrated by the sculpture—the symmetries of the tetrahedron and cube—the E6 affine and E7 affine subgroups of SU(2). The multiplication of symmetries in SU(2) and the corresponding vector fields and fibrations of S3, and finally
  • cutting and folding 3d cardboard kits into 4d solids - an illustrated step by step video guide for 4d kids
Thursday, October 27 Professor Richard Montgomery (University of California, Santa Cruz)
2:30 p.m. New Solutions to the N-body Problem
ABSTRACT I will start off with a tour of the gravitational N-body, leading into the rediscovered figure eight orbit of Chenciner and the myself. I will then focus on the mathematical methods on which the discovery of the new orbits were based: calculus of variations, homotopy theory, group theory, and spherical geometry, assuming no knowledge of these topics. I will discuss the host of new choreography in which the N bodies chase each other around curves having the shape of flowers, whales, spirograph patterns, etc. I will end with a summary of some of the (many) open problems within the N-body problem.
Thursday, November 3 Professor Krishnaswami Alladi (University of Florida, Gainseville)
2:30 p.m. How many prime factors does a number have?
ABSTRACT Although prime numbers have been studied since Greek antiquity, the first systematic analysis of the number of prime factors of integers is due to Hardy and Ramanujan who showed in 1917 that almost all numbers have about loglogn prime factors. The significance of this observation was realized later when Turan gave a simpler proof in 1934 which indicated that there might be probabilistic connections. Then when Erdos and Kac established the Gaussian law for the distribution of the number of prime factors, the subject of Probabilistic Number Theory was born. We will begin by tracing the development of Probabilistic Number Theory by focusing on the number of prime factors, and discuss some problems of current interest on the number of prime factors in which difference-differential equations and sieve methods play a crucial role.
Thursday, November 10 Professor Carla Savage (North Carolina State University)
2:30 p.m. Venn Diagrams and Symmetric Chain Decompositions
ABSTRACT A Venn diagram for n sets is a drawing of n simple closed curves in the plane which, in the regions created by their intersections, represents all of the 2n possible ways that n sets can intersect. The familiar two and three circle Venn diagrams from grade school have the appealing property of being rotationally symmetric. It has long been known that Venn diagrams can be constructed for any number of sets, but that rotational symmetry is not possible if n is not prime. We describe our discovery, with Jerrold Griggs at USC and Chip Killian, an undergraduate at N.C. State, that symmetric Venn diagrams can be constructed for every prime n.
Thursday, November 17 Professor Louis Kauffman (University of Illinois at Chicago)
2:30 p.m. Knots, Tangles, DNA and Quantum Physics
ABSTRACT: This talk is an introduction to knot theory and its relationships to research in Natural Science, specifically DNA and quantum physics. We begin with a deceptively simple problem about understanding unknots and let it lead to the theory of rational tangles and the classfication of these tangles in terms of rational fractions, the classification of rational knots, applications to DNA and relations with physics via the structure of knot invariants (bracket model of the Jones polynomial, statistical mechanics, quantum amplitudes, Witten's work, …
Thursday, December 1 Professor Boris Khesin (University of Toronto)
2:30 p.m. Different facets of linking numbers
ABSTRACT We discuss two different generalizations of the Gauss linking number of two curves in the three-space. The first generalization is helicity: linking of trajectories of a divergence-free vector field and Arnold's theorem on the asymptotic Hopf invariant, The second generalization is the holomorphic linking number for complex curves in complex three-folds. The applications range from energy estimates in magnetohydrodynamics to topological and holomorphic Chern-Simons theory on real and complex three-folds.