MASS Colloquia 2006

Thursday, September 7 Professor Vitaly Bergelson (Ohio State University)
2:30 p.m. Uniform Distribution: from Diophantine approximations to ergodic theorems
Thursday, September 14 Professor Sergei Chmutov (Ohio State University)
2:30 p.m. Virtual links and ribbon graphs
ABSTRACT Regions of a link diagram can be colored in black and white in a checkerboard manner. Putting a vertex in each black region and connecting two vertices by an edge if the corresponding regions share a crossing yields a planar graph. In 1987 Thistlethwaite proved that the Jones polynomial of the link can be obtained by a specialization of the Tutte polynomial of this planar graph. I will explain this theorem and its generalization to virtual links. In this case the graph will be a ribbon graph, which means that it will be embedded into a (higher genus, possibly non-oriented) surface. For such graphs we use a generalization of the Tutte polynomial discovered recently by B. Bollobas and O. Riordan.
Thursday, September 21 Professor Ronald Graham (University of California, San Diego)
2:30 p.m. Packing discs in the plane
ABSTRACT How many non-overlapping discs can be placed into a square of side a? This apparently simple question turns out to be surprisingly difficult. The related question which asks for the maximum number of non-overlapping unit squares which can be packed into a square of side a seems to be even harder! In this talk, I will summarize what is known and what is not known about these (and related) classical geometrical problems.
Thursday, September 28 Professor David Savitt (University of Arizona)
2:30 p.m. Noncrossing partitions, meanders, and the fundamental theorem of algebra
ABSTRACT In recent decades, noncrossing partitions have been a subject of intense interest in combinatorics. Since their introduction by Germain Kreweras in 1972, noncrossing partitions have not only yielded many interesting enumerative problems, they have been shown to have fascinating and deep links with algebra and geometry. I will give a tour of some of this theory, and describe a relationship between noncrossing partitions and the geometry of polynomials.
Thursday, October 12 Professor Paul Rabinowitz (University of Wisconsin)
2:30 p.m. Direct methods in the calculus of variations
ABSTRACT This talk will begin with an introduction to the calculus of variations. Then we will describe some so-called direct methods of the subject and discuss how they can be used to solve problems in dynamics ranging from the existence of periodic solutions to the construction of chaotic solutions.
Thursday, October 19 Professor Lisa Traynor (Bryn Mawr College)
2:30 p.m. Classification of Legendrian Knots and Links
ABSTRACT A basic problem in topology is to construct a list of topological knots and links. This infinite list should contain every knot and link, and no object should appear more than once. There are beautiful tables of topological knots and links with small crossing numbers. I will discuss attempts to develop such a table for Legendrian knots and links. These are topological knots and links that satisfy an additional geometric condition imposed by a contact structure.
Thursday, November 9 Professor Yuri Burago (Steklov Institute, St. Petersburg, Russia)
2:30 p.m. Bending polyhedra
ABSTRACT One can consider two types of bending polyhedra. In one case faces of a polyhedron can only move as rigid plates. Such bendings exist (proved by R. Connelly) and the volume of the polyhedron does not change (this is the Bellows Conjecture proved by I. Sabitov). In the latter case we allow new edges to arise and move. In such a case a polyhedron becomes much more flexible, in particular its volume can always be increased (as shown by D. Bleecker, I. Pak, V. Zalgaller and the speaker).
Thursday, November 16 Professor Derek Smith (Lafayette College)
2:30 p.m. The Integral Octonions (That Funny Number System)
ABSTRACT The real numbers, the complex numbers, the quaternions, and the octonions comprise the four composition algebras. The first three are usually seen to be the well-behaved members of the family; the octonions, to quote John Baez, are the crazy old uncle nobody lets out of the attic.

I would like to let the octonions out for just a little while to introduce some of its fundamental properties, with an eye toward its most important ring of integers, the integral Cayley numbers of Coxeter. A beautiful and remarkably simple algorithm of Rehm leads to a unique factorization theorem for this ring, despite its non-associativity. Just as remarkably, there are several elementary questions about factorization in this and related rings that remain unanswered.

Thursday, November 30 Professor Nikita Netsvetaev (St. Petersburg University, Russia)
2:30 p.m. Topology of algebraic curves and surfaces (Hilbert's 16th problem and beyond)
ABSTRACT Algebraic curves and surfaces are among the most natural objects in mathematics: they are sets determined by (systems of) equations. Curves and surfaces of degree 1 and 2 are objects familiar from Calculus: they are lines, planes, conics, and quadrics. Unfortunately, in degrees greater than 2 their structure becomes too diverse to be included into basic undergraduate (and even graduate) courses.

Hilbert's famous 16th problem concerns topological arrangement of algebraic curves (surfaces) in the real [projective] plane (space). Though many important general theorems have been proved here, there remain open questions even for curves of degree 8 and surfaces of degree 5.

Of much help here is information coming from the complex domain, despite (but maybe also owing to) the fact that the situation with complex curves and surfaces turns out to be radically different: the topology of a (nonsingular projective) complex curve (surface) is completely determined by the degree of the defining equation. (Note that complex cubic curves play a role in one of the Millenium $1,000,000 Problems!)

Allowing isolated singularities, we return to the diversity of the real case.