Algebra and its Applications

The linear orbit of a plane curve is its orbit under the action of the group of projective linear transformations of the plane. The linear orbit of a curve is a quasi-projective variety which depends subtly on the geometry of the curve. We will describe the closure of this variety in its ambient projective space, with particular attention to methods computing its degree.

The Mahler measure of an integer polynomial is the product of roots outside the unit circle. In 1933, Lehmer asked if there is a gap on the real line between one and the rest of the Mahler measures of monic integer polynomials. He also presented a candidate 10 degree polynomial, which to date has the smallest known Mahler measure greater than one.

One can restate Lehmer's problem as a question about fibered links. The question of whether Lehmer's polynomial has smallest Mahler measure greater than one, translates to questions about the minimality of the -2,3,7 pretzel knot K_2,3,7. Using a result of C. McMullen, Coxeter theory can be applied to answer Lehmer's problem for a class of fibered links which contains K_2,3,7. Further properties of K_2,3,7 are discussed.

I will prove and then comment on the polar decomposition theorem, which states that every invertible matrix over R can be expressed in a unique way as a product of an orthogonal matrix and a positive definite symmetric matrix. Using related ideas, I will do some Morse theory on O(n).

Starting with a finite graph, one can build various groups, including the well-known Coxeter group and Artin group, the less well-known Mumford group, and monodromy groups coming from associated isometries of surfaces. I'll describe these groups, relations between them, and discuss cases when their growth series may be derived from the combinatorics of the graph.

Lattices and their theta functions; Kac-Moody algebras and their characters; the Monster and the Hauptmoduls… Over the centuries we've accumulated several examples of relatively simple algebraic structures, which are directly associated to modular forms and related functions. In this lecture I'll propose that the braid group provides the ultimate explanation.

Bost and Connes constructed a dynamical system with phase transition and spontaneous symmetry breaking, which has remarkable arithmetic properties. I will discuss Planat's recent interpretation of this system as a model for the phenomenon of quantum phase-locking in lasers. I will also describe how the result of Bost and Connes may lead to a general approach, via noncommutative geometry, to the Hilbert XII problem of explicit class field theory (this is work in progress with N. Ramachandran).

The talk will discuss the use of the logarithmic derivative for the problem of factoring polynomials in two variables over a finite field.

The Tame Symbol (or Tate Symbol) is an operation performed on a pair of rational functions on a curve, or, more generally, on a field with valuation. It has interesting properties such as the classical Weil reciprocity.

A nice geometric picture is obtained in terms of certain cohomology theories introduced by P. Deligne and, later, A. Beilinson. Far-reaching developments connect to Polylogarithms, Motives, K-Theory.

I will present an informal introduction to some of the simplest aspects of these ideas, and some applications to hermitian and arithmetic geometry.