Welcome to the *Algebra and its Applications* seminar
home page!

The seminar is organized by Ettore Aldrovandi. Please send an email to contact me.

The seminar meets on Thursdays, 2:00-3:15pm in 104 LOV

September 4, 2003 | Paolo Aluffi, FSU | Linear orbits of plane curves |
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September 11, 2003 | Paolo Aluffi, FSU | Linear orbits of plane curves II |
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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.

September 18, 2003 | Eriko Hironaka, FSU | Lehmer's problem and Coxeter links |
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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.

September 25, 2003 | Eric Klassen, FSU | Polar Decompositions and Morse Theory on
O(n,R) |
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October 2, 2003 | Eric Klassen, FSU | Morse Theory on O(n,R) |
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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*).

October 9, 2003 | Eric Klassen, FSU | Morse Theory for algebraists: gradient flows on
O(n,R) |
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October 16, 2003 | Eriko Hironaka, FSU | Growth series for "graph groups" |
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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.

October 24, 2003 | No meeting |
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October 30, 2003 | Terry Gannon, University of Alberta | The braid group and modular invariance |
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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.

November 6, 2003 | Matilde Marcolli, MPI Bonn & FSU | From laser physics to class field theory via non commutative geometry |
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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).

November 13, 2003 | Mark van Hoeij, FSU | Factoring bivariate polynomials |
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The talk will discuss the use of the logarithmic derivative for the problem of factoring polynomials in two variables over a finite field.

November 20, 2003 | Ettore Aldrovandi, FSU | Taming Tame Symbols |
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December 4, 2003 | Ettore Aldrovandi, FSU | Taming Tame Symbols II |
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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.