Algebra and its Applications (Spring 2004)

January 15	Organizational meeting

January 22	No meeting (Departmental ext. review)

January 29	Ettore Aldrovandi, FSU	Tame Symbols

After the break we resume our discussion of the Tame Symbol. We will review what has been discussed in the previous meetings and finish the Deligne construction in terms of the Dilogarithm function.

The following is the general abstract for these talks:

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.

February 5	No meeting (pre-empted by GPC meeting)

February 12	Ettore Aldrovandi, FSU	Tame Symbols, II

We complete Deligne construction and prove some important properties of the Tame Symbol, hopefully including its holonomy characterization.

We also mention a few more applications: regulator maps from K-theory, hermitian-holomorphic classes.

February 19	Ettore Aldrovandi, FSU	Tame Symbols III

We sketch Bloch's construction of the global regulator map from K₂ to Deligne cohomology for a complete curve X. We also outline extensions to the hermitian case, higher tame symbols, and possible applications and open problems.

February 26	Sam Huckaba, FSU	Integral closures of ideals, normality, and blowup algebras
March 4	Sam Huckaba, FSU	Integral closures of ideals, normality, and blowup algebras II

Integrality over an ideal plays an important role in commutative algebra and algebraic geometry (specifically, in the process of blowing up) as does the more commonly known concept of an integral ring extension. These talks will review the definitions and backgrounds of both, and will work towards descriptions of some recent research on the topics.

March 11	Spring break

March 18	Matilde Marcolli, MPI & FSU	Quantum Statistical Mechanics of Q-lattices

In this joint work with Alain Connes we generalize the Bost-Connes dynamical system with arithmetic spontaneous symmetry breaking to a system for GL₂ of adèles. The underlying noncommutative space is the set of 2-dimensional Q-lattices up to scaling, modulo commensurability.

March 25	Sam Huckaba, FSU	Integral closures of ideals, normality, and blowup algebras III
April 1	Sam Huckaba, FSU	Integral closures of ideals, normality, and blowup algebras IV
April 8	Sam Huckaba, FSU	Integral closures of ideals, normality, and blowup algebras V

This is the second series (III, IV, and V) after the interruption

Integrality over an ideal plays an important role in commutative algebra and algebraic geometry (specifically, in the process of blowing up) as does the more commonly known concept of an integral ring extension. These talks will review the definitions and backgrounds of both, and will work towards descriptions of some recent research on the topics.

April 15	Dimitre Tzigantchev, FSU	Linear orbits of line configurations
April 22	Dimitre Tzigantchev, FSU	Linear orbits of line configurations II

We study the action of the group of linear transformations on spaces of plane curves, and in particular on curves consisting of unions of lines. The main goal is the computation of a projective enumerative invariant in terms of the combinatorics of the line configuration.