Keefe San Agustin

Ph.D student, Mathematics

116 Goldsmith, Department of Mathematics, Brandeis University Email address: syzygy@br-.edu

Teaching, Fall 2009: MATH 10a Techniques of Calculus log-on required

Research Interests: Algebra, Geometry and Topology.

More specific interests include the mapping class group, operads, matroids, manifold bundles, dessins d'enfants, braid groups and configuration spaces.
One of the ways these are all related is through the work of Kiyoshi Igusa, who is my dissertation advisor.

A seminar I organized: Grothendieck's theory of Dessins d'Enfants
Talks:
(1) From Matroids to Manifold Bundles and The Mapping Class Group In unpublished work and private communication, Kiyoshi Igusa has defined a "category of oriented matroids". There are several strong motivations for this and the one I want to focus on here is a series of implications regarding the mapping class group of surfaces. This talk outlines the current state of affairs on this front. We define the category of regular oriented matroids and examine its homotopy type and cohomology for low rank. We then change course and give a brief outline of the Igusa-Klein higher torsion invariants for smooth manifold bundles and relate these via the Riemann zeta function to the Miller-Morita-Mumford classes for the mapping class group of punctured surfaces. Finally, we discuss the interdependence between all of these objects and end with a few facts and a few conjectures involving matroids, higher torsion classes in the Torelli group and the work of Soren Galatius on the homology of automorphism groups of free groups.
(2) Operads, A-infinity Algebras, Trees and Infinite Loop Spaces I'll give an anti-historical talk on the connections between the items of the title. This is just the preface to a very modern book, the end chapters of which have yet to be written. The stable mapping class group, Artin braids, some homological algebra, and the ubiquitous pictures of "little disks" and their homology will all make an appearance, but really this talk could alternatively be titled simply "The Associahedron" and we'll stick to the underlying simplicity that the definition of an operad brings to the picture. The primary thrust will be in presenting simple examples that act as advertisements for the further study of the viewpoint presented here. But for the number theorists in the crowd, I'll try to briefly explain the relevance of this set-up to the study, due to Grothendieck and, later, Kontsevich, of the absolute Galois group over the rationals via its "homotopical action" on a certain moduli space whose definition naturally involves an operad over an A-infinity algebra.
(3) Applications of the lower algebraic K-groups I'll define the first three groups in algebraic K-theory and then present some applications to classical number theory, topology and algebra; algebraic geometry and C*-algebras fit in here, too, but those will be ignored; and at risk of offending just about everyone, Bott Periodicity will never be mentioned. Specifics that will be touched upon, however, include: Number Theory: Minkowski's theorem on the finiteness of the class number of a number field, Dirichlet's theorem on the structure of the ring of integers of a number field, and the Quillen-Lichtenbaum Conjecture relating the K-theory of the integers to odd zeta values; Topology: Wall finiteness obstruction, Whitehead torsion,and the Hatcher-Wagoner and Igusa Stability Theorems; Algebra: Mercurjev-Suslin's K_2 result on the Brauer group (via Matsumoto and Wedderburn).
(4) A vertex and an edge I, II In the past ten years, there has been a flurry of results relating to the graph with one vertex and one edge. I'll give an over-view of what I know about this graph. No prior knowledge of Hall algebras, coherent or perverse sheaves, Kac-Moody Lie algebras, quantum groups, elliptic curves, or Calabi-Yau categories will be assumed, although by the end of the second talk I hope to convey some sense of how these topics are all related and why there is still a lot to be learned from such simple objects as finite graphs.
(5) Dessins d'Enfants One of the more ambitious programs laid out by Alexander Grothendieck in his Esquisse D'un Programme is the complete characterization of the absolute Galois group over Q via its actions on topological spaces. The jumping-off point for this is the study of dessins d'enfants ("children's drawings") and this talk be an introduction to them. Dessins are certain types of simple finite graphs drawn on Riemann surfaces - i.e., combinatorial objects that can be explicitly and concretely described. Now two remarkable things happen: each dessin is naturally defined over a number field and the absolute Galois group over Q acts faithfully on the set of dessins! This yields a simple bijection between a certain set of finite graphs and surfaces defined over the algebraic closure of Q. Following this thread of ideas leads to "combinatorial invariants of geometric Galois theory" and I'll try to explain both what is known and what is not in this still mostly unexplored terrain.
Conferences I am involved in:
Braids National University of Singapore International Conference on Quantum Topology Institute of Mathematics, VAST, Hanoi Teichmuller Theory and Kleinian Groups MSRI, Berkeley Connections For Women: Geometric Group Theory MSRI Midwest Number Theory Conference for Graduate Students V University of Wisconsin Algebraic and Geometric Topology: a conference in honor of the retirement of Bob Stong (Charlottesville, Virginia) Nov 10-11, 2007 Maxwell Institute Colloquium on Khovanov Homology (Edinburgh) 16 November, 2007 New Topological Contexts for Galois Theory and Algebraic Geometry March 9-14, Banff New Paths Towards Quantum Gravity, Summer School May 12-16, 2008, Roskilde University Homotopical Group Theory and Topological Algebraic Geometry WORKSHOP June 16-20, 2008, Copenhagen Homotopical Group Theory and Topological Algebraic Geometry: conference 23-27 June 2008, MPI, Bonn Workshop on Algebraic Structures in Geometry and Physics 21-25 July 2008, Leicester Geometry and Physics May-Aug 2008, Hausdorff Research Institute for Mathematics
AIM Workshop: Higher Reidemeister Torsion 19-23 Oct 2009
Course Notes

Homotopy Course Notes, A Bestiary of Topological Objects, and Homology and Manifolds all by N.P. Strickland You can't afford not to read these

Fibre Bundles by Burt Totaro and 2nd semester Algebraic Topology and Cup products and intersections by M. Hutchings Ditto

An introduction to affine Kac-Moody algebras Workshop notes Lie groups and Lie algebras A year-long course by Reshetikhin, Serganova and Borcherds

Simplicial Objects and Homotopy Groups a book by Jie Wu and Categorical Aspects of Topological Quantum Field Theories a thesis by BB

The Birth of Homological Algebra and Stable algebraic topology 1945-1966 some history by Peter's Hilton and May

Links to (math) articles I'm reading, have read or want to read:
On the Fundamental Group of the Complement of a Complex Hyperplane Arrangement L. Paris and Configuration Spaces by Sam Gitler. Short and excellent introductions

The homology of the little disks operad and A pairing between graphs and trees A great one-two punch by Dev Sinha

On a universal mapping class group of genus zero, Grothendieck's Reconstruction Principle and 2-dimensional Topology and Geometry, Teichmuller spaces, triangle groups and Grothendieck dessins , Braid groups and Galois Theory (Grothendieck-Teichmuler Theory) and The Grothendieck-Teichmuller group and automorphisms of braid groups An initial fleshing out of one of the programs sketched in l'Esquisse

From Operads to `Physically' Inspired Theories, Point Configurations and Coxeter Operads, and Braids, trees, and operads

On braid groups and homotopy groups and Representations of the braid group B3 and of SL(2,Z) Cyclic Homology Theory Jean-Louis Loday and Deformations of algebras over operads and Deligne's conjecture Kontsevich and Soibelman

Duality in Khovanov Homology, On combinatorial link Floer homology and Following Lin: Expansions for Groups Hand-outs of talks I recently attended

All what I wanted to know about Langlands program and was afraid to ask and Generalised Monstrous Moonshine and Genus Two Conformal Field Theory

Music