Chile can't keep my sister away: Love across the Skype-equator.

Alyson Lin Burchardt

Hello! This is my webpage, last updated September 19, 2012.
I'm a 5th year Mathematics PhD student at Brandeis University, studying geometric topology with Daniel Ruberman. My thesis work involves extending the research of Chuck Livingston (IU) and Paul Kirk (IU) in the calculation of the Hausmann-Weinberger 4-manifold invariant of free abelian groups to the class of right-angled Artin groups.
Office: Goldsmith 118b, Department of Mathematics, Brandeis University
Email address: aburchar "at" brandeis "dot" edu


Research Interests: Algebraic and Geometric Topology

I've been known to read/think/learn/talk about knots, braids, surgery theory, K(G,1) spaces, Heegaard-Floer theory, Combinatorial Heegaard-Floer theory, and 4-manifolds.


Teaching Experience

MATH 10a Differential Calculus log-on required (Summer 2012)
MATH 10b Integral Calculus log-on required (Spring 2011)
MATH 10a Differential Calculus log-on required (Fall 2010)
MATH 10a Differential Calculus log-on required (Spring 2010)
MATH 5a Precalculus log-on required (Fall 2009)


Talks I've given in the past

An introduction to the classification of 4-manifolds and Kirby calculus.

4/3/12 and 4/17/12, Brandeis Topology Seminar

The Hausmann-Weinberger 4-manifold invariant for right-angled Artin groups.

3/31/12, Indiana Graduate Student Topology Conference

An introduction to calculating the Hausmann-Weinberger 4-manifold invariant for right-angled Artin groups.

11/17/11, Graduate Student Seminar

On the Tristram-Levine signatures of a knot.

10/18/11, Brandeis Topology Seminar

Algebraic obstructions to concordance.

10/4/11, Brandeis Topology Seminar

An introduction to Spectral Sequences.

11/4/10, Graduate Student Seminar

Abstract: I'll be giving an introduction to spectral sequences from the (co)homological standpoint, showing how they are useful for computing homology from a chain complex, especially in the case where our chain complex perhaps carries extra structure besides just a boundary map. Such cases arise with filtered chain complexes, double complexes, exact couples, etc. Once we go through the introduction and purpose of spectral sequences I hope to use the theory with double complexes to give a cool proof of the Snake Lemma, and I promise no diagram chasing will be necessary! I think this will be especially accessible to first year graduate students, since only knowledge of chain complexes and exact sequences are really necessary.

Some knots with surgeries yielding lens spaces.

4/26/10, Second-year-seminar

Abstract: John Berge wrote an unpublished paper on knots with surgeries yielding lens spaces in which he characterizes these knots in a genus 2 Heegaard splitting of S^3. He conjectured that all knots that have surgeries yielding lens spaces are of this characterization. He also gives a method for finding the exact lens space resulting from surgery of a particular knot, given that we know the homology class the knot represents in the Heegaard splitting. I'll give some introductory definitions/explanations on handlebody decompositions, Heegaard splittings, 0- and 1-bridge knots, and Dehn surgery before I start to talk about the theorems in the paper.

Braid Groups on Surfaces and the Fadell-Neuwirth Fibration.

4/19/10, Graduate Student Seminar

Abstract: I'll give a brief introduction to the braid group as the fundamental group of a configuration space (which normally is taken to be D^2 with n punctures), but will generalize it to braids on an arbitrary orientable surface M of genus g. Fadell and Neuwirth form a short exact sequence from the fibration of the n^th configuration space to the (n-1) configuration space, and the existence of sections leading to bijective maps from the pure braid group to a semi-direct product of free groups depends on the surface M, genus, and number of punctures n. From this we can see that the pure braid groups are residually torsion-free nilpotent.

An introduction to the Braid Group.

2/9/10, Brandeis Topology Seminar


Conferences

Indiana Graduate Student Topology Conference, Indiana University, March/April 2012
Homological Invariants in Low-Dimensional Topology Workshop, Simons Center for Geometry and Physics, June 2011
Low-Dimensional Topology and Categorification Stony Brook University, New York, June 21-25, 2010.