I am a 5th year mathematics graduate student at Stanford University. My advisor is Ralph Cohen. I will be graduating in the spring of 2012. I am interested in algebraic topology and symplectic geometry. More specifically, I am interested in using algebraic topology to study various moduli spaces.
My thesis topic:
Segal studied the relative homotopy type of the space of holomorphic maps from a complex curve to a complex projective space inside of the space of all continuous maps. He proved that the inclusion map is a homology equivalence though a range tending to infinity as degree increases. I am investigating if Segal's theorem is still true if one considers general almost complex structures on the target. I have proven that the inclusion map is a homology surjection through a range tending to infinity with degree in the case when the curve is genus 0 and the target is CP^2. The main idea of the proof is to use ''automatic transversality'' as well as gluing. One of the key insights is to view gluing as an operation and consider the induced operations on homology, instead of just viewing it as a converse to Gromov compactness.
Homological stability properties of spaces of rational J-holomorphic curves in P^2
Email:
jkmiller at stanford dot edu
My resume:
Student Topology Seminar:
Also know as the no progress seminar.