Professor Cherry invites you to attend the Master's thesis defense of

Yingyu Zhang.

WHEN: Friday, July 30, 2021, at 10:00am

"Counting Plane Tropical Curves via Lattice Path in Polygons"

ABSTRACT:

The thesis being defended explores two important theorems about projective plane tropical curves. A projective plane tropical curve is a proper immersion of a graph into the real Cartesian plane subject to some conditions such as that the images of all the edges must be lines with rational slopes. Two important combinatorial invariants of a projective plane tropical curve are its degree d and genus g. First, we explore Gathmann and Markwig's approach to the the study of the moduli spaces of such curves and explain their proof that the number of projective plane tropical curves, counting multiplicity, passing through n = 3d + g -1 points does not depend on the choice of points, provided they are in general position. This number of curves is called a Gromov-Written invariant. Second, we discuss the proof of a theorem of Mikhalkin that allows one to compute the Gromov-Written invariant by a purely combinatorial process of counting certain lattice paths. This defense will focus on the second theorem and illustrate the combinatorial lattice path technique for explicitly computing the Gromov-Witten invariants.