## Departmental PhD Thesis Exam: Jihyeon Jessie Yang

Everyone welcome.  There will be refreshments in the Math Lounge
before the exam.

DEPARTMENTAL PHD THESIS EXAM

Wednesday, April 25, 2012, 2:10 p.m., in BA 6183, 40 St. George St.

PhD Candidate:  Jihyeon Jessie Yang

Thesis Title:  Tropical Severi Varieties and Applications
(http://www.math.toronto.edu/jyang/PhDThesisJYANG.pdf)

Thesis Abstract:

The main topic of this thesis is the tropicalizations of Severi varieties,
which we call tropical Severi varieties. Severi varieties are classical
objects in algebraic geometry. They are parameter spaces of plane nodal
curves. On the other hand, tropicalization is an operation defined in
tropical geometry, which turns subvarieties of an algebraic torus
into certain polyhedral objects in real vector spaces. By studying
the tropicalizations, it may be possible to transform algebro-geometric
problems into purely combinatorial ones.  Thus, it is a natural question,
"what are tropical Severi varieties?" In this thesis, we give
a partial answer to this question: we obtain a description of tropical
Severi varieties in terms of regular subdivisions of polygons. Given
a regular subdivision of a convex lattice polygon, we construct an
explicit parameter space of plane curves. This parameter space
is much simpler object than the corresponding Severi variety and it is
closely related to a flat degeneration of the Severi variety, which in
turn describes the tropical Severi variety.

We present two applications. First, we understand G. Mikhalkin's
correspondence theorem for the degrees of Severi varieties in terms
of tropical intersection theory. In particular, this provides a proof
of the independence of point-configurations in the enumeration
of tropical nodal curves. The second application is about Secondary fans.
Secondary fans are purely combinatorial objects which parametrize all
the regular subdivisions of polygons. We provide a relation between
tropical Severi varieties and Secondary fans.