Algebraic Gluing of Holomorphic Discs in K3 Surfaces and Tropical Geometry

Speaker: 

Yu-Shen Lin

Institution: 

Harvard CMSA

Time: 

Monday, October 30, 2017 - 4:00pm

Location: 

RH 340P

We will start from the motivation of the tropical geometry. Then
we will explain how to use Lagrangian Floer theory to establish the
correspondence between the weighted counting of tropical curves to the
counting of holomorphic discs in K3 surfaces. In particular, the result
provides the existence of new holomorphic discs which do not come easily
from direct gluing argument.

Vector bundles and A^1-homotopy theory

Speaker: 

Marc Hoyois

Institution: 

USC

Time: 

Monday, March 12, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

The study of vector bundles on algebraic varieties is a classical topic at the intersection of geometry and commutative algebra. In its algebraic form it is the study of finitely generated projective modules over commutative rings. There are many long-standing conjectures and open questions about algebraic vector bundles, such as: is every topological vector bundle over complex projective space algebraic? In recent years, there have been a number of significant developments in this area made possible using the A^1-homotopy theory of algebraic varieties introduced by Morel and Voevodsky in the late 90s. The talk will provide some background on such questions and discuss some recent joint work with Aravind Asok and Matthias Wendt.

Subgroups of the mapping class group via algebraic geometry

Speaker: 

Nick Salter

Institution: 

Harvard University

Time: 

Monday, November 27, 2017 - 4:00pm

Location: 

RH 340P

This talk will be a discussion of some interesting and novel subgroups of the mapping class group that arise via algebro-geometric constructions. Our talk will focus on the special case of how the theory of plane algebraic curves (essentially just polynomials in two variables!) interacts with the mapping class group in subtle ways. The motivating question can be formulated simply as, ``which mapping classes (of a surface of genus g) arise as one-parameter families of polynomials in two variables?’’ Perhaps surprisingly, the answer turns out to be ``either none at all, or else virtually all of them”. No familiarity with algebraic geometry will be assumed. 

Localizing the Fukaya category of a Stein manifold

Speaker: 

Sheel Ganatra

Institution: 

USC

Time: 

Tuesday, November 28, 2017 - 4:00pm

Location: 

RH 306

We introduce a new class of non-compact symplectic manifolds called
Liouville sectors and show they have well-behaved, covariantly functorial
Fukaya categories.  Stein manifolds frequently admit coverings by Liouville
sectors, which can be used to understand the Fukaya category of the total
space (we will study this geometry in examples). Our first main result in
this setup is a local-to-global criterion for generating Fukaya categories.
Our eventual goal is to obtain a combinatorial presentation of the Fukaya
category of any Stein manifold. This is joint work (in progress) with John
Pardon and Vivek Shende.

The realization problem of prism manifolds

Speaker: 

Yi Ni

Institution: 

Caltech

Time: 

Monday, October 9, 2017 - 4:00pm

Location: 

RH 340P

Prism manifolds are spherical 3-manifolds with D-type finite fundamental
groups. They can be parametrized by a pair of relatively prime integers p>1
and q. The realization problem of prism manifolds asks which prism manifolds
can be obtained by positive Dehn surgery on a knot in S^3. This problem has
been solved in the cases q<0 and q>p. We will discuss the basic idea of the
proof. This talk is based on joint work with Ballinger, Hsu, Mackey, Ochse
and Vafaee, and with Ballinger, Ochse and Vafaee.

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