For Current and Prospective PhD Advisees:
My research covers a range of topics in algebraic geometry. Much of my
research deals with geometric aspects of string theory and related areas of
physics, or with algebraic geometry questions inspired by string theory.
Currently (2018) I am working most on modern enumerative theories for Calabi-Yau threefolds,
Donaldson-Thomas-type theories incluing stable pairs, moduli of sheaves, and
their motivic variants, the role of modular forms in the associated generating functions
(topological string partition function),
F-theory in physics
(elliptically fibered Calabi-Yau manifolds in algebraic geometry), and geometric aspects of supersymmetric gauge theories.
Supervision of PhD students
I advise students in both stringy topics and in algebraic geometry proper.
I expect PhD advisees to have a solid foundation
in algebraic geometry in any case. I also expect to be able to learn from
I run a seminar for students and postdocs.
I am always willing to supervise reading courses while students are
acquiring a foundation in algebraic geometry. I prefer working with
groups of students, both for efficiency's sake and because it is
usually better for students to have other students to learn with. The
book that students interested in my research have read most often is
my book Enumerative Geometry and String Theory, which was
written for advanced undergraduates. If you read the book, be warned
about typos and check out the errata page where the typos are
Background: Books and Courses
There are many ways to acquire the background in algebraic geometry that
I am looking for in PhD students. Before listing some paths, I want to
emphasize that algebraic geometry does not exist in isolation, and it is
also important to gain familiarity with related fields, including algebra
algebra), topology, and differential geometry. Students interested in
working on "physical" algebraic geometry would do well do gain familiarity with
geometric physics, e.g. Yang-Mills theory or string theory.
Hartshorne's book Algebraic Geometry is a classic,
covering algebraic geometry via
the theory of schemes. Students who are well-versed in this book, with their knowledge
supplemented by examples from classical algebraic geometry, are generally
ready to begin
research with me in algebraic geometry. Many of the important ideas in the book are contained
in the exercises. I encourage all students interested in mastering algebraic
geometry to try to do all of the exercises (but be warned that some
exercises are unsolved problems).
Forming study groups and working together is an efficient way to do this.
Students will do best with this book if they already have facility with
Hartshorne is usually used for Math 512, Modern Algebraic Geometry, at least
when I teach it. Some students have told me that they prefer Ravi Vakil's notes
A good and more introductory book is
Shafarevich, Basic Algebraic Geometry. The first part of the book
provides an introduction
to classical algebraic geometry (with no schemes) and contains many beautiful
examples of classical constructions. I use this book when I
teach Math 511, Introduction to Algebraic Geometry, which is offered as
a comps course every spring.
Another good introductory book is
Miranda's Algebraic Curves and Riemann Surfaces. This is algebraic
geometry over the complex numbers, in complex dimension 1, from both the algebraic
and transcendental points of view. I use this book
when I teach Math 510, Riemann Surfaces and Algebraic Curves, which is
offered as a comps course every fall.
Griffiths and Harris, Principles of Algebraic Geometry,
covers complex manifolds and algebraic geometry over the complex numbers.
Math 514, Complex Algebraic Geometry is taught every two years, next
in Fall 2018. I usually use Griffiths and Harris when I teach the
course but I sometimes use Claire Voisin's Hodge Theory book.
In addition to the core curriculum, reading courses in algebraic geometry
are often offered. You are encouraged to take advantage of the opportunities
presented by these (typically) advanced courses whenever you have the
Reading Griffiths and Harris, and Shafarevich
(or taking the appropriate courses), supplemented by
facility with the language of schemes (either from e.g. Hartshorne or later chapters of
Shafarevich) will prepare students for doing research with me. I didn't
include Miranda on this list only because
Griffiths and Harris contains many of the ideas from Miranda; but reading
Miranda's book could help many students prepare to study Griffiths and Harris.
Former PhD students with PhD year, with current locations when known
- Martha Waggoner 1994, Simpson College
- Thomas Zerger 1996, Saginaw Valley State University
- Artur Elezi 1999, American University
- Mutaz Al-Sabbagh 2002, University of Damman, Saudi Arabia
- Jonathan Cox 2004, SUNY Fredonia
- Xinyun Zhu 2005, University of Texas, Permian Basin
- Joshua Mullet 2006 (co-advised with Dan Grayson), Ohio National Financial
- Josh Guffin 2008, Energy Solutions Forum Inc.
- Mehmet Sahin 2009
- Yong Fu 2010
- Artan Sheshmani 2011 (co-advised with Tom Nevins),
Centre for Quantum Geometry of Moduli Spaces, Aarhus University, Denmark
- Jinwon Choi 2012, Sookmyung Women's University, Korea