For Current and Prospective PhD Advisees:

My Research

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, especially 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 my students.

Seminar

I run a seminar for students and postdocs.

Reading Courses

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 corrected.

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 (particularly commutative 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 commutative algebra. 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 background.

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