Xiaolei Zhao's website

PhD Students:

Joel Pion (Ph.D. 2026)
PhD Thesis: Categorical resolutions of singularities of type A_n
Chris Dare (Ph.D. 2025)
PhD Thesis: A Thurston compactification of stability manifolds for some local Calabi-Yau varieties
Qingjing Chen (Ph.D. 2023)

PhD Thesis: Equivalence between Kuznetsov components of cubic fourfolds and Gushel-Mukai fourfolds

Ethan Robinett (Ph.D. 2022, co-advised by Laura Pertusi)
PhD Thesis: Stability conditions on Kuznetsov components of Gushel-Mukai threefolds and Serre functor

To start learning algebraic geometry:

First you should read some classical algebraic geometry, like in the notes by Gathmann (link). This provides an idea of the foundational examples and methods in the area.
At the same time, you need some knowledge on commutative algebra: Atiyah & MacDonald's book (link) will suffice. This is usually covered in Math 220B.
After you feel comfortable with the previous steps, you can start learning the modern language of algebraic geometry: schemes, sheaves and cohomology. You can follow Vakil's notes (link) or Hartshorne's book (link).

If you are interested in algebra (algebraic geometry, algebraic number theory, representation theory, mathematical physics, etc.), you should plan on learning this as early as possible. Every year we have strong undergraduates finishing Math 237 successfully. For Ph.D. students at UCSB interested in working with me, it is important to finish this by the end of your second year.

We have a weekly Seminar on Geometry and Arithmetic, on topics related to algebraic geometry, number theory and etc. Students are strongly encouraged to regularly attend this as early as possible! Having difficulty following the talks? I recommend Vakil's "three things" exercise and other advice on going to talks.

To know my research:

In recent years, my research focuses on the study of moduli spaces, derived categories, and hyperkähler geometry. The main tool is stability conditions. To learn about this subject:

Derived categories in algebraic geometry (Huybrechts' book link).
Stability conditions: they were introduced in Bridgeland's papers (link and link), while the notes by Macrì-Schmidt (link) provide a good summary.

To do a reading course:

The way it works is that you pick a topic and read some books/notes/papers. Then you will lecture me and other colleagues on what you read. It should be well-prepared lectures with clear definitions, examples, results and proofs. We will shoot many questions and get better understanding through this process. You can choose a foundational theory:

Hodge theory: read Voisin's book (link).
Birational geometry: read Debarre's book (link) or Kollár-Mori (link).

Here are some topics close to my research:

K3 surfaces: read Huybrechts' book (link).
Hyperkähler geometry: read Huybrechts' book chapters (link) and Debarre's notes (link). This requires basic complex differential geometry.
Moduli of sheaves: my favorite are the books by Le Potier (link) and Huybrechts-Lehn (link).

You don't need the whole package of schemes and sheaves to read these topics. Quickly passing through the foundation and getting to research topics is part of graduate study. I will help you find a geodesic to the starting line of your reading.