A. These days I mainly work on geometry of symplectic manifolds (in even dimensions) and contact manifolds (in odd dimensions), with an emphasis on low dimensions. I am particularly interested in quantitative questions about the dynamics of Reeb vector fields on contact manifolds, and symplectic embeddings of symplectic manifolds. To study these questions, I use and develop various forms of contact homology. Most of this development so far has been on "embedded contact homology" in three dimensions, which is related to gauge theory (Seiberg-Witten and Heegaard Floer homology).
Q. Where can I get more of an introduction to these research topics?
A. Here are some links to get started. You are also welcome to email me with questions, or we can discuss over Zoom.
Q. Are you accepting new PhD students?
A. Yes.
Q. What are the prerequisites for becoming your student?
A. I would say a solid understanding of algebraic topology and smooth manifolds, a bit of differential geometry, and strong interest in research areas that I might be able to help you with. Some skills in analysis and/or computer programming are a plus but not required. You don't need to be ready to get started working with me right now, as at Berkeley you don't have to be set with an advisor until the end of your second year. Also you might come in wanting to work with one advisor and end up working with another. The faculty in our department cover a very broad range of research areas, so there is a lot of flexibility here.
Q. What kind of research do your students do?
A. All sorts of things. Current and recent students have worked on foundations of contact homology including embedded contact homology, four-dimensional Seiberg-Witten theory, symplectic embedding problems, dynamics of Reeb vector fields and area-preserving surface diffeomorphisms, and more.
Q. How often do you meet with your students?
A. For one-on-one meetings, this depends on the student and on what stage they are at. Some like to discuss things frequently, while others like to work more independently. I also organize a weekly seminar for my students and anyone else who is interested. This is currently disrupted, but I plan to restart this soon in some online format, and if you are interested you are welcome to join my mailing list for this.