For prospective graduate students

1. Coursework: Before I engage in any capacity with direct supervision, I would expect that you have passed the preliminary exams, taken 202AB and have taken or are intimately comfortable with the material of 206 (also 208 and/or 209 would be ideal). The best possible preparation would be to go through Chapters I through IX in the book ``A Course in Functional Analysis'' by John B. Conway, and independently do all the exercises in them. I expect this to take one summer (could be good to consider after completing the 202AB sequence in your first year). 

2. Disclaimer: While I am interested in taking on new PhD students, it is not always a certainty that I will accept a new student as this may depend on various factors. Also each student various and there is no fixed format or model for supervision that I follow. One common intermediary step which I will likely recommend is to do a reading course on an advanced topic with me. 

3. Seminar: As a student interested in this field, I consider it important to attend the operator algebras research seminar (Tuesdays 3:30PM) weekly, regardless of how advanced you are. 

4. Breadth: Outside of research and reading credits, I strongly advice to take a wide variety of graduate courses in the department (at least one or two every semester), particularly including the following (in no particular order) which are highly relevant to operator algebras: 205, 214, 241, 250AB, 261AB, 240, 229,  215AB, 218AB, 258, 235AB.

My main area of research is in the field of C* and von Neumann algebras, which is a subfield of operator algebras. Jesse Peterson has a substantial collection of resources for books to read in the field. However I isolate two advanced ones below, that are very relevant to my own particular interests. It is essential to obtain these either online or in print, and begin reading them carefully. 

1. ``C*-Algebras and Finite Dimensional Approximations'' by N. Brown and N. Ozawa. 

2. ``An Introduction to II_1 Factors'' by C. Anantharaman and S. Popa. 

Some cornerstone topics that I like include: subfactors, free entropy, deformation rigidity, amenable actions, strong convergence, noncommutative L^p spaces, etc.

Here are some lists containing open problems of interest to me. These may be out of date, so it's best to consult with me or other experts regarding the status of a problem before working on them. 

1. Schafhauser-Tikuisis-White's 99 open problems on the structure of nuclear C*-algebras: https://arxiv.org/pdf/2506.10902

2. My 50 open problem list on ultraproducts of II_1 factors: https://arxiv.org/abs/2511.20377

3. J, Peterson's open problem list in operator algebras: https://math.vanderbilt.edu/peters10/problems.html

4. Hannes Thiel's problem list: http://hannesthiel.org/category/open-problems

Before embarking on this journey, it might be helpful for you to read this. These points may appear idealistic (and they are to a large extent), but are beneficial to strive towards. To be clear, I have picked up most of this from watching my former PhD advisor. I disclose that I at many points in my career so far I've struggled with understanding these points and have behaved in contradictory ways. In hindsight I felt it was detrimental to my growth as a person and as a mathematician whenever I veered away from these ideals. 

1. Trust: I believe that mathematics is a great equalizer. It will fully reveal its beauty and greatly empower any human being who is honest, patient, disciplined and dedicated. As you progress in your mathematical journey, it is very likely that this joy will become larger and larger and completely consume you, let it! Everyone has their own unique and beautiful relationship with mathematics. Throughout your journey, try your best to not let negativity (in the form of competition, peer pressure, job market, rejections, failure etc) obscure the core feeling of bliss that you experience when a piece of beautiful mathematics has revealed itself to you. My former PhD advisor always used to tell me: "focus only on developing your mathematics, and everything else will take care of itself". It's hard to process, but I think he is right. 

2. Community: I think it is a great idea to make a lot of friends, and be willing to talk and listen to everyone. Treat everyone else with compassion, and share in their happiness. I believe that mathematics can inspire us to be better to each other. Just like how it takes time to fully grasp a theorem, give a lot of time and chances for the people around you. Be fully honest with yourself and each other, much like how you would with math. Be invested in your community, and try to find ways make yourself and everyone else happy and excited to be doing mathematics. Most importantly, I would recommend to enrich and enlarge your life outside mathematics. While mathematics is truly extraordinary, you only have one life and there are many other beautiful things to experience. 

3. Health: Your ability to succeed in this profession, much like every other one, crucially relies on your mental and physical health. Navigating life is difficult for everyone, so I highly recommend that you seek out advice and help especially from people who are older and more experienced. Never feel ashamed to do this. Try to get your annual physical checkups done, and try to follow your doctor's advice. Mental health is equally, if not more, important. Take care of yourself first. Make sure you are financially stable, and are living in a safe and secure condition. 

4. Work: Having mentioned all of the above, it is very important to stress that pursuing a career in mathematics demands an intense amount of hard work. Have a systematic routine everyday and keep track of your progress. Try to prioritize learning deep new theorems every day, read a lot of papers with honesty and care. Knowing one theorem in full and intimate details is much better than knowing 100 theorems in a wishy washy way. Try to focus on solving problems that other mathematicians have asked and have worked on. This not only exposes you to the current trends in mathematical research, but also will help you seek out collaborators and pursue a collective mathematical vision. If you prove a theorem or even an exercise, write it down in excruciatingly many details. This will help build technical strength which will be crucial in future research. Aside from working on other people's questions, try to ask and pursue questions that you genuinely find are aesthetically pleasing. Do not compete with other mathematicians, just work on your own pace and aim for perfection. If you cite or need a theorem of someone else in your work, make sure you fully read what you need and understand all the mathematics behind it. In mathematics, reinventing the wheel is the only way you can make sure you are being honest, and being honest is the only way to make sure you are making meaningful progress in our collective endeavour as mathematicians.  

Additionally, please seriously consider the following:

Sacrifice:  As a graduate student, it is likely that you will spend at least five years in your mid twenties engaged in deep work that requires a tremendous amount of patience and dedication. Note that in these years, you will not be earning a sizable income, and you might additionally experience social and peer pressures. Further on in your career, it is likely you will have to move several times for postdocs and early career positions. These moves will likely involve several sacrifices and might force you to have a later start in a stable life than usual. However an academic job offers you with an enourmous amount of freedom and flexibility and this might be worth the effort for some. I recommend the reader to also read Greg Hjorth's advice which concerns the statistical side of things, which is very relevant.