If you are not a PhD student at UIUC, admissions at UIUC are centralized, and I cannot comment on individual applications. You are, however, welcome to apply in the usual PhD application cycle.

If you are a PhD student at UIUC, please read the information below, and feel free to send me an email with some of your interests.

In order to succeed in a PhD with me, you must have a strong background in mathematics and statistics.
While you do not have to have the following background immediately upon embarking on research, the following references are most useful:

• Matrix Analysis (I learned from Horn and Johnson, but I like Bhatia's the best)
• High-dimensional probability (Vershynin's high-dimensional probability book) or statistics (Wainwright's high-dimensional statistics book)
• Spectral Methods for statistics (see this monograph)
• (Non)convex Optimization (e.g., gradient descent, KKT conditions for constrained optimization)
• Asymptotic theory (at the level of Van der Vaart's asymptotic statistics book)

I recommend students interested in a PhD with me be familiar with most, if not all, of these topics by their fourth year or so (depending on research interests).

Topics of interest
I have several topics that I am interested in, so it helps if you have a general idea of what you may be interested in working on. Fundamentally, my research focuses on any types of high-dimensional problems with matrix and tensor data. Potential areas of research are:

• Network Asymptotics and Hypothesis Testing (see this or this)
• Methodology for Mulilayer Networks (this,this, or this)
• Spectral methods (this,this, or this)
• PCA/Covariance Estimation (this or this)
• Optimization for matrices and tensors (e.g., this, but more coming soon!)
• Precise statistical guarantees for matrix-valued data (this or this, but more are coming soon!)

I am also open to students with ideas that may align with my interests. I have also recently developed an interest in exact asymptotics for high-dimensional regression and classification motivated by problems in machine learning.

Finally, I strongly believe that PhD students should learn to be good speakers. If you choose to do a PhD with me I will require that you give an informal talk (e.g., to other PhD students) once per year, and ideally 2+ talks at a conference like JSM. I am still figuring out how to implement this logistically.