I’ve been pretty busy lately with writing and researching with visitors. These announcements serve as a quick summary of what I’ve been up to:
1. Tables of the existence of equiangular tight frames (with Matthew Fickus). Today, there’s quite a bit known about equiangular tight frames (ETFs), but what is known seems to be scattered across different papers. This paper surveys everything that is known, and tabulates all of the known real and complex ETFs with sufficiently few vectors in sufficiently small dimension. The tables were generated by coding up existence theorems in MATLAB so as to minimize errors. This serves as a “solution” to problem 21 in this documentation of the open problems discussed at the AIM workshop Frame theory intersects geometry. Recently, Matt and I have made a few ETF discoveries with John and Jesse, so you can expect this table to be updated after we announce these discoveries in the coming months.
Continue reading Three Paper Announcements
I recently finished Nate Silver‘s famous book. Some parts were more fun to read than others, but overall, it was worth the read. I was impressed by Nate’s apparently vast perspective, and he did a good job of pointing out how bad we are at predicting certain things (and explaining some of the bottlenecks).
Based on the reading, here’s a brief list of stars that need to align in order to succeed at prediction:
Continue reading The Signal and the Noise
I’m pretty excited about Afonso‘s latest research developments (namely, this and that), and I’ve been thinking with Jesse Peterson about various extensions, but we first wanted to sort out the basics of linear and semidefinite programming. Jesse typed up some notes and I’m posting them here for easy reference:
Many optimization problems can be viewed as a special case of cone programming. Given a closed convex cone in , we define the dual cone as
Examples: The positive orthant and the positive semidefinite cone are both self-dual, and the dual of a subspace is its orthogonal complement. Throughout we assume we have , , closed convex cone , closed convex cone , and linear operator . We then have the primal and dual programs
Notice when and are both the positive orthant, this is a standard linear program. Further, considering the space of real symmetric matrices as , when and is the positive semidefinite cone, this is a standard semidefinite program. We consider several standard results (weak duality, strong duality, complementary slackness) in terms of the general cone program.
Continue reading Cone Programming Cheat Sheet
I’ve been on the job market full-time for the last 6 weeks or so, and I’ve finally settled on my destination: Starting this August, I’ll be a tenure-track assistant professor at The Ohio State University. My wife and I are very excited to move to Columbus!
I wanted to document my process for applying, interviewing and negotiating. I’ll probably refer to this blog post later when I give advice to a future PhD student or postdoc.
1. More offers make a better selection
My goal was to get the most attractive offer possible. Of course, different people have different notions of attractiveness, but there’s still an objective function to optimize. The main point is that you will be more satisfied if there are more options on the table. Not only are you maximizing over a larger set, the offers will compete with each other, and so you can auction for better offers. If you have more than two offers, it might be a little confusing how to maintain the auction — more on that later.
Continue reading Interviews and offers
Readers of this blog are probably already aware that Alexander Grothendieck died on Thursday. He is widely regarded as one of the most influential mathematicians in the twentieth century. Since his is not my field of study, I felt that now was a good time to learn a little about why he is so well regarded — I took the day to read a couple of articles from 10 years ago that provide an overview of his life, research, personality, and philosophies. I highly recommend the read: here and here.
Continue reading Alexander Grothendieck