I am starting this webpage to maintain and update information regarding my research group and the Algorithms group at BU.
Disclaimer: At all times, these pages are a work in progress and are not necessarily in sync with my current opinions or beliefs.
If you are interested in doing research with me, you should consult the following pages for some of the background knowledge that you may find useful:
See this webpage for milestone of the PhD program at BU. Alina and Lorenzo are in the process of redesigning the in-depth oral exam in the area of Algorithms. We will post here what the requirements for the new exam will be.
Things I am trying to learn
My approach to research (and the way I like doing it) is to learn some new math and connect it with techniques I already know about to make progress on important problems. Here are some areas that I am trying to learn about:
- Polynomial Optimization: while this is hard in general, there are very interesting classes of relaxations that have beautiful connections to algebra and algebraic geometry. See Pablo Parrillo’s course at MIT. For a quick intro, you can also watch the Simons bootcamp lectures. For this, my goal is not really to absorb all the material but to get a working knowledge of the methods (e.g. SOS relaxations) to allow me to combine them with techniques I understand, say iterative methods, to get some interesting results. There are lots of open problem that I think could be tackled with this approach, including within the next bullet.
- Discrepancy Theory: Lots of fundamental open questions at the intersection of combinatorics, geometry and optimization. All the constructive results have a strong iterative flavor, but the correct underlying formulations are not clear. This blog post is a good place to start.
- Analytical Mechanics: I have always found the multitude of online and offline iterative algorithms confusing. I have worked on a simpler primal-dual interpretation of these methods based on discretizing natural continuous trajectories. While this is nice and helps a lot with intuition, I think more insights can be gained by relating this picture with the physical picture coming from Analytical mechanics and, in particular, by using the Hamiltonian formalism. This has natural connections with convex optimization but also introduces additional geometric structures thanks to the underlying symplectic structure. Where to start? Look up all these words on Wikipedia: Lagrangian mechanics, Euler-Lagrange Equations, Hamiltonian mechanics. A college-level classical mechanics textbook is also good to learn the basics of Hamiltonians.