Thesis work
I am working on my doctoral thesis under Professor George Glauberman and my thesis will (if things go as planned) be about problems related to groups of prime power order. My topic proposal (a formal requirement before getting started on doctoral work) can be viewed here (PDF).
Here are some problem areas that I am or have been working on. I will put up notes on these areas once my ideas are in better shape. If you would like to know more about my research, please feel free to email me at vipul [AT] math.uchicago.edu.
Extending the Lazard correspondence to more cases: The Lazard correspondence is a correspondence between finite p-groups of small 3-local nilpotency class (i.e., any 3 elements generate a subgroup of class at most p - 1) and finite p-Lie rings of small 3-local nilpotency class. My work (not yet public) extends this to some 2-groups of class two and possibly some 3-groups of class three. This work is based on discussions with John Wiltshire-Gordon (an undergraduate at the University of Chicago) and with advice and suggestions from my adviser. I've also been taking help from MathOverflow for related questions.
Extending Jonah and Konvisser's work on counting abelian subgroups of p-groups: I currently have some very modest results that don't seem to have been published before. I am trying to formalize and simplify the proofs of the results I have and to obtain more results.
Some work on potentially characteristic subgroups: More updates on this later.
Other research
I collaborated on a paper on a research problem in computer science called the "evasiveness conjecture" with Laci Babai, Raghav Kulkarni, and Anandam Banerjee. The journal version of the paper is not yet completely ready, but a short version has been submitted and accepted at the STACS 2010 conference and can be accessed here. It is also available on the ArXiV here. Here are some slides on the talk prepared for the STACS conference by my collaborator Raghav.
Online contributions to mathematics
As a complement to research in mathematics, I am also interested in mathematical exposition and making mathematical content widely available in different ways. As part of such efforts, I started a Group Properties Wiki that I have been working on over the past three years (though it's collaborative, I am currently the main contributor). Other related wikis that I've started are Topospaces, The Topology Wiki and Commalg, The Commutative Algebra Wiki. The subject wikis reference guide provides a starting point to explore these subject wikis.
I am also a participant on Math Overflow. My user page is here.
Expository writings
Some miscellaneous articles I have written and have stored online are listed below. For the articles I wrote before coming to Chicago, refer the Write-ups and presentations section of my CMI page.