Homepage of Apoorva Khare

Contact information

Teaching

In the Spring of 2010, I am teaching a section of Math 370: Fields and Galois Theory. Also of use is the reference: classesv2.

Relevant Publications

1. Category O over a deformation of the symplectic oscillator algebra. Journal of Pure and Applied Algebra, Volume 195 No. 2 (15 January, 2005), pages 131-166; math.RT/0309251. Here is the PDF without the errata.
   (The erratum appeared in the same JPAA, in Volume 199 (1 July, 2005), pages 319-320.)
2. Category O over the symplectic oscillator algebra, Ph.D. Thesis, University of Chicago, June 2006.
3. Quantized symplectic oscillator algebras of rank one, joint with Wee Liang Gan. Journal of Algebra Volume 310 No. 2 (15 April 2007), pages 671-707; math.RT/0405176. Here is the PDF.
4. Functoriality of the BGG Category O; Communications in Algebra Volume 37 No. 12 (December 2009), pages 4431--4475; arxiv:0811.2073 (math.RT). Here is the PDF.

Other publications

1. Divisibility Tests, Furman University Electronic Journal of Undergraduate Mathematics FUEJUM Volume 3 (1997), pages 1-5. PDF
2. Divisibility tests and recurring decimals in Euclidean domains, JP Journal of Algebra, Number Theory, and Applications JPANTA, Volume 7 No. 1 (February 2007), pages 1-32; preprint, math.NT/0406025. Here is the PDF.
3. Vector spaces as unions of proper subspaces; Linear Algebra and its Applications Volume 431 No. 9 (1 October 2009), pages 1681--1686; arxiv:0803.2746. Here is the PDF.

Accepted

1. I have been asked to write an entry, titled The BGG Category O, for the Encyclopedia of Mathematics (Michiel Hazewinkel, Editor), Kluwer Academic Publishers. (I have submitted the same.)
2. Center and representations of infinitesimal Hecke algebras of sl_2, joint with Akaki Tikaradze. Accepted for publication in Communications in Algebra; arxiv:0807.4776 (math.QA). Here is the PDF.

Preprints and work in progress

1. The sum of a finite group of weights of a Hopf algebra, preprint, math.RA/0701467 v2. Here is the PDF, that is currently being revised.
2. Drinfeld-Hecke algebras over cocommutative algebras, preprint, arxiv:0705.2067 (math.RA). Here is the PDF.
3. Axiomatic framework for the BGG Category O; preprint, arxiv:0811.2080 (math.RT). Here is the PDF.
4. Abelian groups as unions of proper subgroups, preprint, arxiv:0906.1023 (math.AC). Here is the PDF.
5. Faces of polytopes and Koszul algebras, joint with Vyjayanthi Chari and Tim Ridenour, in preparation.
6. Apoorva Khare and Akaki Tikaradze, in preparation.

Lectures and talks

1. Here are the transcripts of a talk I gave at various places in the Fall of 2008 - the slides.
2. Here are the transcripts of a talk I gave at Claremont McKenna College, at the 2008 Spring AMS Western Section Meeting, in May 2008 - the slides.
3. Here are the transcripts of a talk I gave at Louisiana State University, at the 2008 Spring AMS Southeastern Sectional Meeting, in March 2008 - the slides.
4. Here are the transcripts of a talk that I gave at the University of Arkansas, Fayetteville, at the 2006 Fall AMS Southeastern Sectional Meeting, in the Special Session on Combinatorial Representation Theory, in November 2006.
5. Here are the transcripts of a talk I gave in the conference in hono(u)r of J. Leposwki and R. Wilson, in NCSU, Raleigh, in May 2005 - here are the transparencies.

Other documents

1. Here is my topic proposal, titled Semisimple Lie algebras and their representations from 2002; I went through the text and the problems of Introduction to Lie algebras and representation theory (Graduate Texts in Mathematics, Volume 9, Springer Verlag), by James E. Humphreys.
2. I participated in the Directed Reading Program in Fall 2005. This was a one-on-one interaction with an undergraduate; I was paired with Philip Brunetti, and we studied Mobius functions on partially ordered sets. Here are some notes we made.
3. Here are some notes on various things, thanks to various people. They talk about simple modules and complete reducibility over products of algebras, and also about block decomposition using central characters.
4. Here is a crash course on algebras, coalgebras, bialgebras, Hopf algebras, and convolutions of weights. The course is taken by solving the exercises. Needless to say, any mistakes are mine.
5. Here is an explanation of the workings behind verifying the Diamond Lemma by Bergman, in a practical situation (the PBW Theorem for Lie algebras). No proofs given.

Quantum Gravity Seminar

I was taking (TeXing!) notes for the Cohomology and Quantization module of Professor John Baez's Quantum Gravity Seminar in 2006-07.

These notes are only a preliminary version; they have to be polished up, edited, and figures added. In the meantime, here they are. (Note that each new lecture will start on a new page, so it's convenient for "updating".)


1. Fall 2006
2. Winter 2007
3. Spring 2007