Homepage of Apoorva Khare

Contact information

Conference Organization

• Lie Theory Workshop on Quantum Groups, Stanford University, February 4-5, 2012. Co-organized with Daniel Bump (Stanford) and Anne Schilling (UC Davis).
• Young Researchers Meet (Mathematics and Computer Science), Stanford University, May 26-27, 2012. Co-organizing with Govindan Rangarajan (Indian Institute of Science, India).

Publications

1. Divisibility Tests, Furman University Electronic Journal of Undergraduate Mathematics FUEJUM Volume 3 (1997), pages 1-5. PDF
2. 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.)
3. Category O over the symplectic oscillator algebra, Ph.D. Thesis, University of Chicago, June 2006.
4. Divisibility tests and recurring decimals in Euclidean domains (final section with Pieter Moree), JP Journal of Algebra, Number Theory, and Applications JPANTA, Volume 7 No. 1 (February 2007), pages 1-32; math.NT/0406025. Here is the PDF.
5. 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.
6. Vector spaces as unions of proper subspaces; accepted for publication in Linear Algebra and its Applications Volume 431 No. 9 (1 October 2009), pages 1681--1686; arxiv:0803.2746. Here is the PDF.
7. 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.
8. Center and representations of infinitesimal Hecke algebras of sl_2, joint with Akaki Tikaradze. Communications in Algebra Volume 38 No. 2 (February 2010), pages 405--439; arxiv:0807.4776 (math.QA). Here is the PDF.

Accepted

1. Faces of weight polytopes and a generalization of a theorem of Vinberg, joint with Tim Ridenour, accepted for publication in Algebras and Representation Theory; arxiv:1005.1114 (math.RT). Here is the DOI and the PDF.
2. Faces of polytopes and Koszul algebras, joint with Vyjayanthi Chari and Tim Ridenour, accepted for publication in Journal of Pure and Applied Algebra; arxiv:1105.2840 (math.RT). Here is the DOI and the PDF.
3. 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.)

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.
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. The covering problem for Chinese rings, preprint, arxiv:0906.1023v2 (math.AC). Here is the PDF.

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.
6. Here are some notes on a few basic identities by Weyl and Macdonald.
7. When is every complex irreducible character of a compact group G a nonvanishing function of G? For finite groups, this is an old result by Burnside that uses algebraic integers - as does his Solvability Theorem.
8. A few notes on translation-invariant rational functions in finitely many variables over a finite field, that I wrote up while teaching Galois Theory at Yale in Spring 2010.

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