Niranjan Balachandran

I am currently a Harry Bateman Research instructor in Mathematics at Caltech. I did my Phd in Mathematics at The Ohio State University under the supervision of  Dr. Dijen Ray-Chaudhuri. I also hold a Bachelors(B. Stat.(Hons)) and Masters degree(M. Stat.) in Statistics from The Indian Statistical Institute (Kolkata and Bangalore), India.

Here is my CV. And here is a summary of my research.

Research Interests: Combinatorics. My current research interests include Extremal Combinatorics, Probabilistic methods, and Design theory. I have more recently become very interested in the applications of Probabilistic methods in Combinatorics especially in Graph-theoretic applications such as coloring problems.

TeachingI taught Math 121 during Fall 09. I taught the same course in Fall and Spring in 2008 as well. I also taught Ma/CS 6b  during Winter 09 and Winter 2010 though I covered different topics. As for Math 121c,  I taught a course in Design theory in 2009, and a course on Probabilistic methods in Combinatorics in 2010. Here is a link to lecture notes (This should see some changes soon as some parts of it are not complete) from this course, thanks to the students of this course who consented to act as scribes. The nature of the notes is attempt to solve the corresponding 'Exposition Problem' (see this blog of Gowers for a definition of this term, not to mention a fantastic exposition of Razborov's magical theorem!) for the Probabilistic method. Two important features of this course that are not covered in other texts/lecture notes are the result of Pippenger and Spencer on chromatic index of uniform hypergraphs, and Jeff Kahn's asymptotic solution to the Erdős-Faber-Lovász conjecture. This is my Teaching philosophy.
Check the Teaching page for more information. Also, see an update on the upcoming course for Winter, Ma/CS6b.

Some of my talks:
  A new proof of Assmus-Mattson Theorem on Designs from Codes by discrete Harmonics(04/11/02) (Seminar - Combinatorics Seminar, The Ohio State University)
 An upper bound for K-intersecting families(10/10/02) (Seminar - Combinatorics Seminar, The Ohio State University)
 Rooted Forest Set Systems and Steiner Designs.  (2005 AMS Spring Eastern Sectional Meeting, Newark, DE, April 2-3, 2005)
 Steiner 3-designs: Results Old and New (28th Ohio State - Denison Mathematics conference, May 19-21, 2006)
 Simple $3$-designs and $PSL(2,q)$, $q\equiv 1\pmod 4$ (Wright State University, Dayton, OH, November 6, 2006). Additional results in this direction were later presented at the Nanyang Technological University, Singapore and National University of Singapore (NUS) on the 26th and 27th of September, 2007, respectively.
 Infinite families of Steiner $3$-designs with block size $6$  (44th MIdwestern GrapH TheorY conference, Wright State University, Dayton OH, May 12, 2007)
 A $ \lambda$ -Large Type Theorem for Candelabra Systems: ( OSU-Denison 29 conference), The Ohio State University, Columbus, May 17, 2008.
 Towards a Lambda-large Theorem for Candelabra Systems: At the 5-day workshop on Combinatorial design theory at The Banff International Research Station (BIRS), November 13, 2008, Banff, Canada.
 Graphs with Bounded degree and matching size: Combinatorics Seminar, Caltech, Pasadena, December 4, 2008.
 Simple Forbidden Configurations and Steiner designs: Combinatorics Seminar, Caltech, Pasadena, May 11, 2010.
Some coloring and packing problems for Graphs: 3 SURFs of Summer 2010, Combinatorics Seminar, Caltech, Pasadena, October 19, 2010.
Forbidden configurations, extremal set systems, and Steiner designs, Mathematics Seminar, Indian Institute of Sciences (IISc), Bangalore,  December 14, 2010.

  Simple 3-designs and PSL(2,q) (Jointly with Dijen Ray-Chaudhuri) : Designs, Codes, and Cryptography, Volume 44 (2007) .
 Graphs with restricted valency and matching number (jointly with Niraj Khare) :  Discrete Math., Volume 309, Issue 12, (2009).
 New infinite families of Candelabra systems with block size 6 and Steiner designs J. Comb. Inform. Sys. Sci., Volume 34, (2009).
A transform of complementary aspects with applications to entropic uncertainty  (with P. Mandayam,  S Wehner) : 
J. Math.Phy.,
Volume 51, Issue 8,(2010). Also here (online journal copy).
Forbidden Configurations and Steiner Designs : submitted.
A lambda-large theorem for Candelabra systems, preprint.
A Short Proof of Brooks' theorem for List Colorings, preprint.
 Baranyai's theorem: An exposition, in preparation.

On optimal planar packings of some trees, (with E. Cohen), in preparation.

On a list coloring problem for graphs embedded on the torus, in preparation.

SURF Mentoring:
Over the past two years, I have had the occasion of mentoring undergrad students for a Summer Research Fellowship (SURF). I mentored one student in 2008, and 4 in 2009.

Other interesting sites (Combinatorial):
The Electronic Journal of Combinatorics.
MathWorld (a quick reference in case you forget the definition of something!)
Design Theory : A very interesting site for design theorists.
Noga Alon's papers:  I find most of his papers extremely well organised, lucidly presented and very interesting combinatorially-especially some of his survey papers.
Tim Gowers:  Apart from his own very exciting material, he also has a link to some very interesting webpages of equally interesting Combinatorists!
Terrence Tao's Blog: What can I say - Everything he writes is fantastic!
The Papers of Paul Erdős: The Renyi Institute seems to be doing a great service to all Combinatorists by having an online repository of all(all? well, perhaps not all!) the papers of Paul Erdos. As far as I see, the list looks pretty comprehensive till 1989. 
Douglas West' Open Problem Page.
How to give a talk: I saw this link on the homepage of Paul Nevái; I have tried to find the original article but all the links I have found online seem dead or lead elsewhere. This is apparently based on a talk delivered by Gian-Carlo Rota on the occasion of the Rotafest in April, 1996.

Other Links:
Cricket: I think this is a vastly superior game to baseball, though my American friends may not agree. 

Other (Non-mathematical) Interests:
: I have now moved samples of my work here
My Blog: I wrote more regularly at one point of time. Some of them were just random rants/musings. But some of them were written purely for form and structure rather than content.