Niranjan Balachandran
About me: I am a Phd
student in the
Mathematics department at The Ohio State University. My academic
advisor is Dr. Dijen Ray-Chaudhuri.
Education: I hold a
Bachelors(B. Stat.(Hons)) and Masters degree(M. Stat.) from The Indian
Statistical Institute (Kolkata and Bangalore),
India.
Research Interests: Combinatorics-specifically,
Design Theory, Finite geometry and Extremal Combinatorics. I also
enjoy the forays into Probabilistic methods in Combinatorics.
Design Theory: The Holy Grail
of Design Theory is the problem of existence of designs. A t-design on
a set X of v points with block size k is simply a set system of certain
subsets of size k such that every set of size t of X is contained in
the same number of blocks(called the replication number, conventionally
denoted by the greek letter lambda). When this replication number is 1,
the design is called a Steiner
System. One also considers designs with
repeated blocks(here we regard the design as a non-negative integer
-valued function) which are not of much interest to the
combinatorist/design theorist(though they are of value to the
Statistician).
The existence of a simple design( a design with non repeated
blocks) is equivalent to the existence of a 0-1 solution vector to an
appropriate matrix equation. While the general problem in this setting
is computationally hard(NP complete), this problem has its special
features and there is a lot of heuristic cause as to the general belief
in the existence of Steiner designs. For instance, the existence
problem for 2-designs has a very
satisfactory `asymptotic' theory(R.M.Wilson). There have been some
recent very interesting developments settling the "v-large" existence
problem for Orthogonal Arrays (equivalently, Transversal Designs)( J.L.
Blanchard). The corresponding problem for minimal covers has the
'correct' limiting
number of blocks(conjectured by Hanani-Erdős), i.e., if M(v,k,t)
denotes the number of blocks in a minimal cover for all the t-sbsets of
a v-set, then M(v,k,t)/(vCt/kCt) approaches 1 as v tends to infinity(
here vCt is the binomial coefficient)(Rödl and later by Jeff Kahn).
Talks:
Let $K$ be a subset of $k$ positive integers. By a
$K$-intersecting
family $\mathcal{F}$, we mean a family of subsets of $S=\{1,2,...,n\}$
such that for
any $A,B$ in the family $\mathcal{F}, |A\cap B|$ is an element of $K$.
The purpose
of the talk is to present a result due to Snevily: for such a family
$\mathcal{F},
|\mathcal{F}\leq$ the sum of the binomial coeffs $\binom{(n-1)}{i}, 0
\leq i\leq k$.
This generalizes an earlier proved result for the set $K=\{1,2,...k\}$.
3.
Rooted Forest Set Systems and Steiner Designs.
(2005 AMS Spring Eastern Sectional Meeting, Newark, DE, April 2-3, 2005)
4.
Steiner 3-designs: Results Old and New (28th Ohio State -
Denison Mathematics conference, May 19-21, 2006)
5. 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.
6. Infinite families of Steiner $3$-designs with block size $6$
(44th MIdwestern
GrapH TheorY conference, Wright State University, Dayton OH, May
12, 2007)
(*: these are not original work)
Papers:
1. Simple
3-designs and PSL(2,q) (Jointly with Dijen Ray-Chaudhuri) :
This paper appeared in the special volume of Designs, Codes and
Cyptography on the occasion of Dan Hughes' 80th birthday. You
can download the paper from here.
2. Graphs
with restricted valency and matching number (jointly with Niraj
Khare) : Submitted to Discrete Math, 2006.
3. New
infinite families of Candelabra systems with block size 6 and stem size
2 : Submitted to J. Comb. Designs, 2007.
4. A lambda-large theorem for Candelabra systems : In preparation.
Other preprints:
1.
Other
interesting sites
(Combinatorially):
EJC:
The Electronic Journal of Combinatorics.
ArXiV
MathSciNet
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: One of the great combinatorists of our times. I
find most of his papers extremely well organised, lucidly presented and
very interesting combinatorially-especially some of his survey papers.
Tim
Gowers: Another of those greats. Some of his survey articles in
this link are truely fantastic-especially the one titled, "The two
cultures of Mathematics". Besides being a mathematician of his calibre,
I find Gowers to be an extremely articulate and brilliant writer.
Coming from a Fields' medalist, people are bound(nay, advised!) to give
his musings some serious thought and not merely dismiss them as the
rantings of a combinatorist.
A=B : A
fantastic book by Wilf/Zeilberger. And moreover, it's free! (as in this book
by Wilf-another great book for any combinatorist)
Other Links:
Dictionary
Weather
Wikipedia
Cricket:
I think this
is a vastly superior game to baseball, though my American friends may
not agree.
Other Interests:
Music
Sudoku