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\author{\small{ Navin Singhi}\\

\small{School of Mathematics}\\

\small{Tata Institute of Fundamental Research}\\

\small{Mumbai 400 005, India}}

\title{studying designs via  multisets}  

\date{}

\begin{document}

\maketitle

\thispagestyle{empty}

\begin{abstract} 

A design is a collection of subsets (called blocks) of a set $X$ (usually
 finite). Such a design is called a $t$-design if all $t$- subsets of $X$
 are contained in the same number of  blocks. Examples of $t$-designs inc
lude set of lines or  higher dimensional subspaces etc. in  affine, proje
ctive or some polar spaces. The existence conjecture for $t$-designs stat
es that such designs with  large $\left|X\right|$ exist, whenever necessa
ry parametric conditions are satisfied.  The degree of an element of $X$,
 in a given design, is the number of blocks containing it. The sequence o
f all such degrees is called the degree sequence of the design. A design 
is said to be simple if any subset of $X$  occurs as a block in it at mos
t once. Classifying degree sequences of all simple designs is another wel
l known unsolved problem. A lot of Discrete Mathematics developed, specia
lly in the later half of last century, around problems of these types of 
parametric characterizations. 



 A new method, developed recently to study these types of problems by loo
king at multisets obtained as  projections of blocks, will be discussed. 
Some other methods  which have been used to study such problems, include 
viewing them as problem of finding integer point in a convex polytope wit
h some symmetry and using Minkowski type theorems, using box splines and 
related approximations or viewing parameters of such designs as f- vector
s of some convex polytopes and  studying Hilbert Poincar\'e series for co
rresponding Cohen Maccalay rings, toric varieties or Erhart polynomials e
tc. Such  alternative viewpoints will also be discussed.   

\end{abstract}





\end{document}