#### Alladi Ramakrishnan Hall

#### Polynomial IP van der Waerden Theorem for Nilpotent Groups

#### Dibyendu De

##### University of Kalyani

*A set of natural numbers is called syndetic if gaps in it are bounded*

and called thik if it contains arbitrary long block. Further a set

will be piece wise syndetic if it can be expressed as an intersection

of

Using a dynamical approach, Furstenberg and Weiss extended van der

Waerden\textquoteright s theorem to arbitrary abelian groups and restricted

the arith- metic structure to IP-sets \cite{key-3}. In fact they

introduced the notion of IP mapping from the partial semigroup of

all finite subsets of set of natural numbers (takin union as semigroup

operation) to a commutative group. They proved that any piecewise

syndetic set contains contains IP progression of arbitrary length.

It is natural to ask if there are extensions of abovf Theorem to non-

abelian groups. However, in the case of nilpotent groups it is. By

interpreting IP map- pings as \textquotedblleft polynomial mappings

of degree 1\textquotedblright , Bergelson and Leibman in \cite{key-2=00005B2=00005D}

used this insight to prove a powerful polynomial extension of the

aboveTheorem for nilpotent groups.

Depending on their work we shall pose in this lecture some open problems

and conjecture.

Done