Teaching Statement and Statement of Research Plans - GWDG

Index 3
Introduction
In this short article, I describe briefly the projects that I am currently working
on, and the projects that I’d like to work on in the near future. My
primary research interest is in automorphic representations and arithmetic
number theory. More specifically, I would like to use Bruhat-Tits Buildings
based on non-archimedian local fields and associated automorphic forms and
representations to solve certain problems in Number Theory, Combinatorics,
and Computer Science.
Most of my research is centered around Ramanujan (graphs) hypergraphs and
associated Zeta and (L)-functions. Before giving a description of projects, I
give an informal overview of my construction of Ramanujan hypergraphs.
Ramanujan hypergraphs
In my thesis (see : http://www.uni-math.gwdg.de/asarveni/index1.html or
enclosed paper) I studied Ramanujan regular hypergraphs, generalizing the
definition of Ramanujan graphs introduced by A.Lubotzky, R.Philips, P. Sarnak
[13], [14]. There are closely related papers by A.Lubotzky B.Samuels and
U.Vishne [15] ,and a beautiful simple construction by W. Lie [12]. The main
difference between these and my construction is that I am using a very special
kind of algebra, namely the skew polynomial ring F q d{τ} and its quotient
field F q d(τ), as the main tools throughout my thesis to arrive at an explicit
construction of Ramanujan hypergraphs. These skew polynomial rings also
are very useful in other branches of Mathematics, such as Drinfeld Modules,
or Invariant theory. Though the explicit construction is beautiful, it
is not possible to understand this construction without technical arguments
and a deep knowledge of the geometric nature of Buildings of type Ãd−1. I
would like to illustrate the rich connections of the notion of Ramanujan hypergraph
with several parts of mathematics, for example with the theory of
Bruhat -Tits Buildings, with representation theory, with combinatorics, and
with ‘non-commutative algebraic-geometry’ as the natural search for zeros of
skew polynomials over finite fields. This viewpoint shows that my construction
is more practical for applications to Computer Science and is the most
natural extension of the notion of Ramanujan graph. Apart from the use of
the arithmetic skew polynomial ring as specified above, the other important
difference is in making use of Lafforgue’s and Arthur-Clozel’s results in the
representation theory of automorphic forms. Below we quickly review my
explicit construction (for proofs and further details see [21]). The explicit
construction shows that the Cayley graph of a certain complicated group