Teaching Statement and Statement of Research Plans - GWDG

lack 1
Teaching Statement and Statement of
Research Plans
Alireza Sarveniazi
March 15, 2005

2 Index
Teaching Statement:
I enjoy teaching mathematics and gain great satisfaction from seeing my
students learning different ways of thinking about mathematical notions and
problems. My prefered method of teaching is to allow students the time so
they can attain the fundamental understanding of a particular subject. This
process of learning is completed by stressing the essential elements of mathematics.
To understand Mathematics at the university level, one must develop
the ability to think abstractly and also geometrically. It is also my belief that
my students should recognize abstract algebraic notions and geomemetrical
objects. The teaching philosophy that I adhere to is based on the theory, of
how students want to learn mathematics. Students develop their mathematical
dexterity best by building relationships between new definitions and their
capacity of abstract understanding and geometrical imagination. Students
must be active participants in the learning process by trying to design nontrivial
questions. Further, they must be prepared to apply their knowledge
in order to solve mathematical problems.
I try to give in my lectures many examples and analogies that make the often
abstract concepts more clear. The most important aspect of my teaching is
to provide the core abilities mixed with new concepts that allow student to
develop their problem solving skills. I belive that students can be provided
with a deeper understanding of mathematics, in particular under graduate
level of mathematics, if they know applications of certain contexts in the real
world, such as computer science, communication networks or even in physics.
Teaching Interest:
-Calculus
-Number theory (elementary course)
-Galois theory and Finite fields
-The first and second course in Algebra
-The first and second course in Analysis
-Differential Geometry (elementary course)
-Complex functions and Analysis
-General Topology
-Discrete Mathematics
-The first and second course in Graph theory
-Seminars for graduate students

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

4 Index
(with respect to a certain set of generators), is the same as the Cayley graph
of a well-known group PGL(d) over a finite field, with respect to a set of
easily-described generators.
Property (τ), Selberg Property, Ramanujan-Peterson conjecture
It is well known that every arithmetic lattice Γ (at least over fields of characteristic
0 and conjecturally always) has the Selberg property. If in addition
Γ has the congruence subgroup property, then Γ has also (τ). This leads to
following conjecture: see [16] for more detail.
Conjecture 1. Let Γ be a lattice in a semisiple group.
congruence subgroup property if and only if Γ has (τ).
Then Γ has the
There is a close relationship between the congruence subgroup property, the
Selberg property and the Ramanujan-Peterson conjecture. One of my main
goals is to study these relationships more deeply with the viewpoint of obtaining
more information related to the above conjecture. I have some ideas
how to generalise the known case to the higher dimensional case and I am
interested in arithmetic groups having property (τ) over a division ring and
I want to consider the effect of the (τ) property on the lifting of the representations
to the similar corresponding linear groups such as GL or SL under
the Langlands corespondence.
Group action on the set of galleries of a Bruhat-Tits Building
Let D be a division algebra over a function field K i.e, a finite extension
of F q (t) with dim D K = d 2 , and π = (π) a place which is unramified in D,
i.e,
D π := D ⊗ K K π
∼ = Md (K π )
and π is an irreducible polynomial in OsupsetF q [t] .
Let M 0 and M π 0 be the set of maximal orders in D and D π respectively. We
can see first that,M π 0 can be interprated as the set of vertices of a Bruhat-
Tits building X π ∗ associated to PGL d (K π ). We can also consider another
Bruhat-Tits building X (global case) with M 0 as the set of vertices. Let R
be a fix maximal order in D. Now consider the set:
The unit group Γ (1)
R
X 0 (R) π = {M ∈ M 0 | M v = R v for all v ≠ π}
R is equal to the number of orbits of Γ (1)
R
:= R ⊗ O[ 1
π ]× acts on X 0 (R) π and the class number of
under this action. For any oder S

Index 5
of level π n in D, we denote its nth associated unit group with
U (n)
S
:= (S ⊗ O[ 1 π ])× .
I will generalise the work of three “Son” (Rosson, H J. and Ellison,B J.
and Wilson,J B.), TREES, HECKE OPERATORS, AND QUADRATIC
FORMS, http://darkwing.uoregon.edu/ jwilson7/math/PrePrintTree.pdf, 2001
as follows:
I will show there is a group action of U (n)
S
:= (S ⊗O[ 1 π ])× on the set of galeries
of length n on the Bruhat-Tits building X ∗ which gives us the class number
of S as the number of orbits of U (n)
S
:= (S ⊗ O[ 1 π ])× under this action. Also I
will study Brandt matrices and show that there is some recursive formulas to
compute these matrices and also to compute the number of solutions arised
from reduced norm equation.
Bruhat-Tits Building and the Zeta functions of nilpotent groups
Using the combinatorial geometry of the Bruhat-Tits building for SL(d,Q p ),
C. Voll introduced a new method to compute local normal zeta functions of
nilpotent groups (see [24]). This motivates me to study more the nature of
the Bruhat-Tits Building arising in my thesis for the same situation but over
the local function field instead of Q p in a more technical point of view.
Zeta and (L)-functions of finite graphs and finite regular-hypergraphs
Ihara obtained, for a discrete co-compact and torsion-free subgroup Γ of
SL(2,Q d ) and the associated Bruhat-Tits tree (X . ), a zeta function of the
finite graph Γ\(X ∗ ). Ihara, using the theory of automorphic forms, has presented
successfully his beautiful discovery: the identity of the zeta function
of the finite graph Γ\(X ∗ ) and the zeta function of certain Shimura curve
S Γ reduced modulo a prime p (see [10], [9]). One of the main ingredients
required, to generalise the above connection to higher-rank reductive groups
G over a global field K, is to obtain a suitable definition for the zeta function
of a finite hypergraph (= a finite quotient of the Bruhat-Tits building)
associated to the G(K ν ) for a place ν.
There are some ideas by C.Ballantine and W. Li and A.Deitamr (based on
Selberg’s trace formula) in the higher dimensional case. But it seems that a
thorough structured theory could be worked out. In my recent work [22], I
have three main goals:
1) Thanks to Garland’s work [6] I define a Zeta function related to the closed

6 Index
galleries of finite quotient (hypergraph) S, i.e., Z S (ρ, u) with associated representation
ρ, (which for dimension = 2 matches with the usual Zeta function
for a finite graph). Actually I use the notion of Bruhat-Tits Building based
on a Tits-system plus the orientation given in Garland’s paper–where the
oriented simplicial has been given in the language of parahoric subgroups
and co-chains which I need for my further study.
2) For a finite dimensional (usually unitary) representation ρ : Γ −→ GL(V )
where Γ is the fundamental group π 1 (X) of a given finite graph X, we can
define
Z(X, ρ, u) = ∏
1
det (Id − ρ([C])u l([C]) )
[C]∈P
(
∑ ∞ )
= exp Nm(X) ρ um m
i=1
where P is the set of all primitive classes of primitiv closed paths, and l the
length of any such class. For the trivial representation, Nm Id (X) = card{ the
set of all primitive closed paths of length m}. Ihara has given a closed formula
for this Zeta function and Bass [3] has generalized this formula further. It
is of great interest to me to try to do the same for the Zeta function of a
hypergraph as in 1) above. In this case we have more adjacency operators,
and the main problem is again to control the oriented simplices so that these
adjacencies occur in the denominator. Also we can use (in the situation of
1) above) the other definition of Bruhat-Tits building related to simplicial
complexes (i.e., using parahoric subgroups ) and for example, for such simple
fundamental domains as in my explicit construction mentioned above, I define
a Zeta function and obtain a formula similar to Ihara’s formula.
3) Let Γ ⊂ SL(2, K) be a discrete, cocompact and torsion-free subgroup
where K is a local field ≠ R and ≠ C, and let X Γ := Γ\(X . ), where X .
is the (q + 1)-regular tree associated to SL(2, K). Further let ρ Γ be the
corresponding representation of the associated Iwahori-Hecke algebra (see
[7] and [8]). Then we have
det L(W, S, ρ Γ , u)
det L(W 1 , S, ρ Γ , u) det L(W 2 , S, ρ Γ , u) = Z(X Γ, u)
which is related to the Tits-system (G, B, N, S) where
G := SL(2, K)
( ) O
×
O
B :=
M O ×

Index 7
where O is the ring of integers of K, M its maximal ideal with the generator
π.
N := the group of monomial matrices
S :=
{
s 1 :=
( 0 −1
1 0
)
, s 2 :=
( 0 −
1
π
π 0
)}
W := The Weyl group
N/B ∩ N
W i := the subgroup generated by s i
Recall the following definition of Gyoja’s for the L-function of any Coxeter
system (W, S) with respect to a finite dimensional representation φ :
H q (W, s) −→ End(V ) (where H q (W, s) is the associated Iwahori-Hecke algebra
with the generator set {e w } (for all these definitions see [4].))
L(W, S, φ, u) := ∑ w∈W
φ(e w )u l(w)
In the above l is the length function defined on W . L(W i , S, φ, u) could be
defined similarly by replacing W with W i . My main goal is to generalise this
beautiful formula ( as mentioned in [8]) to the higher dimensional case based
on definitions discussed in 1) and 2) above. With such a suitable definition
of the Zeta function Z( ˜X, u) for the finite regular hypergraph ˜X : Γ\X . , X .
being the Bruhat-Tits building associated to GL(d, K ν ), we must have:
Z( ˜X, u) = ∏ L(W I , S, ρ Γ , u) (−1)|I|
I⊂S
The proof of the above formula and the definitions it needs are being written
up in detail and will appear as a paper in arXiv soon.
Some interesting research Projects for the near future
In [18] the so-called discrete quantum Fourier-transforms of the number states
i.e. the superpositions:
q−1
∑
|θ p >:= q −1/2
n=0
exp( 2iπpn )|n >
q
I believe that such states live in Ramanujan graphs, especially those associated
to finite groups Z/NZ for suitable N, but the question is :
How can one in general interpret such superpositions as vertices of suitable
Ramanujan graphs? How can one use the deep results from representation
theory and arithmetical group theory lying behind the Ramanujan property
to the study of Cyclotomy or in general to associated physical phenomena?
I also have ideas to generalise these connections even to higher dimensions.

8 Index
References
[1] L. , Lafforgue. Chtocas de drinfeld et ramanujan-petersson. Asterisque,
(243), 1997.
[2] J. Arthur and L. Clozel. Simple Algebras, Base Change, and the Advanced
Theory of the Trace Formula. Princton University Press, 1989.
[3] Hyman Bass. The ihara -selberg zeta function of a tree lattice. Internat.
J.Math. (1992) 717-797.
[4] N. Bourbaki. Lie groups and Lie algebras. Chapter 4 − 6.
[5] K. Brown. Buildings. Springer-Verlag, New York, 1988.
[6] H. Garland. p-adic curvature and cohomology of discrete subgroups of
p-adic groups. Ann. of Math. 97, 375 − 423.
[7] Ki-I. Hashimoto. Zeta functions of finite graphs and represetations of
p-adic groups. Adv. Stud. Pure Math., vol. 15, 1991, pp. 171 − 210.
[8] W.J. Hoffman. Remarks on the zeta function of a graph.
www.math.lsu.edu/ preprint/2002/jwh2002b.pdf.
[9] Y. Ihara. Discrete subgroups of PL(2, k p ). c. Sympos. Pure Math. IX,
Boulder, Colo, 1965. 272 − 278.
[10] Y. Ihara. Algebraic curves mode p and arithmetic groups. Proc. Sympos.
Pure Math. IX, Boulder,Colo, pp.265 − 271, 1965.
[11] S. Lang. Algebra. Addison Wessly, 1965.
[12] W. Li. Ramanujan hypergraphs, preprint. Preprint, 2003.
[13] A. Lubotzky, R. Philips, and P. Sarnak. Ramanujan conjectures and
explicit construction of expanders. Proc. Symp. on Theo. of Camp. Sci
(STOC), 86, 240-246, 1988.
[14] A. Lubotzky, R. Philips, and P. Sarnak. Ramanujan graphs. Combinatorica
8,261 − 277, 1988.
[15] A. Lubotzky, B. Samuels, and U. Vishne. Ramanujan complex of type
à n . http://arxiv.org/pdf/math.CO/0406217, 2004.
[16] A. Lubotzky and A. Zuk. On property (τ). priprint, 2003.

Index 9
[17] Ronan. M. Buildings: Main ideas and applications, I,II. Bulletin of the
London Mathematical Society 24,1 − 57, 97 − 126, 1992.
[18] M. Planat and H.C. Rosu. Cyclotomy and Ramanujan sums in quantum
phase loking. [PDF] arXiv:quant-ph/0304101 v2 17 Jun 2003.
[19] H J. Rosson, B J. Ellison, and J B. Wilson. Trees, hecke operators,
and quadratic forms. http://darkwing.uoregon.edu/ jwilson7/math/PrePrintTree.pdf,
2001.
[20] P Sarnak. Some Applications of Modular Forms. Cambridge Tracts in
Mathematics 99, Cambridge University Press,, 1990.
[21] A. Sarveniazi. P.h.d thesis. Goettingen University publication, 2003.
[22] A. Sarveniazi. Zeta and L functions on regular hypergraphs, preprint.
2003.
[23] J. Serre. Trees. Springer-Verlag, Berlin-Heidelberg-New York, 1980.
[24] C. Voll. Zeta functions of groups and enumeration in Bruhat-Tits building.
Ph.D. thesis, university of Cambridge, 2002.