Room: D511
Time:
12:00-12:50
PM
Speaker:
Habiba Kadiri (University of Lethbridge)
Title: The Ihara Zeta function of graphs
Abstract: In this talk, I will give an introduction to the Ihara
Zeta function of graphs, which has analogous behavior to the Riemann
Zeta function. For example, we will discuss its analytical properties,
the explicit formula, the Riemann Hypothesis and the Graph Prime Number
Theorem.
This talk is accessible to students.
Wednesday, January
21, 2009
Room: D511
Time:
12:00-12:50
PM
Speaker:
Habiba Kadiri (University of Lethbridge)
Title: About the distribution of the eigenvalues of
Ramanujan's graphs
Abstract:
In the last lecture, we discovered the notion of arithmetic for graphs
and how the Ihara Zeta function was used to count the number of paths
and prime paths of arbitrarily large length. This type of relation is
analogous to the relation between the distribution of prime numbers and
the location of the zeros of the Riemann Zeta function.
This lecture is a follow up to the previous lecture. I will focus on
some particular regular graphs, called Ramanujan graphs. These graphs
appear in various domains of mathematics and computer science. I will
discuss the distribution of the eigenvalues for Ramanujan graphs, the
Riemann Hypothesis for graphs and compare it to the case of the
classical Riemann Zeta function. If time permits, I will also discuss
the problem of the distribution of the spacings of the zeros.
Wednesday,
February 11, 2009
Room: D511
Time:
12:00-12:50
PM
Speaker: Amir Akbary (University of Lethbridge)
Title: Ramanujan Graphs
Abstract: A
Ramanujan graph is a connected regular graph whose non-trivial
eigenvalues are relatively small in absolute value. This talk
introduces these graphs and describes some basic constructions of them.
The talk will be accessible to people familiar with basic elements of
graph theory.
Wednesday, February
25, 2009
Room: D511
Time:
12:00-12:50
PM
Speakers:
Amir Akbary and Dave Morris (University of
Lethbridge)
Title: Expander
Graphs
Abstract: This
talk will introduce the class of expander graphs, and describe
some of their basic properties. We will start the seminar with
some applications of Ramanujan graphs, which are a special case.
The talk will be accessible to people familiar with basic elements of
graph theory.
Wednesday,
March 4, 2009
Room: D511
Time:
12:00-12:50
PM
Speaker:
Dave Morris (University of
Lethbridge)
Title: More on Expander Graphs
Abstract: This
talk will have three parts: a proof that random k-regular graphs are
very likely to be expanders, a discussion of the eigenvalues of the
adjacency matrix of an expander graph, and a brief explanation of
Margulis' explicit construction of expander graphs.
Wednesday, March 18,
2009
Room: D511
Time:
12:00-12:50
PM
Speaker: Kaneenika
Sinha (PIMS/University of Alberta)
Title: A
trace formula for Hecke operators on spaces of newforms
Abstract:
Fourier coefficients of some appropriately chosen modular forms can be
interpreted as eigenvalues of Hecke operators. We derive a trace
formula for Hecke operators acting on spaces of newforms of given level
and weight. This explicit formula can be applied to study the
distribution of Fourier coefficients of newforms. We will also
derive arithmetic information about newparts of Jacobians of modular
forms.
Wednesday, April 8,
2009
Room: D511
Time:
12:00-12:50
PM
Speaker:
Behruz Tayfeh-Rezaie (Institute for
Research
in Fundamental Science/Iran)
Title: On the sum of Laplacian
eigenvalues of a graph
Abstract: Let $k$ be a natural
number and let $G$ be a graph with at least $k$ vertices. A. E. Brouwer conjectured
that the sum of $k$ largest Laplacian eigenvalues of $G$ is at most
$e+{k+1\choose 2}$, where $e$ is the number of edges of $G$. We prove this conjecture for
$k=2$. We also show that if $G$ is a tree, then the sum of $k$ largest
Laplacian eigenvalues of $G$ is at most $e+2k-1$.
This is a joint work with W.
H. Haemers and A. Mohammadian.
Wednesday, August
19,
2009
Room: C630
Time:
11:00-11:50
PM
Speaker:
Fabien Pazuki (Paris 7, U of L, Bordeaux 1)
Title: Bounds on torsion for abelian
varieties and reduction properties
Abstract: Let $k$ be a number
field and $A/k$ be an abelian variety. The Mordell-Weil theorem implies
that there are only finitely many torsion points defined over $k$, and
finding a uniform upper bound on this number is still an open question
for abelian varieties of dimension $g>1$. We will see how properties
of reduction of the variety are linked with getting good upper bounds
on the cardinality of the torsion subgroup. We will try to avoid
technicality.
Wednesday, August
26,
2009
Room: C630
Time:
11:00-11:50
PM
Speaker:
Fabien Pazuki (Paris 7, U of L, Bordeaux 1)
Title: Bounds on torsion and
reduction properties (II). The height theory strikes back.
Abstract: We will focus on the height
theory in this second talk. We will study a conjecture formulated by
Lang and Silverman, predicting a precise lower bound to the canonical
height on an abelian variety. The goal is to understand the link
between these height inequalities and uniform bounds on the number of
torsion points. We will briefly recall the key facts from the first
talk, present the conjecture and the known results.