Number Theory and Combinatorics Seminar

Fall 2007, Spring 2008, Fall 2008, Spring 2009, Fall 2009, Spring 2010, Fall 2010

Wednesday,
January 14, 2009

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.

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.