Number Theory and Combinatorics Seminar

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

Thursday, January10,
2008

Room: C756

Time: 3:05-3:55 PM

Speaker: Dragos Ghioca (University of Lethbridge)

Title: The Skolem-Mahler-Lech theorem

Abstract: We present an old result for linear recurrence sequences whose proof relies on p-adic analysis.

Based on it we derive an interesting result for automorphisms of the affine space.

Thursday, January 24, 2008

Room: C756

Time: 3:05-3:55 PM

Speaker: Nathan Ng (University of Lethbridge)

Title: Lower bounds for moments of L-functions

Abstract: Hardy and Littlewood initiated the study of high moments of the Riemann

zeta function. They were initially interested in this problem because of its

connection to the Lindelof hypothesis. In recent years, the moments of central

values of families of L-functions have attracted much attention.

In this talk I will explain a recent method of Rudnick and Soundararajan

which establishes lower bounds for moments of L-functions of the

correct order or magnitude.

Thursday, January 31, 2008

Room: C756

Time: 3:05-3:55 PM

Speaker: Amir Akbary (University of Lethbridge)

Title: On the reduction mod p of a rational point of an elliptic curve

Abstract: Let E be an elliptic curve over Q. For any prime p of good reduction, let E_p

be the elliptic curve over the finite field F_p obtained by reducing E modulo p. We investigate

that for a point of infinite order in the Mordell group E(Q), how the order of the reduction of

this point mod p varies as p goes to infinity. We study this problem by comparison with reduction

of integers mod p.

Thursday, February 7, 2008

Room: C756

Time: 3:05-3:55 PM

Speaker: Nathan Ng (University of Lethbridge)

Title: Upper bounds for moments of the Riemann zeta function

Abstract: I will explain a recent method of Soundararajan that proves an upper

bound for moments of the Riemann zeta function which is nearly

as sharp as the expected order of magnitude. This technique assumes

the Riemann hypothesis and relies on obtaining an upper bound for

the frequency of values for which the zeta function is large on the

critical line.

Thursday, February 14, 2008

Room: C756

Time: 3:05-3:55 PM

Speaker: John Irving (Saint Mary's University)

Title: Lattice Paths Under a Shifting Boundary

Abstract: The generalized ballot theorem gives a well-known formula for

the number of lattice paths in the first quadrant lying weakly under the

line x=ay, where a is an arbitrary positive integer. While there is

almost certainly no simple formula for the number of paths under an

arbitrary piecewise linear boundary, we show that nice enumerative

results are available if we allow for cyclic shifts of such a general

boundaries. A refinement of this result allows for the counting of paths

with a specified number of corners, and we also examine paths dominated

by periodic boundaries. This is joint work with A. Rattan.

Thursday, February 28, 2008

Room: C756

Time: 3:05-3:55 PM

Speaker: Nathan Ng (University of Lethbridge)

Title: Upper bounds of the Riemann zeta function

Abstract: Hadamard and Weierstrass proved a product formula for entire

functions of order one. This factorization theorem leads to a formula for the

logarithmic derivative of the zeta function in terms of the zeros of the zeta function.

I will explain how such formulae can be applied to obtain upper bounds for the

Riemann zeta function on the critical line. This talk is based on recent work of Soundararajan.

Thursday, March 13, 2008

Room: C756

Time: 3:05-3:55 PM

Speaker: Matthew Greenberg (University of Calgary)

Title: Modular forms, Stark units and Heegner points

Abstract: In this talk, we will discuss constructions and conjectures concerning

units in rings of algebraic integers and algebraic points on elliptic

curves over number fields. Modular forms, in various guises, are the

key ingredients in all known systematic constructions of such objects.

Thursday, March 20, 2008

Room: C756

Time: 3:05-3:55 PM

Speaker: Brandon Fodden (PIMS/University of Lethbridge)

Title: Fractional moments of the Riemann zeta function

Abstract: We discuss the Heath-Brown method to find upper and lower bounds

for the fractional moment of the Riemann zeta function for s with real

part 1/2.

Thursday, March 27, 2008

Room: C756

Time: 3:05-3:55 PM

Speaker: Hadi Kharaghani (University of Lethbridge)

Title: The energy of matrices

Abstract: Let M be an m by n matrix , m<= n. Let \lambda_i, i=1, 2, ..., m

be the eigenvalues of the matrix MM^t. Motivated by an application in theoretical chemistry,

Gutman defined the energy of M to be the sum of the square roots of \lambda_i's.

The (0,1)-matrices attaining maximum energy will be discussed.

Thursday, April 3, 2008

Room: C756

Time: 3:05-3:55 PM

Speaker: Habiba Kadiri (University of Lethbridge)

Title: A bound for the least prime ideal in the Chebotarev density problem.

Abstract: A classical theorem due to Linnik gives a bound for the least prime number

in an arithmetic progression. Lagarias, Montgomery and Odlyzko gave a generalization

of this result to any number field. Their proof relies on some results about the distribution

of the zeros of the Dedekind Zeta function (zero free regions, Deuring Heilbronn phenomenon).

In this talk, I will present some new results about these zeros. As a consequence, we are able to

prove an effective version of the theorem of Lagarias et al.

Thursday, April 17, 2008

Room: B650

Time: 12:00-12:50 PM

Speaker: Harald Helfgott (University of Bristol)

Title: Growth in SL_3

Abstract: Let K be R, C or a Z/pZ. Let G = SL_2(K). Not long ago, I proved the following theorem:

for every subset A of G that is not contained in a proper subgroup, the set A A A is much larger than A.

A generalization to groups of higher rank was desired by many, but seemed hard to obtain.

I have obtained a generalization to SL_3(K). The role of both linearity and the group structure of G should

now be clearer than they were at first. Bourgain, Gamburd and Sarnak derived various consequences on

expander graphs from my SL_2 result; analogous consequences should follow in the case of SL_3.

Friday, May 9, 2008

Room: E575

Time: 11:00-11:50 AM

Speaker: Nathan Jones (CRM, Montreal)

Title: A refined version of the Lang-Trotter conjecture

Abstract: Let E be an elliptic curve defined over the rational numbers and r a fixed integer. Using a probabilistic model consistent with the Chebotarev density theorem for the division fields of E and the Sato-Tate distribution, Lang and Trotter conjectured an asymptotic formula for the number of primes up to x which have Frobenius trace equal to r. However, when one sums the main term in their asymptotic over r in a fixed residue class modulo q, one does not recover the main term in the Chebotarev theorem for the q-th division field, but rather 8/\pi times the main term.

In this talk, I will state a refinement of the Lang-Trotter conjecture and demonstrate consistency of this refinement with the Chebotarev Theorem for a fixed division field. This is based on joint work with S. Baier.

Time: 3:05-3:55 PM

Speaker: Dragos Ghioca (University of Lethbridge)

Title: The Skolem-Mahler-Lech theorem

Abstract: We present an old result for linear recurrence sequences whose proof relies on p-adic analysis.

Based on it we derive an interesting result for automorphisms of the affine space.

Thursday, January 24, 2008

Room: C756

Time: 3:05-3:55 PM

Speaker: Nathan Ng (University of Lethbridge)

Title: Lower bounds for moments of L-functions

Abstract: Hardy and Littlewood initiated the study of high moments of the Riemann

zeta function. They were initially interested in this problem because of its

connection to the Lindelof hypothesis. In recent years, the moments of central

values of families of L-functions have attracted much attention.

In this talk I will explain a recent method of Rudnick and Soundararajan

which establishes lower bounds for moments of L-functions of the

correct order or magnitude.

Thursday, January 31, 2008

Room: C756

Time: 3:05-3:55 PM

Speaker: Amir Akbary (University of Lethbridge)

Title: On the reduction mod p of a rational point of an elliptic curve

Abstract: Let E be an elliptic curve over Q. For any prime p of good reduction, let E_p

be the elliptic curve over the finite field F_p obtained by reducing E modulo p. We investigate

that for a point of infinite order in the Mordell group E(Q), how the order of the reduction of

this point mod p varies as p goes to infinity. We study this problem by comparison with reduction

of integers mod p.

Thursday, February 7, 2008

Room: C756

Time: 3:05-3:55 PM

Speaker: Nathan Ng (University of Lethbridge)

Title: Upper bounds for moments of the Riemann zeta function

Abstract: I will explain a recent method of Soundararajan that proves an upper

bound for moments of the Riemann zeta function which is nearly

as sharp as the expected order of magnitude. This technique assumes

the Riemann hypothesis and relies on obtaining an upper bound for

the frequency of values for which the zeta function is large on the

critical line.

Thursday, February 14, 2008

Room: C756

Time: 3:05-3:55 PM

Speaker: John Irving (Saint Mary's University)

Title: Lattice Paths Under a Shifting Boundary

Abstract: The generalized ballot theorem gives a well-known formula for

the number of lattice paths in the first quadrant lying weakly under the

line x=ay, where a is an arbitrary positive integer. While there is

almost certainly no simple formula for the number of paths under an

arbitrary piecewise linear boundary, we show that nice enumerative

results are available if we allow for cyclic shifts of such a general

boundaries. A refinement of this result allows for the counting of paths

with a specified number of corners, and we also examine paths dominated

by periodic boundaries. This is joint work with A. Rattan.

Thursday, February 28, 2008

Room: C756

Time: 3:05-3:55 PM

Speaker: Nathan Ng (University of Lethbridge)

Title: Upper bounds of the Riemann zeta function

Abstract: Hadamard and Weierstrass proved a product formula for entire

functions of order one. This factorization theorem leads to a formula for the

logarithmic derivative of the zeta function in terms of the zeros of the zeta function.

I will explain how such formulae can be applied to obtain upper bounds for the

Riemann zeta function on the critical line. This talk is based on recent work of Soundararajan.

Thursday, March 13, 2008

Room: C756

Time: 3:05-3:55 PM

Speaker: Matthew Greenberg (University of Calgary)

Title: Modular forms, Stark units and Heegner points

Abstract: In this talk, we will discuss constructions and conjectures concerning

units in rings of algebraic integers and algebraic points on elliptic

curves over number fields. Modular forms, in various guises, are the

key ingredients in all known systematic constructions of such objects.

Thursday, March 20, 2008

Room: C756

Time: 3:05-3:55 PM

Speaker: Brandon Fodden (PIMS/University of Lethbridge)

Title: Fractional moments of the Riemann zeta function

Abstract: We discuss the Heath-Brown method to find upper and lower bounds

for the fractional moment of the Riemann zeta function for s with real

part 1/2.

Thursday, March 27, 2008

Room: C756

Time: 3:05-3:55 PM

Speaker: Hadi Kharaghani (University of Lethbridge)

Title: The energy of matrices

Abstract: Let M be an m by n matrix , m<= n. Let \lambda_i, i=1, 2, ..., m

be the eigenvalues of the matrix MM^t. Motivated by an application in theoretical chemistry,

Gutman defined the energy of M to be the sum of the square roots of \lambda_i's.

The (0,1)-matrices attaining maximum energy will be discussed.

Thursday, April 3, 2008

Room: C756

Time: 3:05-3:55 PM

Speaker: Habiba Kadiri (University of Lethbridge)

Title: A bound for the least prime ideal in the Chebotarev density problem.

Abstract: A classical theorem due to Linnik gives a bound for the least prime number

in an arithmetic progression. Lagarias, Montgomery and Odlyzko gave a generalization

of this result to any number field. Their proof relies on some results about the distribution

of the zeros of the Dedekind Zeta function (zero free regions, Deuring Heilbronn phenomenon).

In this talk, I will present some new results about these zeros. As a consequence, we are able to

prove an effective version of the theorem of Lagarias et al.

Thursday, April 17, 2008

Room: B650

Time: 12:00-12:50 PM

Speaker: Harald Helfgott (University of Bristol)

Title: Growth in SL_3

Abstract: Let K be R, C or a Z/pZ. Let G = SL_2(K). Not long ago, I proved the following theorem:

for every subset A of G that is not contained in a proper subgroup, the set A A A is much larger than A.

A generalization to groups of higher rank was desired by many, but seemed hard to obtain.

I have obtained a generalization to SL_3(K). The role of both linearity and the group structure of G should

now be clearer than they were at first. Bourgain, Gamburd and Sarnak derived various consequences on

expander graphs from my SL_2 result; analogous consequences should follow in the case of SL_3.

Friday, May 9, 2008

Room: E575

Time: 11:00-11:50 AM

Speaker: Nathan Jones (CRM, Montreal)

Title: A refined version of the Lang-Trotter conjecture

Abstract: Let E be an elliptic curve defined over the rational numbers and r a fixed integer. Using a probabilistic model consistent with the Chebotarev density theorem for the division fields of E and the Sato-Tate distribution, Lang and Trotter conjectured an asymptotic formula for the number of primes up to x which have Frobenius trace equal to r. However, when one sums the main term in their asymptotic over r in a fixed residue class modulo q, one does not recover the main term in the Chebotarev theorem for the q-th division field, but rather 8/\pi times the main term.

In this talk, I will state a refinement of the Lang-Trotter conjecture and demonstrate consistency of this refinement with the Chebotarev Theorem for a fixed division field. This is based on joint work with S. Baier.