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.