Number Theory and Combinatorics Seminar

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

Wednesday, September
10, 2008

Room: C583

Time: 12:00-12:50 PM

Speaker: Brandon Fodden (PIMS/University of Lethbridge)

Title: A lower bound for fractional moments of certain L-functions

Abstract: We extend the method of Heath-Brown to find a lower bound for the fractional moments of a certain class of L-functions.

Wednesday, September 17, 2008

Room: C583

Time: 12:00-12:50 PM

Speaker: Dragos Ghioca (University of Lethbridge)

Title: Algebraic Dynamics

Abstract: The classical Mordell-Lang conjecture (proven by Faltings and Vojta) describes the intersection between a finitely generated subgroup of a semiabelian variety G defined over the field of complex numbers with a subvariety V of G. We may view this subgroup of G as the image of 0 under the action of a finitely generated semigroup S of automorphisms of G (each automorphism being a translation). We present extensions of the Mordell-Lang conjecture in which S is any finitely generated semigroup of endomorphisms of G.

Wednesday, September 24, 2008

Room: C583

Time: 12:00-12:50 PM

Speaker: Amir Akbary (University of Lethbridge)

Title: Rankin-Selberg Convolutions

Abstract: We describe how the study of the analytic properties of the convolution of the Dirichlet series leads to the results on the size of the coefficients of the Dirichlet series.

Wednesday, October 1, 2008

Room: C583

Time: 12:00-12:50 PM

Speaker: Nathan Ng (University of Lethbridge)

Title: Non-vanishing of L-functions and application to a Fermat equation

Abstract: In recent years, a popular research topic in analytic number theory has the been the non-vanishing of L-functions. In this talk I will discuss some non-vanishing results which imply the Fermat equation A^4+B^2=C^p for p a prime larger than 5 has no non-trivial solutions.

Wednesday, October 22, 2008

Room: C583

Time: 12:00-12:50 PM

Speaker: Nathan Ng (University of Lethbridge)

Title: The Mobius function summed over short intervals

Abstract: The Central Limit Theorem in probability determines that a sum of independent identically distributed random variables is normally distributed. A number theoretic model for a sequence of such random variables is the Mobius function. In this talk we discuss the distribution of the sum of the Mobius function in short and long intervals. We will see that in short intervals the Mobius function behaves like a sum of independent random variables. However, over longer intervals its behaviour depends on the zeros of the zeta function.

Wednesday, October 29, 2008

Room: C583

Time: 12:00-12:50 PM

Speaker: Dave Morris (University of Lethbridge)

Title: Introduction to Ratner's Theorems on unipotent flows (I)

Abstract: Unipotent flows are very well-behaved dynamical systems. In particular, Marina Ratner has shown that every orbit is uniformly distributed (on some invariant submanifold). The first talk will present some important number-theoretic consequences of this theorem, and the second talk will explain a few of the ideas of the proof.

Wednesday, November 5, 2008

Room: C583

Time: 12:00-12:50 PM

Speaker: Dave Morris (University of Lethbridge)

Title: Introduction to Ratner's Theorems on unipotent flows (II)

Abstract: Unipotent flows are very well-behaved dynamical systems. In particular, Marina Ratner has shown that every orbit is uniformly distributed (on some invariant submanifold). The first talk will present some important number-theoretic consequences of this theorem, and the second talk will explain a few of the ideas of the proof.

Monday, November 10, 2008

Room: C620

Time: 12:00-12:50 PM

Speaker: Chantal David (Concordia University)

Title:*Almost prime orders of elliptic curves over
finite fields*

Abstract: Let*E* be an elliptic curve over the rationals. A conjecture of Neal
Koblitz predicts an exact asymptotic for the number of primes *p*
such that the order of *E* over the finite field with *p*
element is prime. This conjecture is still open. Using sieve
techniques, one can find a lot of primes *p* such that the order *p*+1-*a*_{P}(*E*) is almost
prime. The best result that one may hope to achieve by sieve techniques
was obtained by Iwaniec and Jimenez Urroz for complex multiplication
curves using Chen's sieve. They showed that there are infinitely many
primes *p* such that *p*+1-*a*_{p}(*E*)=*P*_{2},
where *n*=*P*_{k} means that the integer *n*
has at most *k* prime factors. For elliptic curves without
complex multiplication, it is not known how to apply the switching
principle of Chen's sieve to get such a result.

For curves without complex multiplication, we show that there are many primes*p* such that *p*+1-*a*_{p}(*E*)=*P*_{8}
with an explicit lower bound (in terms of the constant *C*(*E*)
of Koblitz's conjecture), using Greaves' sieve and under the GRH. This
improves previous work of Steuding and Weng. One can also show that
there are many primes such that *p*+1-*a*_{p}(*E*)
has at most 6 *distinct* prime factors, but still cannot
improve the number of (not necessarily distinct) primes from 8
to 6. This surprising result is related to the difficulty of
sieving square-free numbers in the sequence *p*+1-*a*_{p}(*E*).

This is joint work with Jie Wu (CNRS, Institut Elie-Cartan, Nancy).

Friday, November 21, 2008

Room: D630

Time: 12:00-12:50 PM

Speaker: Harald Helfgott (University of Bristol)

Title: Escape and incidence: their role in growth in groups

Abstract: There is, so far, one tool that geometric group theory (largely on infinite groups) and recent work on non-commutative group combinatorics (largely on finite groups) have in common: the idea of escape.

After a brief discussion of what escape is and how it can be used, we shall pass to the possibility of restating much of "additive combinatorics" as the combinatorics of an abstract projective plane. There is a basic statement in the latter that does not seem to have a clear analogue in classical additive combinatorics; we shall see how the main idea of the proof is, again, escape.

Wednesday, November 26, 2008

Room: D633

Time: 12:00-12:50 PM

Speaker: Hadi Kharaghani (University of Lethbridge)

Title: On mutually unbiased Hadamard matrices

Abstract: Two Hadamard matrices $H$ and $K$ of order $n$ are called {\it unbiased} if $HK^t=\sqrt n L$, where $L$ is a Hadamard matrix of order $n$. Mutually unbiased Hadamard matrices have applications in quantum measurement, quantum cryptography and design theory. I will try to give a simple survey talk on the the existence, structure and applications of these matrices.

This talk will be accessible to the senior undergraduate mathematics students.

Wednesday, December 3, 2008

Room: C583

Time: 12:00-12:50 PM

Speaker: Pablo Spiga (University of Padova)

Title: Synchronization and homomorphisms

Abstract: An automaton is a machine which can be in any of a set of internal states which cannot be directly observed. A synchronizing automaton is an automaton admitting a sequence of transitions which take the automaton from any state into a known state. In this talk we present some recent connections between synchronizing automatons, permutation groups and graph homomorphisms. All relevant definitions would be given during the talk.

Time: 12:00-12:50 PM

Speaker: Brandon Fodden (PIMS/University of Lethbridge)

Title: A lower bound for fractional moments of certain L-functions

Abstract: We extend the method of Heath-Brown to find a lower bound for the fractional moments of a certain class of L-functions.

Wednesday, September 17, 2008

Room: C583

Time: 12:00-12:50 PM

Speaker: Dragos Ghioca (University of Lethbridge)

Title: Algebraic Dynamics

Abstract: The classical Mordell-Lang conjecture (proven by Faltings and Vojta) describes the intersection between a finitely generated subgroup of a semiabelian variety G defined over the field of complex numbers with a subvariety V of G. We may view this subgroup of G as the image of 0 under the action of a finitely generated semigroup S of automorphisms of G (each automorphism being a translation). We present extensions of the Mordell-Lang conjecture in which S is any finitely generated semigroup of endomorphisms of G.

Wednesday, September 24, 2008

Room: C583

Time: 12:00-12:50 PM

Speaker: Amir Akbary (University of Lethbridge)

Title: Rankin-Selberg Convolutions

Abstract: We describe how the study of the analytic properties of the convolution of the Dirichlet series leads to the results on the size of the coefficients of the Dirichlet series.

Wednesday, October 1, 2008

Room: C583

Time: 12:00-12:50 PM

Speaker: Nathan Ng (University of Lethbridge)

Title: Non-vanishing of L-functions and application to a Fermat equation

Abstract: In recent years, a popular research topic in analytic number theory has the been the non-vanishing of L-functions. In this talk I will discuss some non-vanishing results which imply the Fermat equation A^4+B^2=C^p for p a prime larger than 5 has no non-trivial solutions.

Wednesday, October 22, 2008

Room: C583

Time: 12:00-12:50 PM

Speaker: Nathan Ng (University of Lethbridge)

Title: The Mobius function summed over short intervals

Abstract: The Central Limit Theorem in probability determines that a sum of independent identically distributed random variables is normally distributed. A number theoretic model for a sequence of such random variables is the Mobius function. In this talk we discuss the distribution of the sum of the Mobius function in short and long intervals. We will see that in short intervals the Mobius function behaves like a sum of independent random variables. However, over longer intervals its behaviour depends on the zeros of the zeta function.

Wednesday, October 29, 2008

Room: C583

Time: 12:00-12:50 PM

Speaker: Dave Morris (University of Lethbridge)

Title: Introduction to Ratner's Theorems on unipotent flows (I)

Abstract: Unipotent flows are very well-behaved dynamical systems. In particular, Marina Ratner has shown that every orbit is uniformly distributed (on some invariant submanifold). The first talk will present some important number-theoretic consequences of this theorem, and the second talk will explain a few of the ideas of the proof.

Wednesday, November 5, 2008

Room: C583

Time: 12:00-12:50 PM

Speaker: Dave Morris (University of Lethbridge)

Title: Introduction to Ratner's Theorems on unipotent flows (II)

Abstract: Unipotent flows are very well-behaved dynamical systems. In particular, Marina Ratner has shown that every orbit is uniformly distributed (on some invariant submanifold). The first talk will present some important number-theoretic consequences of this theorem, and the second talk will explain a few of the ideas of the proof.

Monday, November 10, 2008

Room: C620

Time: 12:00-12:50 PM

Speaker: Chantal David (Concordia University)

Title:

Abstract: Let

For curves without complex multiplication, we show that there are many primes

This is joint work with Jie Wu (CNRS, Institut Elie-Cartan, Nancy).

Friday, November 21, 2008

Room: D630

Time: 12:00-12:50 PM

Speaker: Harald Helfgott (University of Bristol)

Title: Escape and incidence: their role in growth in groups

Abstract: There is, so far, one tool that geometric group theory (largely on infinite groups) and recent work on non-commutative group combinatorics (largely on finite groups) have in common: the idea of escape.

After a brief discussion of what escape is and how it can be used, we shall pass to the possibility of restating much of "additive combinatorics" as the combinatorics of an abstract projective plane. There is a basic statement in the latter that does not seem to have a clear analogue in classical additive combinatorics; we shall see how the main idea of the proof is, again, escape.

Wednesday, November 26, 2008

Room: D633

Time: 12:00-12:50 PM

Speaker: Hadi Kharaghani (University of Lethbridge)

Title: On mutually unbiased Hadamard matrices

Abstract: Two Hadamard matrices $H$ and $K$ of order $n$ are called {\it unbiased} if $HK^t=\sqrt n L$, where $L$ is a Hadamard matrix of order $n$. Mutually unbiased Hadamard matrices have applications in quantum measurement, quantum cryptography and design theory. I will try to give a simple survey talk on the the existence, structure and applications of these matrices.

This talk will be accessible to the senior undergraduate mathematics students.

Wednesday, December 3, 2008

Room: C583

Time: 12:00-12:50 PM

Speaker: Pablo Spiga (University of Padova)

Title: Synchronization and homomorphisms

Abstract: An automaton is a machine which can be in any of a set of internal states which cannot be directly observed. A synchronizing automaton is an automaton admitting a sequence of transitions which take the automaton from any state into a known state. In this talk we present some recent connections between synchronizing automatons, permutation groups and graph homomorphisms. All relevant definitions would be given during the talk.