Date

Speaker 
Title 

Sept 10

everyone

Open problem session

at noon in E575 
(University of Lethbridge) 
Please bring your favourite (math) problems. Anyone with a problem to share will be given about 5 minutes to present it. We will also choose most of the speakers for the rest of the semester.


Sept 17

Joy Morris

Calculating partition numbers

at noon in E575 
(University of Lethbridge) 
The partition number \(p(n)\) is the number of ways that \(n\)
can be partitioned into a sum of smaller positive integers. At the SIAM
Discrete Math conference in June, I attended a plenary talk by Ken Ono of
Emory on how to calculate partition numbers. This topic incorporates both
combinatorics and number theory. Ken Ono was kind enough to give me a copy
of his slides so that I could present this topic in our seminar, and I
will be using those slides for this talk.


Sept 24

Amir Akbary

On a Conjecture of Erdös

at noon in E575 
(University of Lethbridge) 
Let \(m\) be an integer bigger than 1 and let \(P(m)\) denote the
largest prime divisor of \(m\). In 1962, Erdös conjectured that
$$\lim_{n\rightarrow \infty} \frac{P(2^n1)}{n}=\infty.$$
In 2000, Ram Murty and Siman Wong conditionally resolved this conjecture,
under the assumption of a celebrated conjecture in number theory. In this
talk I will describe their work.


Oct 1

Dave Morris

Hamiltonian paths in solvable Cayley digraphs

at noon in E575 
(University of Lethbridge) 
Cayley graphs are very nice graphs that are constructed from finite groups. If the group is abelian, then it is easy to show that the graph has a hamiltonian cycle. It is conjectured that the nonabelian Cayley graphs also have hamiltonian cycles.
We will discuss a few recent results (both positive and negative) on the related problem where the graph is replaced by a directed graph, and the finite group is assumed to be solvable (which means it is not too far from being abelian).


Oct 15

Soroosh Yazdani

Local Szpiro Conjecture

at noon in E575 
(University of Lethbridge) 
The Szpiro conjecture is one of the big conjectures in number theory and Diophantine equations. It is equivalent to the ABC
conjecture, and so it implies many interesting results. In this talk I will mention a conjecture that is motivated by the Szpiro
conjecture, which seems much less strong than the Szpiro conjecture, even though it still has many interesting Diophantine
applications. We will also present a few cases where we can prove this conjecture.


Oct 22

Nathan Ng

Additive Divisor Sums

at noon in E575 
(University of Lethbridge) 
The divisor function \( d(n) \) equals the number of divisors of an integer \( n \).
In this talk I will discuss what is known about additive divisor sums of the shape
$$D(N,r)=\sum_{n \le N} d(n) d(n+r)$$
where \( r \) is a fixed positive integer. These sums were introduced by Ingham in 1926, who proved an upper bound
for \( D(N,r) \). This was later refined to an asymptotic formula by Estermann and over the years was further
sharpened by a succession of authors, including HeathBrown, Deshouilliers and Iwaniec, Motohashi, and Meurman.
More recent evaluations of \( D(N,r) \) makes use of the spectral theory of automorphic forms.
I will also discuss more general additive divisor sums of the shape
$$D_k(N,r) = \sum_{n \le N} d_k(n) d_k(n+r)$$
where \( k \) is a natural number larger than 2 and
where \( d_k(n) \) equals the number of ordered \( k \)tuples \( (n_1, \ldots, n_k) \) such that
\( n = n_1 \cdots n_k \).


Oct 29

Majid Shahabi

Weil Conjectures

at noon in E575 
(University of Lethbridge) 
In 1949, Weil proposed a set of conjectures about the generating functions
which are derived from counting the number of points on an algebraic
variety over a finite field. Solving Weil's conjectures was one of the
central mathematics projects of the twentieth century. These problems were
totally solved by a group of people including Dwork, Grothendieck, and
Deligne.
In this talk, we present a historical background and state the assertions
of Weil conjectures. We further explain some sentences about the ideas of
the proofs.


Nov 5

Farzad Aryan

The distribution of \(k\)tuples of reduced residues

at noon in E575 
(University of Lethbridge) 
Let \(q\) be a natural number, and write \(P = \varphi(q)/q\),
that is \(P\) is the probability that a randomly chosen
integer is relatively prime to \(q\). Let
$$ 1 = a_1 < a_2 < \cdots < a_{\phi(q)} < q $$ be the reduced residues mod \(q\)
(integers coprime to \(q\) in increasing order).
A quantity of central interest is
$$V_\gamma (q) = \sum_{i=1}^{\phi(q)} (a_{i+1} a_i )^ \gamma .$$
In 1940, Erdős conjectured that
$$V_{\gamma }(q) \ll qP^{1\gamma }.$$
Let \(\mathcal{D}=\lbrace h_1, h_2 , \cdots, h_s \rbrace\) be an admissible
set. We call \(a+h_1,\ldots, a+h_s\) an \(s\)tuple of reduced residues, if each
of these numbers is coprime with \(q\). Study of \(s\)tuples of reduced
residues is an analogue to the study of \(s\)tuples of primes. In this talk
we prove estimates about the distribution of \(s\)tuples of reduced residues
and finally we prove an extension of Erdős's conjecture for \(s\)tuples:
$$V^{\mathcal{D}}_{\gamma }(q):=\sum_{a_i < q} ( a_{i+1}  a_i )^ \gamma \ll qP^{s(\gamma1) }, $$
where the sum runs over the integers \(1 = a_1 < a_2 < \cdots < q \)
for which \(a_i+h_1,\ldots, a_i+h_s\) is an \(s\)tuple of reduced residues.


Nov 23

Chris Godsil

Continuous Quantum Walks on Graphs

at noon in room B650 
(University of Waterloo) 
If \(A\) is the adjacency matrix of a graph \(X\), then the matrix
exponential \(U (t) = \exp(itA)\) determines what physicists term a continuous
quantum walk. They ask questions such as: for which graphs are there vertices
\(a\) and \(b\) and a \(t\) such that \( U (t)_{a,b}  = 1\)? The basic
problem is to relate the physical properties of the system with properties of
the underlying graphs, and to study this we make use of results from the
theory of graph spectra, number theory, ergodic theory.... My talk will
present some of the progress on this topic.


Nov 30

Heinz Bauschke

An Invitation to Projection Models

at 1pm in room D610 
(UBC Okanagan) 
Feasibility problems, i.e., finding a solution satisfying certain constraints, are common in mathematics and the natural sciences. If the contraints have simple projectors (nearest point mappings), then one popular approach to these problems is to use the projectors in some algorithmic fashion to approximate a solution. In this talk, I will survey three methods (alternating projections, Dykstra, and DouglasRachford), and comment on recent advances and remaining challenges.


Dec 3

Mark Thom

Squarefree Values of Trinomial Discriminants

at noon in E575 
(UBC) 
The discriminant of a trinomial of the form \(x^n \pm x^m \pm 1\)
has the form \(\pm n^n \pm (nm)^{nm} m^m\) when \(n\) and \(m\) are
coprime. We determine necessary and sufficient conditions for identifying
primes whose squares never divide the discriminants arising from coprime
pairs \((n,m)\). These conditions are adapted into an exhaustive search
method, which we use to corroborate a heuristic estimate of the density of
all such primes among the odd primes. The same results are used to produce a
heuristic estimate of the density of squarefree values of these discriminants.
We'll also look at an unlikely seeming family of divisors of the
discriminants, arising from an elementary identity on them. This is joint
work with David Boyd and Greg Martin.
