at noon
in E575

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.

at noon
in E575

(University of Lethbridge)

Shimura-Taniyama conjecture (now known as modularity theorem) states that all rational elliptic curves are modular. Despite its technical sounding statement, this conjecture became famous due to its application to Fermat's last theorem. Thanks the work of Wiles and others, not only we know that all rational elliptic curves are modular, we also know that many other generalizations (for example, Q-curves) are also modular. However, what do we mean by "all rational elliptic curves are modular", and how would one go about proving such a statement?

In this talk I will explain what we mean by modularity of an elliptic curve using Galois representation, and provide some (a.k.a. epsilon amount) hint on how one can prove the modularity theorems.

at noon
in E575

(University of Lethbridge)

A square \(\pm1\)-matrix with orthogonal rows is called a Hadamard matrix. It is conjectured that a Hadamard matrix of order \(4n\) exists for each natural number \(n\). A result of Seberry [1976] states that:

For any positive integer \(p\) there is a Hadamard matrix of order \(2^n p\) for every \(n \ge [2 \log_2(p - 3)]\).

Rob Craigen, while in our department in 1994, improved this result considerably by showing that:

There exists a circulant signed group Hadamard matrix of every even order \(n\), using a suitable signed group. This in turn would imply the existence of Hadamard matrices of order \(2^n p\) for every \(n \ge 4[(1/6) \log_2 \bigl( (p-1)/ 2 \bigr)] + 2\).

These and other asymptotic results will be discussed.

at noon
in E575

(University of Lethbridge)

The normalized error term in the prime number theorem is intimately related to the location of the zeros of the zeta function. In 1901, Von Koch proved that the Riemann hypothesis implies that this error term is less than \(\log^2 x \). In 1914, Littlewood famously proved that the error term is infinitely often larger than \( \log \log \log x \). Which function is closer to the truth? In important work, Montgomery suggested that the truth lies near \( (\log \log \log x)^2 \). His work depends on modelling the normalizing error term by a sum of independent random variables. He then derives sharp estimates for large deviations of this sum of independent random variables. I will attempt to explain the main ideas behind his conjecture.

at noon
in E575

(University of Lethbridge)

A primitive root modulo a prime \(p\) is an integer which generates the group of non-zero residues modulo \(p\). For primes \(p\), we can always find a primitive root modulo \(p\). In 1927, Artin conjectured that a density for the set of primes for which a fixed integer is a primitive root modulo \(p\) exists. Hooley showed that this is true upon the generalized Riemann hypothesis. I will give two conditional proofs of this result: Hooley's original proof and a proof which generalizes to other problems related to Artin's conjecture.

at noon
in E575

(University of Lethbridge)

Recently, there's been a lot of talk about a result of Fiorilli, relating Chebyshev bias for elliptic curves and ranks of elliptic curves, during coffee hour. At the time of discussion all we had at our disposal was a vague abstract from a talk that Fiorilli gave at AMS in San Diego, so I had a hard time figuring out what was going on. As luck would have it, few weeks ago, I saw a talk of William Stein relating Chebyshev bias and ranks of elliptic curves, and this past week Fiorilli sent a preprint to Farzad. This talk will try to summarize all of these recent developments. In particular, I will describe what we mean by Chebyshev's bias for elliptic curves, and how this bias is related to the (analytic) rank of an elliptic curve.

at noon
in E575

(University of Lethbridge)

A graph is vertex-transitive if its automorphism group acts transitively on the set of vertices. (In other words, every vertex looks exactly like all of the other vertices.) Such graphs can be very complicated in general, but we will use some group theory to see that they are easy to describe when the number of vertices is assumed to be prime. There are similar results when the number of vertices is the square or cube of a prime, but larger powers are harder to understand.

at noon
in B660

(University of Lethbridge)

Additive combinatorics has recently attracted a lot of attention in the mathematics world. A famous conjecture in this field, known as Erdős and Szemerédi's conjecture, concerns the sums and products of integers. It asserts the following:

Conjecture. For any fixed \(\delta>0\) the lower bound $$\max \lbrace |A+A|, |A\cdot A| \rbrace \gg_{\delta} |A|^{2-\delta}$$ holds for all finite sets \(A \subset \mathbb{Z}\).

Here \(A+A=\{a+a^{\prime} : a, a^{\prime} \in A \}\) and \(A\cdot A=\{aa^{\prime} : a, a^{\prime} \in A \}\).

Roughly speaking, the conjecture states that: For a fixed a set of integers, both the sum-set and product-set cannot be small. There are two major achievements towards this conjecture which we discuss during the talk:

1. Chang in 2003 showed that the sum-set must be large whenever the product-set is sufficiently small. More precisely, she has shown that $$|A+A| > 36^{-\alpha}|A|^2$$ if \(|A\cdot A| < \alpha|A|\) for some constant \(\alpha\).
   
   
2. The best known bound today, achieved by Solymosi, follows from his more general inequality $$|A+A|^2 |A\cdot A| \geq \frac{|A|^4}{2\log |A|}.$$ The Solymosi's result is valid for \(A\) a finite subset of the real numbers. From the inequality we have $$\max \lbrace |A+A|, |A\cdot A| \rbrace \geq \frac{|A|^{4/3}}{2(\log |A|)^{1/3}}.$$

at noon
in B660

(University of Lethbridge)

A complex orthogonal design of order \(n\) and type \( (s_1, \ldots, s_k) \), denoted \(COD(n; \ s_1, \ldots, s_k)\), is a matrix \(X\) with entries from \(\{0, \epsilon_1 x_1, \ldots, \epsilon_k x_k\}\), where the \(x_i\)'s are commuting variables and \(\epsilon_j\in\{\pm1, \pm i\}\) for each \(j\), that satisfies $$ XX^{*}= \Bigg(\sum_{i=1}^k s_i x_i^2 \Bigg)I_n,$$ where \(X^{*}\) denotes the conjugate transpose of \(X\) and \(I_n\) is the identity matrix of order \(n\). A complex orthogonal design in which \(\epsilon_j\in\{\pm 1\}\) for all \(j\) is called an orthogonal design, denoted \(OD(n; \ s_1, \ldots, s_k)\). An orthogonal design (=OD) in which there is no zero entry is called a full OD. Equating all variables to \(1\) in any full OD results in a Hadamard matrix. In this seminar, we show that for any \(n\)-tuple \((s_1, \ldots, s_{k})\) of positive integers, there exists an integer \(N\) such that for each \(n\ge N\), there is an \(OD\Big(2^n(s_1+\cdots +s_{k}); 2^ns_1, \ldots, 2^ns_{k}\Big)\). This is a joint work with professor Hadi Kharaghani.

at noon
in B660

(University of Lethbridge)

For a subset \(S\) of natural numbers we consider its set of multiples \(M(S)\). So $$M(S)=\{ms;~m\in \mathbb{N}, s\in S\}.$$ In many cases we can see that \(M(S)\) has an asymptotic density \(\delta(M(S))\). For example if \(S=\{2, 3\}\) then \(\delta(M(S))=2/3\).

Question Is it true that \(\delta(M(S))\) exists for any \(S\subseteq \mathbb{N}\)?

The following conjecture was formulated around 1930's.

Conjecture \(\delta(M(S))\) exists for any \(S\subseteq \mathbb{N}\).

By 1934 the answer to the above question was known. In this talk we study this question.

at noon
in B660

(University of Lethbridge)

Consider the set \(M_n\) of all matrices of order \(n\) with entries \(-1\) and \(1\). The set \(D_n=\{\det(A): A\in M_n\}\) is a finite subset of integers. The maximum determinant problem deals with \(\alpha_n\); the largest possible value of the set \(D_n\). The study of this (relatively old) problem has led people to some very interesting results and questions in number theory, combinatorics and statistics.

The following inequalities and the cases where equalities are attained will be discussed.

1. For \(n\equiv 0 \!\pmod{4}\), \(\alpha_n \le n^{n/2}\). Equality occurs iff there is a \((1,-1)\)-matrix \(H\) of order \(n\) with \(HH^t=nI_n\).
2. For odd \(n\), \(\alpha_n \le \sqrt{2n-1} \,(n-1)^{(n-1)/2}\). Equality occurs iff there is a \((1,-1)\)-matrix \(A\) with \(AA^t = A^tA = (n-1)I_n + J_n\).
3. For \(n\equiv 2 \!\pmod{4}\), \(\alpha_n \le (2n-2)\,(n-2)^{(n-2)/2}\). Equality occurs iff there is a \((1,-1)\)-matrix \(B\) with $$BB^t = B^tB = \left(\begin{array}{cc}M&0\\0&M\end{array}\right),$$ where \(M=(n-2)I_{n/2}+2J_{n/2}\).

at noon
in B660

(University of Lethbridge)

A circulant (di)graph is a (di)graph that can be drawn with its \(n\) vertices equally spaced around a circle, in such a way that rotation by \(360/n\) degrees is a symmetry. It is not hard to see that in the case of a graph, reflection is also a symmetry, so the automorphism group (the group of all of its symmetry operations) must contain the dihedral group of order \(2n\).

I will present results showing that for almost all circulant (di)graphs, these are the only symmetries. I will then look at what we can say about the automorphism group of a circulant (di)graph that has more symmetry than this.

This will draw on work by Babai, Godsil, Dobson, Bhoumik, and myself.