at noon
in UHall B660

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 UHall B660

University of Lethbridge

A Cayley graph \(\mathrm{Cay}(G;S)\) has the elements of \(G\) as its vertices, with \(g \sim gs\) if and only if \(s \in S\). There is a natural colouring of the edges of any such graph, by assigning colour \(s\) to an edge if it came from the element \(s\) of \(S\). For a Cayley digraph, any graph automorphism that preserves this colouring has to be a group automorphism of \(G\). For a Cayley graph, this is not the case. I will present examples of Cayley graphs that have automorphisms that do not correspond to group automorphisms of \(G\). I will also show that for some families of groups, such examples are not possible.

I will also discuss the more general problem of automorphisms that permute the colours, rather than necessarily preserving all of them.

at noon
in UHall B660

University of Lethbridge

Let \(d(n)=\sum_{d|n} 1\). This is known as the divisor function. It counts the number of divisors of an integer. Consider the following shifted convolution sum $$ \sum_{an-m=h}d(n) \, d(m) \, f(an, m), $$ where \(f\) is a smooth function which is supported on \([x, 2x]\times[x, 2x]\) and oscillates mildly. In 1993, Duke, Friedlander, and Iwaniec proved that $$ \sum_{an-m=h}d(n) \, d(m) \, f(an, m) = \textbf{Main term}(x)+ \mathbf{O}(x^{0.75}).$$ Here, we improve (unconditionally) the error term in the above formula to \(\mathbf{O}(x^{0.61})\), and conditionally, under the assumption of the Ramanujan-Petersson conjecture, to \(\mathbf{O}(x^{0.5}).\) We will also give some new results on shifted convolution sums of functions coming from Fourier coefficients of modular forms.

at noon
in UHall B660

University of Lethbridge

We study the distribution of primes for which the index and order of \(a\) modulo \(p\) have a fixed common divisor. We will also motivate this problem through previously known results related to Artin's conjecture for primitive roots.

at noon
in UHall B660

University of Lethbridge

Let \(\pi(x;q,a)\) denote the number of primes up to \(x\) that are congruent to \(a \pmod{q}\). A "prime number race", for fixed modulus \(q\) and residue classes \(a_1,\ldots,a_r\), investigates the system of inequalities $$\pi(x;q,a_1)>\pi(x;q,a_2)> \cdots >\pi(x;q,a_r). $$ We expect that this system should have arbitrarily large solutions \(x\), and moreover we expect the same to be true no matter how we permute the residue classes \(a_j\); if this is the case, the prime number race is called "inclusive". Rubinstein and Sarnak proved conditionally that every prime number race is inclusive; they assumed not only the generalized Riemann hypothesis but also a strong statement about the linear independence of the zeros of Dirichlet L-functions. We show that the same conclusion can be reached with a substantially weaker linear independence hypothesis. This is joint work with Greg Martin.

at noon
in UHall B660

University of Lethbridge

Let \(G\) be a compact semisimple Lie group, and let \(H\) be a closed subgroup of \(G\). Given a linear representation \(\tau:H\to\mathrm{End}(V)\) of \(H\), one can form an associated vector bundle \(\mathcal{V}_\tau \to G/H\) over the homogeneous space \(G/H\), and define an induced representation of \(G\) on the space of \(L^2\) sections of \(\mathcal{V}_\tau\), after the method of Frobenius. Despite the resulting representation of \(G\) being infinite-dimensional, Berline and Vergne showed that it is possible to give a formula for its character. If we assume that \(H\) is a maximal torus in \(G\), then \(G/H\) is a complex manifold, and the vector bundle \(\mathcal{V}_\tau\) can be equipped with a holomorphic structure. In this case one can define the holomorphic induced representation of \(G\) by restricting to the space of holomorphic sections of \(\mathcal{V}_\tau\), which is a finite-dimensional vector space. We'll show that both extremes can be viewed as special cases of a family of induced representations, whose characters can be computed as the index of a transversally elliptic operator on the homogeneous space \(G/H\).

at noon
in UHall B660

We discuss the solvability of certain linear equations in sparse subsets of the squares. Specifically, we investigate equations of the form $$\lambda_1 n_1^2 + \dotsb + \lambda_s n_s^2 = 0,$$ where \(s \geq 7\) and the coefficients \(\lambda_i\) sum to zero and satisfy certain sign conditions. We show that such equations admit non-trivial solutions in any subset of \([N]\) of density \((\log N)^{-c_s}\), improving upon the previous best of \((\log\log N)^{-c}\).

at noon
in UHall B660

University of Lethbridge

We describe heuristics for several well-known density conjectures in prime number theory and elliptic curves theory. In each case we show how one can arrive at explicit expressions for density constants. The conjectures include twin prime conjecture, Bateman-Horn conjecture, Artin's primitive root conjecture, and Koblitz-Zywina conjecture.

at noon
in UHall B660

University of Lethbridge

Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\). Let \(M_E(N)\) be the function that counts the number of primes \(p\) of good reduction such that \(\#E_p(\mathbb{F}_p) = N\) where \(N\) is a fixed integer and \(E_p(\mathbb{F}_p)\) denotes the group of points on the elliptic curve modulo \(p\). We consider this function on average and discuss recent results related to the constant in the asymptotic result in the average.

at noon
in UHall B660

University of Lethbridge

Let \(R\) be an integral domain and let \(A\) be a finitely generated free abelian group. An elliptic net is a map \(W \colon A\longrightarrow R\) with \(W(0)=0\), and such that for all \(p,q,r,s \in A\), \begin{align*} W(p+q+s) \, W(p-q) \, W(r+{} & s) \, W(r)\\ {} +W(q+r+s) & \, W(q-r) \, W(p+s) \, W(p)\\ & {} +W(r+p+s) \, W(r-p) \, W(q+s) \, W(q)=0. \end{align*} In this talk we will give a formula to compute the sign of any term of an elliptic net without actually computing the value of that term.

Friday
at noon
in UHall B660

In 1979 G. Belyi proved that given any smooth curve \(C\) over any number field, there is map \(\beta:C \rightarrow \mathbb{P}^1\) such that \(\beta\) is unramified outside of three points. This is particularly striking since Belyi's theorem is not true over complex numbers, and hence it is an arithmetic result as much as it is a geometric result. In this talk I will give a brief explanation for the proof of this theorem and explain how this theorem can be used to relate arithmetic geometry problems to the ABC conjecture.

at noon
in UHall B660

University of Lethbridge

We will discuss a few basic properties of "arithmetic groups," which are certain groups of \(n \times n\) matrices with integer entries. By definition, the subject combines algebra (group theory and matrices) with number theory (the integers), but it also has connections with other areas, including the theory of periodic tilings. To learn more about these interesting groups, download a free copy of my book from http://arxiv.org/src/math/0106063/anc/