Dalhousie University Mathematics Colloquium, 2012/13

If you would like to be added to (or removed from) the mailing list, please email Peter Selinger. If you would like to schedule a talk, please email Peter Selinger. Here is the schedule:

Abstracts

Abstract: Quantum Computing seeks to exploit quantum effects to solve computational problems more efficiently than would be possible on a classical Turing machine. There are several ways of achieving this; of these, Adiabatic Quantum Computing is arguably the most unusual. I'll try to explain both how it works and how it differs from other approaches to Quantum Computing, all the while assuming a minimal base knowledge of Quantum Physics.

M. Ram Murty (Queen's University): "The partition function revisited"

Abstract: In their celebrated paper of 1918, Hardy and Ramanujan derived an asymptotic formula for the number of partitions of a natural number n, as n tends to infinity. In this paper, they introduced the famous circle method, which later was applied to a galaxy of open questions in additive number theory. In this talk, we will re-visit this function through recent developments, most notably via an algebraic formula discovered by Bruinier and Ono. This allows us to get the same asymptotic as Hardy and Ramanujan but without the circle method. This is joint work with Michael Dewar.

Srinivasa Varadhan (Courant Institute): "Random Graphs"

Abstract: We consider a dense random graph with N vertices. The edge between a pair of vertices can be either present or absent. The probability is p for any edge to be present and independently so for different edges. We can count the number of triangles or other finite graphs that occur as subgraphs in our large graph. Although there is a law of large numbers that provides an estimate of the count, one can have graphs where the count is off. This is a rare event with small probability. The question is to understand how small this probability is and what it implies for the random graphs. For example, if we have more triangles than normal, do we necessarily have more edges?

Rogers Mathew (Dalhousie): "Cube Representation of a Graph: A Probabilistic Approach"

Abstract: A 'k-cube' B=(R₁,R₂,...,R_k) is defined to be the Cartesian product R₁ × R₂ ×... × R_k, where each R_i is a unit length closed interval on the real line. For example, a unit length closed interval on the x axis is a 1-cube, a square with its sides parallel to the x and y axes is a 2-cube and so on. A graph G has a 'k-cube representation' if there exists a function f that maps each vertex in G to a k-cube in ℝ^k such that, for all vertices u, v, the pair uv is an edge if and only if f(u) intersects f(v). Thus, a cube representation gives a geometric representation of a graph. The 'cubicity' of a graph G, denoted by cub(G), is the minimum positive integer k such that G has a k-cube representation. In this talk we use probabilistic techniques to show upper bounds for the cubicity of a graph.

Shannon Ezzat (U. Canterbury, Christchurch, NZ): "Representation Growth: An Introduction"

Abstract: Representation growth is a branch of asymptotic group theory which studies infinite groups by studying the sequence of the number of irreducible representations of dimension n as n grows, usually encoding this data into something called a representation zeta function. This talk will give a general, mostly non-technical introduction to the subject of representation growth, in particular the representation growth of nilpotent groups. Two methods of calculating these representation zeta functions of nilpotent groups will be introduced; a general approximative method based on the Kirillov orbit method, and a case-specific constructive method.

Francis Valiquette (Dalhousie): "Group foliation of differential equations using moving frames"

Abstract: In this talk I will explain how to incorporate the new theory of equivariant moving frames for Lie pseudo-groups into Vessiot's method of group foliation of differential equations. The result is a completely algorithmic and symbolic procedure for finding invariant, partially invariant and non-invariant solutions of differential equations admitting a symmetry group.

Rob Noble (Dalhousie): "Algebraic numbers with rational parameters"

Abstract: The minimal polynomial over the rationals of a rational number is readily determined. Indeed, it is simply the unique monic linear polynomial having the rational number as its root. It is clear that it is not nearly as straightforward to determine the minimal polynomial of an irrational algebraic number, even if it is assumed to have a rational parameter such as real part, imaginary part or modulus. In this talk, we completely characterize such minimal polynomials, as well as those having a root with two or three of these properties. Along the way, we stumble upon a few unexpected and pleasing consequences.

Caroline Cochran (Dalhousie): "Applications of the invariant theory of Killing tensors"

Abstract: Named after the German mathematician Wilhelm Killing (1847-1923), Killing vectors are generators of infinitesimal isometries on a manifold. Their generalization, Killing tensors, have many important applications in mathematical physics, such as Hamilton-Jacobi theory. The properties of Killing tensors or ordered tuples of Killing tensors which remain unchanged under a group action is the central idea in the Invariant Theory of Killing Tensors. In this talk I will describe its application to orthogonal separation of the Hamilton-Jacobi equation, and demonstrate how the theory can be used to give new insight into the geometry of multiseparable and superintegrable Hamiltonian systems.

Robert Milson (Dalhousie): "Exceptional orthogonal polynomials"

Abstract: Classical orthogonal polynomials are defined as polynomial eigenfunctions of Sturm-Liouville problems. By allowing for the possibility that the resulting sequence of polynomial degrees admits a number of gaps we extend the classical families of Hermite, Laguerre and Jacobi to obtain novel polynomial families which are collectively known as Exceptional Orthogonal Polynomials. In this talk we will survey the origins and significant recent developments in this rapidly evolving field of analysis and mathematical physics.

Douglas Stones (Dalhousie): "The trouble with network motifs: an analytical perspective"

Abstract: The real world can often be described through complex networks. Networks are a collection of nodes, joined by edges connecting pairs of nodes. The edges could represent physical connections in a computer network, friendships in a social network, and so on.

One currently popular topic in the study of networks is network motifs: sub-structures which occur with a significantly higher frequency than in randomized networks. It is thought that network motifs play a more important role in networks than arbitrary sub-structures; e.g., in a biological network, a network motif might be a sub-structure conserved by evolution.

Of the many methods used for finding network motifs, the mainstream method counts the number of copies of these sub-structures, and compares these numbers against the same counts in randomized networks. It relies implicitly on assumptions of independence between candidates and a normal distribution of the frequency counts. We show that these assumptions are incorrect. Furthermore, we show that definitions which have been conflated in the literature, e.g., frequency-based vs. concentration-based statistics and high-frequency vs. low-probability motifs, lead to diametrically opposite conclusions and, indeed, to ill-defined concepts.

In this talk, I will assume familiarity with basic probability theory (Poisson, binomial, multimodal distributions) and statistics (p-values, Z-scores). Only elementary graph theory knowledge will be required.

Karl Dilcher (Dalhousie): "The mathematics of Paul Erdős"

Abstract: Paul Erdős was born on March 26, 1913 in Budapest, Hungary. He was one the the 20th century's most prominent mathematicians, and was without doubt the most prolific mathematician since Euler. He made fundamental contributions to various areas of mathematics, including analysis, combinatorics, graph theory, number theory, and probability theory.

In this talk I will give a very brief outline of his main contributions in each of these areas. Obviously, I will not be able to go into details, and the talk will be accessible to a general mathematical audience, including graduate students and upper-year honours students.

I will say little about Erdős's unusual (travelling) lifestyle and the numerous stories and anecdotes about him; this will be left to a documentary, "N is a Number", which will be shown the following day, on his birthday (Tuesday, March 26, 3:30-4:30 pm in the Colloquium Room).

Will Traves (U.S. Naval Academy): "Generalizing Pascal's Theorem"

Abstract: In the fourth century, Pappus of Alexandria wrote Synagogue (The Collection), a textbook on geometry that he hoped would reinvigorate the subject. Unfortunately, it took 1200 years before Blaise Pascal extended one of Pappus's Theorems, anticipating the development of projective geometry. In this talk I'll explain the remarkable theorems of Pappus and Pascal and describe extensions due to Maclaurin, Moebius, and Cayley. I'll also describe my own recent work, an extension of Pascal's theorem to elliptic curves. I hope that the talk will be very accessible; it is partly based on my forthcoming article in the American Math Monthly.

Adam Clay (UQAM): "Ordered groups and 3-manifolds"

Abstract: While the theory of orderable groups is an old subject, it has attracted renewed interest in recent years due to surprising connections with modern topology. Beginning with simple topological examples, I will explain how left-orderable groups came to the attention of 3-manifold topologists. From these examples related to 3-manifolds and from plenty of computational evidence, there was an observed correlation between left-orderability of fundamental groups, existence of certain nice foliations, and having 'nontrivial' Heegaard-Floer homology. This correlation has now been formulated as a precise conjecture. I will explain the conjecture and its significance, and progress towards its proof. The talk will be accessible for a mathematical audience with diverse backgrounds.

Gary Gordon (Lafayette): "From the chromatic polynomial to network reliability"

Abstract: The chromatic polynomial of a graph is a one-variable polynomial that counts the number of ways the vertices of a graph can be properly colored. It was invented in 1912 by G.D. Birkhoff in his unsuccessful attempt to solve the four-color problem. In the 1940's, Tutte generalized Birkhoff's polynomial by adding another variable and analyzing its combinatorial properties. The Tutte polynomial itself has been generalized to other combinatorial objects, with connections to reliability, knot theory, state changes in physics, probability and other areas. We give an overview of the polynomial and describe some recent work in the area.

Ian Wanless (Monash University): "Embedding spherical Latin trades in abelian groups"

Abstract: One of the beautiful aspects of mathematics is the interconnections between apparently different areas. This talk will showcase a problem that combines combinatorial designs (Latin trades), group theory (abelian groups), topological graph theory (triangulations of a sphere), linear algebra (matrices whose permanent and determinant agree) and Euclidean geometry (dissections of equilateral triangles into smaller triangles). The best part is that you don't need to be an expert in any of these areas to appreciate what is going on.

Massimo Caboara (Pisa): "An application of approximate border bases to celestial mechanics"

Abstract: Astronomy has very often required some hard computations in the physical world. Border bases are a tool in computational algebra that has recently been used to deal with large zero dimensional varieties.

I will speak about some work in progress on an application of approximate border bases to celestial mechanics, namely on the determination of a surface describing some limit behavior for Near Earth Object orbits. The talk is introductory, weighted towards the mathematical prerequisites, and assumes no previous knowledge in those subjects.

Vignon Oussa (Bridgewater State University): "The orbit method and its applications"

Abstract: In this talk, we will survey basic properties of Lie groups and their unitary representations. The chief objective of the talk is to introduce the well-known technique known as the Orbit Method. The Orbit Method was invented in the sixties by Kirillov, and is essentially the most powerful tool available in representation theory for a class of groups known as solvable Lie groups. The talk will also cover some modern applications of the orbit method in sampling theory and wavelet theory of non-commutative Lie groups.

Reem Yassawi (Trent University): "A characterization of p-automatic sequences as columns of linear cellular automata"

Abstract: Let p be a prime number. A "p-automatic" sequence is one that is generated by a deterministic finite state machine which is fed base-p expansions of natural numbers. Cobham's theorem characterizes these sequences as the letter to letter projections of fixed points of length p "substitutions": these are generated by a morphism on some finite alphabet A, which replaces letters with words. These sequences generate a rich source of combinatorial and symbolic dynamical systems, whose dynamical, spectral and measurable properties have been studied extensively.

If one further assumes that A has a field structure, then the beautiful theorems of Christol and Furstenberg give other characterizations of p-automatic sequences. Using these characterizations, we show that a sequence over a finite field F of characteristic p is p-automatic if and only if it occurs as a column of the spacetime diagram (with eventually periodic initial conditions), of a linear cellular automaton with 'memory' over F_q. Our proofs are constructive. As a consequence, we show that the dynamical system generated by a length p-substitution can be realized as a topological factor of a linear cellular automaton.

This is joint work with Eric Rowland (UQAM).

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