Dalhousie University Mathematics Colloquium, 2013/14

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Abstracts

Walter Craig (Fields Institute and McMaster): "Waves and wave interactions: from small scales to very large"

Abstract: Wave phenomena occur on an enormous range of scales, from the sub-quantum mechanical to the astrophysical. This talk will discuss some of the common scale independent features of wave propagation and wave interaction, including detailed descriptions of nonlinear wave collisions and a proposal of a kinetic theory for a regime of wave turbulence.

This talk is jointly organized by Oceanography and Mathematics, and co-sponsored by AARMS.

Hui Zhao (Dalhousie, Rowe School of Business and Tianjin University): "Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model"

Abstract: In this paper, we study the optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk model. The insurer is allowed to purchase reinsurance and invest in one risk-free asset and one risky asset whose price process satisfies the Heston model. The objective of the insurer is to maximize the expected exponential utility of terminal wealth. By applying stochastic optimal control approach, we obtain the optimal strategy and value function explicitly. In addition, a verification theorem is provided and the properties of the optimal strategy are discussed. Finally, we present a numerical example to illustrate the effects of model parameters on the optimal investment-reinsurance strategy and the optimal value function. The talk will be accessible to a general audience.

Justin Tzou (Dalhousie): "Slowly varying control parameters, delayed bifurcations, and the stability of spikes in reaction-diffusion systems"

Abstract: We present three examples of delayed bifurcations for spike solutions of reaction-diffusion systems. The delay effect results as the system passes slowly from a stable to an unstable regime, and was previously analysed in the context of ODE's in [P. Mandel, T. Erneux, J. Stat. Phys, 1987]. It was found that the instability would not be fully realized until the system had entered well into the unstable regime. The bifurcation is said to have been "delayed" relative to the threshold value computed directly from a linear stability analysis. We show that the phenomenon of delay is present also in PDE systems while highlighting some novel features not observed for ODE's. Analytic predictions for the magnitude of the delays are obtained through analysis of certain explicitly solvable nonlocal eigenvalue problems. The theory is confirmed by numerical solutions of the full PDE systems.

Larry Ericksen: "Patterns in Art: How mathematics has informed the history of art"

Abstract: This talk will be concerned with the application of certain mathematical patterns to artwork, past and present. Special mathematical topics will be highlighted in works of art, from the Parthenon to patchwork quilts. The golden ratio, which appears in art and nature, has been used in art from classical times. The Fibonacci sequence, first described in the twelfth century, provides a compositional tool used in the golden rectangle and the Fibonacci spiral. Artists from Da Vinci to Dali have employed these mathematical structures.

Several art movements based on mathematical objects have developed in Europe and North America, and special attention will be given to relevant paintings by Canadian artists. In the two dimensional arts and crafts, such as paintings and quilts, coloured block patterns are arranged to evoke visual responses, emotional or intellectual. Coloured squares on a grid can be placed randomly or distributed according to mathematical rules by congruences and periodicities over a number array lattice.

This talk will also cover the geometric constructs of polygon tessellations, Escher designs, and Penrose tilings.

Larry Ericksen is an artist and mathematician living in New Jersey. He is currently visiting Dalhousie for a mathematical research collaboration.

Jay Schweig (Oklahoma State University): "Projective dimension and domination parameters"

Abstract: Projective dimension, an important invariant in commutative algebra, is the shortest length of a projective resolution of an ideal I.  When I is a squarefree monomial ideal, Hochster's Formula allows us to calculate projective dimension via the homology of subcomplexes of an associated simplicial complex known as the Stanley-Reisner complex of I.  

We show how graph domination parameters, invariants which measure how easy it is to "cover" a graph with various subgraphs, can be used to study such problems.  Thus, we can the translate the problem of bounding an ideal's projective dimension into the language of graph theory.  This also allows us to examine the homology of a simplicial complex by studying the associated graph or hypergraph of non-faces.  No prior knowledge of commutative algebra will be assumed, though we will assume a familiarity with simplicial complexes and graphs. (This is joint work with Hailong Dao.)

Russ Woodroofe (Mississippi State University): "Noncontractibility of the coset lattice and fixed points of group actions"

Abstract: Ken Brown asked whether there is any finite group G such that the coset lattice of G is contractible. John Shareshian and I have recently shown that the answer to this question is "no". I'll start by telling you what the question means, then give an overview of our approach. An essential tool of our work is Smith Theory, which examines the fixed point set of the action of a p-group on a topological space. As time allows, I will survey some ideas from this theory.

Christophe Vignat (Orsay and Tulane University): "An introduction to Umbral calculus and its applications"

Abstract: I will introduce the technique of Umbral calculus due to Blissard. This is a symbolic computation technique that allows to easily derive identities involving the usual orthogonal polynomials as well as Bernoulli and Euler polynomials. I will also show how this technique is equivalent, in some cases, to the moment representation technique, a probabilistic approach. A few examples will be given involving Bernoulli and Hermite polynomials.

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