at The University of British Columbia
That year, extreme laziness conducted to an absence of talks in the first term, but it got compensated by weekly talks and our first year with pizza and pop funded by the department! Many thanks to Alex Duncan and David Kohler who organized the seminar that year.
Optimal resource placement - by Alan Lindsay.
Determining strategies for deploying resources which have the most beneficial effect on the recipient is a problem often considered in Ecology. With some basic assumptions this problem can be addressed mathematically and in certain circumstances reduces to an eigenvalue problem with weight function m(x), where m(x) is tailored to represent a distribution of resources. In this talk I will describe a solution to this problem when resources are placed on patches whose areas are relatively small compared to that of the domain which they occupy
Density of maximal divisors of certain quartic recurrences - by Amy Goldlist.
We will look at what linear recurrences are, and develop an algebraic statement to describe whenn primes divide consectutive terms. On the way, we will look at how primes split in various extenstions, and what that tells us about them. Using some simple sums, we will find the density of maximal divisors of a particular sequence.
The Road Coloring Theorem - by Mike LeBlanc.
Given a directed graph of constant outdegree, the road coloring problem asks if there is a coloring of the edges such that there is a word in the alphabet of colors mapping the graph to a single vertex. Adler, Goodwyn and Weiss conjectured in 1970 that the necessary conditions of strong connectivity and aperiodicity are sufficient. This conjecture was proved in 2007 by Trahtman. In this talk I will introduce the problem and discuss the history of partial results, as well as give an overview of the recent proof by Trahtman. No knowledge of graph theory will be required.
Development of a Burst-Death Model for Experimental Evolution - by Jennifer Hubbarde.
Estimating the fixation probability of an initially rare beneficial mutation is fundamental to our understanding of adaptation. Such estimates are critical to studies of evolution under controlled laboratory conditions, and are also essential for predicting the rate of adaptation of natural populations - for example the rate of adaptation in response to environmental change, or the rate of emergence of novel, or drug resistant, pathogens. Recent work has emphasized that fixation probabilities are extremely sensitive to the underlying life history model. In this talk, I will develop a "burst-death" life history model, in analogy to the well-studied birth-death process, in which lifetimes are exponentially-distributed, and when an individual reproduces, a "burst" of a fixed number of offspring is produced. Using this model, we can gain new estimates of the fixation probability of a beneficial mutation in a virus population.
Impulsive Differential Equations via Measures - by Warren Code.
Think of a bouncing ball. It is possible to model its motion by ignoring the deformation of the ball on impact, and simply treat it as a rigid object whose vertical velocity is reversed (and maybe diminished slightly) from down to up at the instant of impact. One way of building this sort of behaviour into a dynamical system is to introduce a measure as part of the dynamics, which permits discontinuities ("jumps") in the state trajectory (in the above case, the velocity of the ball). I will discuss the basic ideas as well as other physical systems where this type of analysis appears to be useful. Assumed background will be an elementary knowledge of dynamical systems (like \dot{x}(t) = f(x(t)) ) and measures.
Algebraic geometry for operation research - by David Kohler.
Solving integer programs in operation research may be done by using tools from algebraic geometry, namely Grobner basis. I'll present both sides of the problem and show the nice relationship linking these two areas of mathematics.
A clever, if obvious, application of a famous theorem - by Simon Rose.
Given a ruler and a protractor, I can do a reasonable job of computing the area of a rectangle. Maybe even a triangle or two. But before the advent of computers, how did one go about computing the area of more exotic shapes? In this talk I will discuss a few such methods, of varying degrees of cleverness, focusing on an explanation of how the Compensating Planimeter works its magic.
Optimal Transportation and Equilibrium in Hedonic Markets - by Bruno L'Esperance.
In the first part of this talk, I expect to introduce some notions of basic economic theory. In the second part, I present a mathematical model for the equilibrium in hedonic markets and show how such problem is related to a specific optimal transportation problem. Finally, I show some of the steps of the proofs, an extension of this model and an application to labor markets. There is no need to know any economic theory, everything needed will be introduced in the first part of the talk. This presentation is mainly based on work done by Dr. Ivar Ekeland.
Continued Fraction Arithmetic - by Alex Duncan.
I will discuss B. Gosper's method for arithmetic with continued fractions. Before his work, calculations with continued fractions "in place" was considered intractable. "Contrary to everybody," as stated in the abstract of his original paper, ". . . continued fractions are not only perfectly amenable to arithmetic, they are amenable to perfect arithmetic."
Computing a partition function with dimer shuffling - by Ben Young.
I will explain how to count (i.e. compute the generating function for) a class of combinatorial objects called "pyramid partitions". The generating function also turns out to be a partition function from probability (or, if you're geometrically inclined, from Donaldson-Thomas theory).
The talk is intended for a general audience; it's the same one I gave to the DT seminar recently. The proof uses a modified version of the domino shuffling algorithm of Elkies, Kuperberg, Larsen and Propp. There will be many pretty pictures, and a surprise ending that nobody (least of all me) expected...