Counterpoint is a western musical tradition with multiple concurrent melody. Often in counterpoint some melodies are derived from others using certain established rules. This type of counterpoint is called _canon_. Many of the most beautiful examples of canon can be found in the music of J. S. Bach (1685-1750).
Group Theory is the mathematical study of the ways in which we can rearrange a set of objects. Although Evariste Galois (1811-1832) was the first to use the term "group", it may be that Bach had an intuitive understanding of some parts of group theory. Using examples from Bach's music we will see how certain groups (specifically the cyclic and dihedral groups) provide a natural language to describe the rules of canon. We might even speculate on a specific calculation Bach himself did to compose this profound music.
Hot Chocolate Dynamics
We all know what happens if we heat milk from below. First heat is simply being diffused away, but if we heat it vigorously enough, convection occurs. But what happens when we throw sugar and cocoa in the mix? When is the resulting mixture stable and when is it unstable? What form do the instabilities take?
This problem comes up in the oceans where both temperature and salinity vary with depth. More importantly, this is an intrinsic part of cooking pasta, a la Italian, and of course of making a decent cup of hot chocolate.
There is in combinatorics a little world made of vertebrates, squids and trees (some of them unrooted, even). This menagerie of structures form the core of the theory of species, a framework for doing enumerative work that is especially good at dealing with tree-like objects. In particular, species are great for proofs by picture.
I'll discuss how species emerged as functors on categories, as well as how they combine together to evolve into other species. Also, every species has a unique decomposition into molecular and atomic subspecies. I'll explain the chemistry of it and give examples.
From Pendulum's to Shear Flow
In my last Spams talk I discussed how oscillating a pendulum vertically can qualitatively change its stability properties. The equation that describes this behaviour is Mathieu's and the rather peculiar type of instability that arises is entitled parametric resonance. This classical work inspired me to ask the question of whether this instability exists in the context of oscillatory shear flow. In this talk, which I promise will be my last Spams talk, I will present the results from this research.
Shifting and Shifted Complexes
Abstract: I will define, motivate, and show results on shifting and shifted complexes.
Property testing is a relatively new field in computational theory. The main question is to test whether an object (e.g., a graph) has a property (e.g., being bipartite) by reading only a few bits from the description of the input. In this talk, I will define the concept of testing and explain it with several examples, some of which involve using powerful tools from extremal combinatorics, such as Szemeredi's regularity lemma. I will also illustrate a technique used for proving non-testability results by an example about graph isomorphism.
Random Walks on Groups
Every child's favourite card game is 52 card pickup. It's not that entertaining, but it can shuffle cards quite well. Mixing cards by repeatedly interchanging the position of two cards chosen at random is slightly more fun - but how productive is it?
We will discuss Fourier Analysis on groups, some representation theory, some probability, and Andrew Greenhalgh's answer to this question. If you can locate 2-338 you will understand the talk.
Do you think you can't hear enough about characters of the symmetric group? If your answer is yes, then I hope this talk won't change it. We will see how these numbers (which I will define purely combinatorially, thus avoiding representation theory) are in some sense polynomials, and how that can be used to compute generating functions for row sums of the character table. No knowledge is assumed or implied, but I will focus on examples that I think are really interesting, with just enough rigor to convince you that I'm probably not lying.
Random Walk with Geometrically Decaying Steps
A simple random walk is the process of moving either left or right by unit steps with equal probability at each time step. Sometimes it is called "the steps of a drunken walker". However, since our drunken man is quite sleepy and losing his energy to walk, he can't maintain equal step length all the time.
A natural revision would be a random walk with geometrically decaying steps such as a, a^2, a^3 ... for a < 1. With this decaying steps, the drunken man eventually will end up at a certain point and fall down. Now there is some probability distribution of the fall-down point over the street.
Colin de Verdiere and Graph Embeddings
I will describe Colin de Verdiere's parameter which is an integer between 0 and n-1 for any graph on n vertices. I will show how this number corresponds to some well-known classes of graphs, and also how it is related to geometric graph embeddings. I will try to prove some results in a slightly different way than usual, which is hopefully more geometrically intuitive. Only basic graph theory and linear algebra is required.
Mechanics of Yeast Growth
Tissue growth has been widely studied experimentally and theoretically. Differential cellular adhesion is believed to be the dominant mechanism for the organization of cells into tissues. Tissue architecture is controlled by the segregation of cells according to their relative adhesions. Then the questions to ask are what controls the kinetics of this sorting? Are there other physical mechanisms that cause cell sorting? There are many models of tissue growth in the literature but none have been quantitatively compared with controlled experiments. We attempt to answer these questions with a simple preliminary mathematical model where different energies in the system is considered. Then quantitatively compare the model to some controlled experiments we explored using baker's yeast from Fink's lab at Whitehead Institute.
"Why is the sky blue?" and Other Random Curiosities
Ever wonder why some cicadas only come out every 13 or 17 years? Or why stock charts are usually plot the stock price on log scale? And why entropy is always defined with that natural log thing? In this light talk, we will take a few moments to explore an eclectic collection of "fun" facts and observations from the real world and beyond which have simple mathematical explanations. Topics may vary. Audience participation is strongly encouraged.
Properties of Graph Drawings in the Plane
We will describe a work in progress motivated by a paper of Tamaki and Tokuyama on 'How to cut pseudo-parabolas into segments'. Given a family of $n$ pseudo-parabolas (curves, every two of which intersect in at most two points, e.g., parabolas in the plane) in the plane, we wish to perform a minimum number of cuts so that every two of the resulting curves intersect in at most one point. This problem turns out to be incredibly interesting with many consequences. We approach it by connecting it to extremal problems in graph-drawings in the plane. For example, assume that a graph $G$ on $n$ vertices can be drawn in the plane in such a way that every two edges, which belong to a cycle of length $4$ in $G$, cross an even number of times. What can be said about the number of edges of $G$? Still there are plenty of open problems.