Fletcher Jones Summer 2010 Research Projects

Project One: Chain Partitions of Normalized Matching Posets

Supervised by Shahriar Shahriari, Department of Math, Pomona College

Consider all 8 subsets of {1, 2, 3} and organize them into the following "chains":

You have now partitioned these subsets into 3 chains with sizes 4, 2 & 2. But could you partition the 1024 subsets of {1, ..., 10} into 16 chains of size 5 and 236 chains of size 4?

The subsets of a finite set are an example of a finite partially ordered set (poset), and partitions of posets into chains have been the object of a number of long standing conjectures in Combinatorics. In fact, an elegant twenty year old conjecture of Jerrold Griggs gives necessary and sufficient numerical conditions for the existence of a partition of subsets of a finite sets into chains with prescribed sizes. This conjecture has recently been generalized to a much larger set of posets and some very special cases have been proved. This larger collection of posets are called normalized matching posets and include the poset of subspaces of a finite dimensional vector space over a finite field as well as the divisors of a positive integer.

During the summer, our team will work on a number of special cases of this conjecture. We will look at many examples and try to come up with patterns and proofs. In addition to reading and deciphering papers and working on new results, students will be writing--in LaTeX--and presenting--in Beamer.

The ideal candidates for this project will have some background in Combinatorial Problem Solving and/or Abstract Algebra.

Project Two: Rack module enhancements of counting invariants

Supervised by Sam Nelson (Claremont McKenna College)

A knotty problem, a tangled web... Two of the knots below are the same; one is diff erent. Can you tell which is which?

Knot Theory involves the search for knot invariants, mathematical machinery for telling different knots apart. Why would we care?

• DNA. DNA molecules are little strings which get tangled up in knots. Many drugs (e.g. certain antibiotics) work by interfering with the cell's ability to untangle DNA
• Protein folding. Protein molecules are strings of amino acids which fold up into a knotted shape. The fi nal shape determines the molecule's chemical properties
• Quantum gravity. Current work on quantum gravity involves mathematical structures called quantum field theories, a certain type of which, known as a topological quantum field theory or TQFT defines a type of knot invariant, known as a quantum invariant
• Much more. Knot theory is relevant for fundamental questions, from "What is the shape of the universe?" to "How can I keep my shoes on?"

Knot invariants have a long history, going back to Gauss’s linking integral and Lord Kelvin’s idea of atoms as knots in the ether. A few examples:

• (1920s) Alexander polynomial
• (1950s) Conway polynomial
• (1950s) Fox coloring
• (1970s) Group system -- a complete invariant
• (1980s) Knot quandle -- a simpli ed complete invariant
• (1980s) Jones polynomial
• (1980s) HOMFLYpt polynomial and Kau man polynomial
• (1990s) CJKLS quandle cocycle invariants
• (2000s) Quandle module cocycle invariants

In this project we will develop a new infinite family of computable knot invariants using a generalization of quandle module invariants called rack modules. Computation of the invariants will involve linear algebra and some python programming; writing up our results will involve LATEX. Side effects may include publication in a professional journal, speaking opportunities at professional conferences, and addiction to low-dimensional topology.

Project Three: Analysis of microfluidic mixing in a drop

Supervised by Ali Nadim (CGU) and Andrew Bernof f (HMC)

One goal of the biotech industry is to build a lab-on-a-chip — that is, develop the ability to manipulate and mix extremely small quantities of fluids for blood tests, DNA sequencing and a host of other applications. Mixing at these small scales is particularly challenging because viscous forces dominate, rapidly damping fluid motion. This project will analyze mixing within a small liquid droplet as it moves along a surface. The three dimensional flow field within the droplet can be approximated using a technique known as lubrication theory. Armed with an analytical expression for the 3D velocity field, mixing by advection and molecular diffusion within such drops will then be modelled and compared with experiments. Based on earlier work on similar problems, it is anticipated that moving the drop along a square or stair-case pattern on the surface will result in chaotic mixing, enabling reagents within the droplet to become well-mixed rapidly.

An example of a somewhat simpler 2D problem that has already been analyzed by such methods is illustrated in the accompanying figure. Here the flow is driven by surface tension gradients along the circular boundary of the 2D drop, generating recirculation zones in the drop that are periodically blinked between east-west and south-north orientations, as shown in the top panels of the figure. Depending on the period, T, of switching, various degrees of mixing can be obtained, as shown in the bottom panels that provide Poincar ́e sections of particle motion.

In this project students will learn about fluid dynamics and the Navier-Stokes equations. They will also learn dimensional analysis and scaling that enable simplifying the equations into a form that makes them analytically tractable. After obtaining the flow field, the students will study and apply methods from the theory of nonlinear dynamical systems to characterize the chaotic advection and mixing within the drops. Finally, to model the spread of diffusive tracers in such flows, the students will learn about Monte Carlo particle methods that are used to simulate mixing.

The students will be co-advised by Professors Andrew Bernoff (HMC) and Ali Nadim (CGU). Both faculty members have a long history of mentoring graduates and undergraduates, many of whom are pursuing/have obtained graduate degrees in mathematics.