### Project 1:: Kaszonyi numbers for edges of snarks

**Faculty Mentor:**

Richard Bradley**Description:**

A ``simple'' graph is a connected (but not necessarily planar) graph (``network'') with finitely many vertices (``nodes'') and edges (``lines'' connecting two ``nodes'') and no ``loops'' and no ``multiple edges''. A simple graph is called ``cubic'' (or ``3-valent'') if every vertex in it is connected to exactly three edges. A ``snark'' is a simple cubic graph which, in addition to satisfying a couple of technical ``nontriviality'' properties, CANNOT be ``edge-3-colored''; that is, its edges cannot each be assigned one of three given colors in such a way that no two edges connected to the same vertex have the same color. If G is a snark and e is an edge of G, then the cubic graph H that one obtains from G by deleting the edge e and ``smoothing'' (or ``eliminating'') the endpoint vertices of e, has the following property: The number of edge-3-colorings of H with three given colors is of the form 18N for some nonnegative integer N. In that factor 18, a factor of 6 occurs trivially from the six permutations of three given colors; the ``remaining'' factor of 3 is an elementary (but not totally trivial) observation of Laszlo Kaszonyi in the 1970s. It is known that for particular edges e of particular snarks G, that integer N can be 0 or 1 and it can be any positive integer whose prime factors are 2, 3, 5, and/or 7; but little else beyond that is known. The main question: What are other possible values of N (for an edge e of a snark G)? There are also numerous variations on that question, involving snarks with special structural properties.**Prerequisites:**

Some group theory and/or combinatorics helpful but not essential### Mathematics, Neuroscience, and Signal Processing

**Faculty Mentor:**

Evie Malaia, Department of Psychological and Brain Sciences, The Cognitive Neuroimaging Laboratory**Description:**In order to understand how the brain functions, neuroscientists employ electroencephalographs (EEGs). These devices record brainwave activity at typically 32 sites. At each site, this activity is recorded in the form of an electromagnetic wave. As an analog signal, this is a real-valued function of time, typically measured in (milli)seconds. Hence, a single trial of an experiment results in 32 graphs of amplitude versus time, or time series. Neuroscientists use EEGs as windows into the inner workings of the brain.

The mathematics involved in analyzing these time series involves: linear algebra, difference equations, Fourier and Laplace transforms, dynamical systems, and statistics, for example.

In this project, students will learn some fundamentals of neuroscience and be exposed to some of the applications of mathematics to problems in neuroscience. Students will make extensive use of the software package Matlab and its toolkits for signal processing. It is expected that acquiring the needed background will take the bulk of the time. If time permits, students will investigate some specific problems posed by researchers concerning correlation between frequency factor at one site and the amplitude at other sites.

**Prerequisites:**

Linear algebra is essential. Differential equations, some exposure to the complex exponential function $\exp(it)$, statistics, and some computer experience are desirable but not essential.### Project 3:: Codes and Designs

**Faculty Mentor:**

Will Orrick**Description:**Error-correcting codes, in addition to their many applications, are intimately related to some intriguing structures in mathematics. The famous binary Golay code, for example, is connected with the Leech lattice (which provides an extremely efficient packing of balls in 24 dimensions) and with some of the sporadic finite simple groups. A certain subset of the codewords in the extended Golay code can be used to form a 24 by 24 matrix of -1s and 1s that has orthogonal rows (a Hadamard matrix) and that, as a consequence, has maximal determinant among all {-1,1} matrices. We will explore the connection between codes and maximal-determinant matrices in various dimensions with the goal of classifying such matrices, and possibly discovering new examples.

**Prerequisites:**

A course or two in linear algebra will be necessary. Some knowledge of group theory, or a desire to learn, would also come in very handy. We may do some calculations using computer algebra packages such as Mathematica or Sage, so interest in computation is desirable. Prior experience with a programming language is a definite plus.### Project 4:: Braids in mathematical physics

**Faculty Mentor:**

Will Orrick**Description:**Certain models formulated on two-dimensional lattices that were originally devised to describe phase transitions or to serve as discrete analogs of quantum field theories turn out to have an amazingly rich mathematical structure, with connections to number theory and knot theory. Diagonalizing the transfer matrix associated with a model gives the energy levels of the physical system, which vary with the physical parameters of the system. The energy levels undergo a reorganization as the parameters pass through a critical point. In each phase, the energy levels can be labeled by elements of a braid group. The goal of this project will be to work out in detail the mapping between energy levels in different phases and their braid labelings.

**Prerequisites:**

A course or two in linear algebra will be necessary. Some complex analysis may be helpful. We will be carrying out some computations in numerical linear algebra, so programming experience and interest in computational issues are desirable.### Project 5:: Complex dynamics and Smale's mean value conjecture

**Faculty Mentor:**

Kevin M. Pilgrim**Description:**Suppose $f(z) = z + a_2z^2 + ... + a_dz^d$ is a complex polynomial having a fixed point of derivative 1 at the origin. One version of Smale's mean value conjecture asserts that there exists a critical point $c$ of $f$ satisfying $|f(c)|/|c| \leq 1-1/d$. Complex dynamics suggests a candidate for this critical point. Does this candidate always satisfy this inequality? It turns out the answer is yes for $d=2, 3$ but beyond that nothing is known.

**Prerequisites:**

Multivariable calclus (e.g. finding minima of functions of more than one variable) is essential. Beyond this, the project can be tailored to suit a variety of backgrounds. Complex analysis and some programming experience would be helpful, but not essential.### Project 6:: Monodromy groups and the classification of critically fixed rational mapss

**Faculty Mentor:**

Kevin M. Pilgrim**Description:**Suppose $f(z)=p(z)/q(z)$ is a rational function of degree $d \geq 2$ with the property that each critical point of $f$ is also a fixed point. There is a conjectural classification of such maps. To test this conjecture, it is of interest to compute various group-theoretic invariants associated to $f$. This project will involve a mixture of complex analysis, topology, and group theory. Development of an algorithm, and an implementation, for computing these invariants is one goal.

**Prerequisites:**

Group theory and a bit of programming experience.### Project 7:: Polymer organization and cell function

**Faculty Mentor:**

Sidney Shaw**Description:**The Shaw laboratory investigates the mechanisms controlling polymer organization and function within cells. The microtubule cytoskeleton provides structure to a cell and vectoral information for intracellular transport. Microtubules are formed through the head-to-tail association of tubulin protein dimers into protofilaments, 13 of which create a hollow tube with important physical properties. Using genetics and live-cell imaging techniques, the Shaw laboratory has amassed substantial data on microtubule polymerization dynamics within living plant cells. Through analytical modeling and large-scale Monte Carlo simulations, we are attempting to reconstitute aspects of the measured biological system in order to determine how the polymer mass comes to steady state and how the microtubules form specific patterns within cells.

**Prerequisites:**

differential equations and numerical analysis or programming experience; Fourier series helpful but not essential.