Faculty Mentor: Hari Bercovici

Description: Some problems in the study of infinite dimensional algebras depend on a good understanding of finite-dimensional spaces. One such problem can be stated as the need to find effective formulas for spaces satisfying a number of intersection requirements. Here is a sample question which does not allow an effective formula:

We are given a six-dimensional complex vector space V, subspaces A,B,C of dimension four, and subspaces A',B',C' of dimension two such that A' is a subspace of A, B' is a subspace of B, and C' is a subspace of C. Find a space M of dimension three which intersects each of A,B,C in a space of dimension two, and each of A',B',C', in a space of dimension 1. (This problem has generically two solutions.)

There is a class of such problems which should however admit effective formulas for their solution, at least generically.

Prerequisites: Studying this problem will require learning the combinatorics associated with such intersection problems (also known as Schubert calculus). Understanding the infinite dimensional implications will require learning some more advanced analysis. Very good understanding of linear algebra is a must. Some aspects of this problem requires quite a bit of experimentation.

Faculty Mentor: Chris Connell 

Description: There are many open conjectures about the geometry and topology of surfaces in Euclidean space which look saddle-like in a neighborhood of every point (negatively curved). Some of these conjectures are fairly old and potentially very difficult. (Milnor's conjecture on asymptotic curvature and the existence of bounded embeddings are two famous examples.) However, questions about the possible asymptotic shapes and possible topologies of properly embedded surfaces are more accessible. For this project, I would like to begin working on building new explicit examples of negatively or nonpositively curved surfaces with certain topological and/or geometric properties. The project may transition into proving general existence or nonexistence results for classes of surfaces. An important aspect of this is to understand some of the possible geometries of the "ends" at infinity of nonpositively curved surfaces.

Prerequisites: Vector calculus and a keen interest in geometry. An introduction to differential geometry (e.g. a course on curves and surfaces in Euclidean space) would be very desirable, but not absolutely necessary. Experience with a computer algebra system such as Mathematica or Maple is also a plus, but also not necessary.

Faculty mentor: Allan Edmonds 

Description: How many elements of order 2 (called "involutions") can a finite group of order n have? Of course, if n is odd, then there are no elements of order 2, by Lagrange's theorem. On the other hand most algebra students have seen the elementary result that if every nontrivial element of a finite group has order 2, then the group is abelian, and therefore isomorphic to (Z/2)^r for some r. In particular the order of the group is a power of 2 in that case. In recent work coming out of an analysis of geometric properties of high dimensional simplices, in particular, their groups of isometries, I needed to know what the maximum number of involutions in a finite group of order n is, where n is necessarily even but not necessarily a power of 2. Although this seems like a natural question, the answer wasn't found in the standard literature. It turns out that I was able to prove that the number of elements of order 2 in a finite group of even order n is at most n/2 + (n_2)/2 - 1, where n_2 denotes the "2-part" of n. Moreover one can characterize the kinds of groups of order n for which this maximum value is achieved. For this project we'll first of all try to understand the idea of the proof of this result. At the same time we will explore possible generalizations. For example, it's clear that if a finite group of order n has the property that n - 2 elements have order 2, then indeed n - 1 elements have order 2 and the group is of the form (Z/2)^r. We'll try to look for the next highest number of elements of order 2 below the maximal number mentioned above. We will at a minimum collect data on the possible numbers of involutions by closely examining the known lists of finite groups of low order. We will expect to use the computer program GAP (at Wikipedia) as part of this study. Indeed we'll be satisfied if we can collect a good amount of data and propose some good conjectures.

Prerequisites: A solid year of abstract algebra with a good emphasis on group theory, including, for example, the Sylow theorems. A willingness and an ability to download and install GAP (official site) and work with it.

Faculty Mentor: Mike Jolly 

Description: Direct simulation of genetic mutation can be done by averaging the results of random walks. This is prohibitively slow, however, even when there are just a handful of gene types involved, and the size of the population modest. In the limit of infinite population size the problem can be expressed as a partial differential equation (PDE), on a right triangle for three gene types, and on a tetrahedron for four gene types. It may surprise you that we can numerically solve the PDE more efficiently than we can carry out direct simulation with random walks. A recent REU student developed a computer code which converges as long as mixed derivative terms are not present. When the mixed derivative terms are introduced, the computed solution relaxes to a steady state, but does not improve with refinement of the mesh. The challenge is to handle these terms so as to avoid referring to grid points outside the tetrahedron, and ultimately converge to the actual solution.

Prerequisites: This project involves a scheme based on finite difference approximations, whose plausibility can be readily accepted by anyone who has seen a Taylor expansion. A student working on this project need not have any prior exposure to genetic modeling or PDEs.

Faculty Mentor: Mike Jolly 

Description: A point vortex is a concentrated type of fluid flow, whose movement is determined by a system of ordinary differential equations. This makes them for several reasons easier to track than general idealized flows governed by Euler's equation . In two dimensions certain configurations of these vortices are known to balance each other. This project concerns computing new configurations, both in the plane, and on a sphere. The technique will be to apply a powerful continuation package AUTO to morph known configurations into new ones. A natural application is to the atmosphere (weather patterns). A general reference is The N-Vortex Problem by P. K. Newton, though we will only touch on a small part of this book.

Prerequisites: programming skills, multivariable calculus, linear algebra

Topics to be discovered: differential equations, complex variables, continuation algorithms

Faculty Mentor: Chris Judge

Description: In topology, a pair of pants is a surface that is homeomorphic to a sphere with three disjoint disks removed. Every surface with finite genus greater than 2 can be decomposed into pairs of pants by cutting along simple closed curves on the surface. One can move from one decomposition to another by replacing certain curves with others. The resulting combinatorial object, the pants complex, is the subject of much current research interest. For example, it was recently shown that the automorphisms of the pants complex exactly correspond to the homotopy classes of all homeomorphisms from the surface to itself, the so-called mapping class group. In this project, the student will explore the geometry of this complex.

Prerequisites: Some familiarity with topology at an intuitive level would be helpful. Though no computer programming experience is required, exploration via the computer is certainly feasible and might be very useful.

Faculty Mentor: Larry Moss

Description: Logic is a discipline going back to ancient times. In its long history, the two outstanding developments are Aristotle's monumental work on the syllogism, and Frege's work on quantifiers leading up to the standard systems of modern logic. Quantificational logic leads to contemporary mathematical logic. The received view is that it 'buried' the syllogism, and math students who have studied some logic may be surprised to hear that there are many interesting problems coming from a resurrected syllogistic-style view. The point is to find fragments of ordinary language and to axiomatize them. This kind of work is called 'natural logic'. It could be applied to computer processing of natural language because the systems coming from natural logic are decidable (whereas standard logic is undecidable).

Prerequisites: Natural logic is a research area just getting off the ground, and there are many mathematical problems of interest. To work in the area, it would be important to have a good background in abstract algebra. It would also be good to have seen logic in some form or other, especially completeness theorems for propositional logic. I can think of several projects that would be good for REU students. Some have a combinatorial flavor, others involve connections to term-rewriting theory, and yet others are connected to probability and statistics.

Faculty Mentor: Larry Moss

Description: This project deals with infinite sequences of numbers, here called 'streams'. There is a natural operation of prefixing a stream by a number. For example,

0. (1 2 1 2 1 2 1 2 . . . )

denotes the stream (0 1 2 1 2 1 2 1 2 . . . )

There is also the operation 'zip' or 'merge', shown in the example below:

zip((0 1 2 3 4 5 . . .), (0 1 4 9 16 . . . )) = (0 0 1 1 2 4 3 9 4 16 . . . )

So 'zip' takes two streams and interleaves their entries, starting with the first entry in the first stream.

Now consider the following system of equations for streams:

(*) x = 1. zip(x,y)

y = 0. zip(y,x)

It turns out that there is a unique solution to this system. The first few entries of the solutions are

x = (1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 . . .)

y = (0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 . . .)

The solution for y is the tail of the famous Thue-Morse sequence t. This sequence t has a number of interesting properties, most notably that for every non-empty finite sequence w, the sequence www is not found in t.

Now, t is usually defined in very different ways from the system (*). This project will attempt to see whether one can prove the basic properties of t using only the representation in (*). Further, one can go on to look at other kinds of fixed-point equations for streams, for fractal sets, and for other mathematical objects. The purpose of this project is to see how far one can go in their study the fixed-point representations as much as possible.

Faculty Mentor: Will Orrick

Description: How large can the determinant of an n by n matrix be if the matrix elements must be chosen from the set {-1,1}? Hadamard proved that the determinant can be no larger than nn/2 and that this bound is reached if and only if the rows of the matrix are orthogonal to each other. This in turn is possible only when n equals 1, 2, or a multiple of 4 because of the restriction on the matrix elements. (Prove this!) The converse of this statement is a major open question in combinatorics: does a {-1,1}-matrix with orthogonal rows (called a Hadamard matrix) exist for every multiple of 4? Another open problem concerns sizes that are not multiples of 4. Some improvements on Hadamard's bound have been found, but relatively little of a constructive nature is known.

An error correcting code is a set of codewords defined over some alphabet (typically {0,1} or {-1,0,1}). In order to correct as many errors as possible, any two codewords should differ in as many positions as possible (so as to be able to distinguish them even if some letters get corrupted). Because of orthogonality, the set of rows of a Hadamard matrix makes a code that is capable of correcting many errors. In recent years, however, the flow of information between disciplines has been mostly in the opposite direction: rather than Hadamard matrices being used to construct good codes, knowledge about codes has been used to understand the structure of Hadamard matrices.

In this project we will explore both of these areas with the aim of shedding light on the structure of Hadamard matrices and other maximal determinant matrices. Some possible avenues to explore are (1) using knowledge about codes to produce maximal determinant matrices that are conjectured to exist, but have not yet been constructed; (2) using the discrete Fourier transform to search for new record determinants; (3) studying two special classes of Hadamard matrices, skew Hadamard matrices and regular Hadamard matrices, with the goals of inventing methods for producing new examples and of understanding the associated codes.

Prerequisites: A solid course in linear algebra. A one semester group theory course and a smattering of number theory would be helpful. A willingness to use the computer in mathematical exploration is highly desirable. Serious programming ability is not necessary, but might come in handy.

Faculty Mentor: Kevin Pilgrim

Description: Iterating the map z --> z^2+c yields fractal Julia sets in the dynamical z-plane and the Mandelbrot set in the parameter c-plane. Associated to certain complex parameter values c are homomorphisms f_c: H --> B where B is a braid group and H is a subgroup of B with finite index. Knowledge of f is closely related to the combinatorics of z --> z^2+c as a dynamical system. The number of examples where f_c has actually been computed is embarrassingly small. Let's remedy this!

Prerequisites: A one-semester course in group theory and a basic familiarity with complex numbers. No computer programming experience is required.