Explicit formulas in Schubert calculusFaculty Mentor:
Recent work has shown that the solutions to certain classes of intersection problems can be solved explicitly by classical projective geometry constructions. The resulting formulas are often quite complex, and there are some open questions about cases when such constructions "commute". (The simplest commutation result is equivalent to mdoularity of the lattice of subspaces in a finite dimesnional space. More complex ones cannot be deduced from modularity.)
The prospective student participating in this project will need to learn some of the basics of the combinatorics involved, and its relations to intersection theory and group representations.
Torsion Subgroups of CAT(0) GroupsFaculty Mentor:
In recent years, CAT(0) groups have generated an enourmous amount of interest among geometers and topologists alike. First, a CAT(0) space is a geodesic metric space whose triangles are at least as "thin" point-wise as the Euclidean triangle with the same side lengths. A CAT(0) group is then a discrete subgroup of isometries of a CAT(0) space whose quotient is compact. A simple example of such a group is the lattice of integers Z^2 which acts on the plane by translations with quotient space R^2/Z^2, a torus. Finally, a torsion subgroup of a group is subgroup whose elements all have finite order. It has been conjectured that torsion subgroups of CAT(0) groups always have finite cardinality. This might sound like hard-core group theory, but it is really mostly about geometry. Motivated in part by this conjecture, we will aim to understand better how these torsion subgroups act on a CAT(0) space and their attendant structures such as the "boundary at infinity".
A course in group theory and/or topology is helpful, but not absolutely necessary.
Computational methods and models in mathematical biologyFaculty Mentor:
Depending on the student's interests, the project will involve:
# 1) the development of statistical / computational methods for estimating (in an unbiased way as possible) levels of DNA sequence variation (and covariation) from whole-genome surveys that are now generating very high-throughput data, but erroneously so;
# 2) developing models for estimating the vulnerability of organisms to cancer with increasing levels of organismal complexity (i.e., more cell divisions); or
# 3) developing models for the evolution of various types of genomic elements in populations of various sizes (which influence the role of chance in evolution).
A background in calculus, differential equations, and some probability theory is needed. Some familiarity with computer coding would be highly desirable.
Studies in natural logicFaculty Mentor:
The term 'natural logic' covers systems of logic that are designed to be as close as possible to natural language. More standard systems of logic such as first-order logic were designed and are currently studied with an eye towards a different field, the foundations of mathematics. The idea with natural logic is to study logical systems that look more like ordinary language, and also systems which are decidable (unlike first-order logic). This leads to an area with connections to linguistics, computer science, philosophy, and psychology. And because it is fairly new, there are still lots of interesting mathematical problems to solve.
The more classes in discrete mathematics, the better. Although it would be good to have had a logic class that presented the completeness theorem of propositional logic in detail, this isn't strictly needed. Classes in theoretical computer science, algebra, and/or combinatorics would be a plus. The REU project itself would depend on the student, so the wider the background, the more possibilities would present themselves.
Complex dynamics and Smale's mean value conjectureFaculty Mentor:
Kevin M. Pilgrim
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?
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.
The Geometry of Tilings
In general, a tiling problem consists of a set of tiles with the purposes of filling a given space with these tiles completely so that the tiles do not seriously overlap. There is a vast literature (Grunbaum/Shephard: Tilings and Patterns is a wonderful book to look at). Many results are exhaustive in the sense that the authors conducted an exhaustive search for tilings with given properties, but there are also stunning results that reveal deep connections to other areas. Let me just mention aperiodic tilings and their relation to quasicrystals, and polyomino tilings related to finitely generated groups, as discovered by Conway and Lagarias.
There are many unsolved problems, and we will consider some of those, both in their original, typically Euclidean context, as well as in other spaces as the hyperbolic plane.
While not essential, the following would be helpful: familiarity with Euclidean, hyperbolic and spherical geometry; basic group theory and the concept of a group action on a set; linear algebra; some familiarity with Mathematica.
From left to right: Garrett Proffitt, Hayley Miles-Leighton, Michael Anselmi, Nathan Dowlin, Hamza Ghadyali, Linh Truong, Elizabeth Kammer, Komi Messan