- Heegaard Floer homology and torus bundles
Faculty Mentor: Corrin Clarkson
Heegaard Floer homology is a topological invariant i.e. it is a way of measuring topological complexity. Given such a tool, it is natural to ask whether or not it is precise enough to distinguish between distinct topological spaces. In general, Heegaard Floer homology is good a distinguishing between spaces, but it is also known to have blind spots. In fact, there are infinitely many examples of distinct manifolds having the same Heegaard Floer homology. Determining whether this invariant distinguishes between a particular pair of manifolds can be challenging due to the difficult nature of computing Heegaard Floer homology. This is why we will focus on torus bundles.
Torus bundles are a particularly nice class of three-manifolds. There is a correspondence between these manifolds and elements of SL(2,Z) which allows us to use linear algebra to describe them. Moreover, there are relatively efficient algorithms for computing the Heegaard Floer homology of these manifolds. The question we will be exploring is the following. Is Heegaard Floer homology a complete invariant of torus bundles i.e. does it distinguish between any two distinct torus bundles?
Linear algebra, some programming experience ideally in Python, a basic understanding of group theory would also be useful, but is not necessary.
- Convex Hulls of Closed Curves
Faculty Mentor: Chris Connell
For a curve c in
Rn, its convex hull is the smallest convex set containing c. There are a number of interesting questions one can ask about the relationship of a curve c to its convex hull. Many of these fall into the mold of an "isoperimetric" type problem where one geometric quantity is being optimized subject to another geometric constraint. A number of authors, including I. Schoenberg and A. Weil, have made important contributions in this area. We will focus on one such question that was related to me by M. Ghomi in 2004. Namely, whether a round planar circle maximizes the surface area of the convex hull among all closed curves c in R3of a fixed length L. (Here the surface area of the degenerate convex hull of a round circle should be interpreted as twice the area of the disk.) Our first plan will be to attack this problem from an existing approach built upon prior work of N. Krabbenhoft and myself. As is often the case, there are a number of analogues to this question in other settings and in higher dimensions which we can also explore depending on student interest.
Multivariable caluculus and enthusiasm is a must. An introductory course in either differential geometry (curves and surfaces) or ODE's would be a welcome bonus, but not necessary.
- Vertical averaged velocity in Bénard convection
Faculty Mentor: Michael Jolly
This project is suitable for a student with strong computational skills and a keen interest in partial differential equations from fluid dynamics. The question is whether the vertical average of a 3D fluid driven by a temperature imbalance on the boundary displays the features of 2D turbulence. Toward an answer for this, we would carry out direct numerical simulation of the 3D problem and then extract from that solution certain quantities which determine the nature of the body force in the equation for the vertically averaged velocity.
The simulation of the 3D problem can be done by Dedalus: http://dedalus-project.org/ a suite of Python scripts which call computational modules. In fact there is an example specifically for the problem of interest. The task then, is to construct the body force for the vertically averaged velocity. Since the relevant quantities are defined in terms of derivatives and integrals, and Dedalus is a spectral code, this should be straightforward. The student would then interpret the results in light of our previous work which explains the criteria for turbulence. We have already done analysis to determine an upper bound on this body force. Without a meaningful lower bound however, we cannot say if the force is strong enough to support 2D turbulence. This is why we turn to simulations.
Good computational skills, some exposure to PDEs and an open mind are essential.
- Inference in Natural Language
Faculty Mentor: Larry Moss
When computers carry out human-level reasoning, they do it in a number of ways. Frequently they do it by translating some natural language (NL) input into a very different form, some language or other that looks much more like formal logic (FL) than NL. This has the advantage of allowing one to use off-the-shelf tools having to do with FL, and this is good. But translation into FL comes with disadvantages: first of all, one is limited to very special kinds of reasoning; second, whatever one gets from FL then has to be re-translated back to NL if one wants to use it. This project is connected to Prof. Moss' project of Natural Logic, an attempt to do reasoning in something that looks more like NL than FL. It is a big interdisciplinary project and has people working on all aspects of it, publishing papers and writing programs.
Next summer would mark the fourth time that someone worked on a summer project related to Natural Logic. There are a number of projects available, mostly having to do with proving theorems about logical systems or about developing and implementing algorithms.
The ideal student would be one who has had courses in one or more of the following: logic, theoretical computer science, combinatorics, linguistics, algorithms, or programming. One would not need all of these (of course!), but the more the better.
- Conformal dimension and energy of graph maps
Faculty Mentor: Kevin Pilgrim and Dylan Thurston
- Cutting and pasting of manifolds and TQFTs
Faculty Mentor: Carmen Rovi
Suppose M is a closed manifold which can be cut open along a codimension 1 submanifold. By doing this cutting operation we obtain two manifolds with the same boundary which we can now glue back together using an automorphism of the boundary which is not the identity. The new object that we obtain is said to be "cut and paste" or SK-equivalent to the manifold M we started with. It turns out that the cut and paste operation doesn't only define an equivalence relation, it defines certain groups called the SK-groups. The SK-groups were first defined some 40 years ago, but they haven't been developed or investigated in depth since then, even though they have interesting connections with very active areas of research. An interesting topic is to understand which topological invariants are invariants of the cut and paste operation. The goal of this project will be to describe which cut and paste invariants are partition functions of topological quantum field theories.
Some familiarity with basic notions of topology would be desirable.
- Homotopy Type Theory
Faculty Mentor: Amr Sabry
Homotopy Type Theory is a new development that establishes surprising connections between logic, algebra, geometry, topology, computer science, and physics. This project will involve formalization of mathematical results in homotopy type theory. The exact topics will depend on the student's interest and background.
Good background in abstract algebra and logic; excellent programming skills; knowledge of some topology is a plus.