June 24-July 27, 2018 research project descriptions
- Volumes of Hyperbolic Simplices
Faculty mentor: Chris Connell
Description: Simplices play an important role in both geometry and topology. In the former context they provide convenient building blocks for spaces, and in the latter context they arise in combinatorial descriptions of objects such as those used in homology and cohomology. For computations in both realms, knowing explicit volumes of simplices in terms of their geometric data become very useful. The volume of Euclidean simplices has been computed from different descriptive data, one important example of this is the Cayley-Menger determinant. On the other hand, sharp computations of the volumes of hyperbolic (constant negatively curved) simplices in high dimension has been much more elusive.
In this project we will be interested in both sharp estimates for the volume of hyperbolic simplices and applications of these estimates to the volume of hyperbolic manifolds and explicit values of an important topological invariant called simplicial volume. For calculation purposes we will be interested in extending Cayley-Menger type formulas to the hyperbolic setting and also estimating existing Schläfli type iterated integral formulas.
Prerequisites: Nothing more than vector calculus and basic linear algebra is necessary, but familiarity with elementary group theory would be helpful for the topological applications. Familiarity with a computer algebra platform such as Mathematica would also be helpful.
- Incidence Structures and Model Theory: Alex Kruckman
Faculty mentor: Alex Kruckman
Description: An incidence structure consists of a set of elements called "points", a set of elements called "lines", and a relation called "incidence" between them: p and l are incident when the point p lies on the line l. These structures are fundamental in the field of combinatorial geometry. A projective plane is an incidence structure such that every pair of distinct points are incident with exactly one line, and every pair of distinct lines are incident with exactly one point. The projective planes admit a natural generalization: say an (m,n)-pseudoplane is an incidence structure such that every m points are incident with exactly (n-1) lines, and every n lines are incident with exactly (m-1) points. So projective planes are (2,2)-pseudoplanes.
Model theory is a subfield of logic which studies definability in mathematical structures and seeks to classify mathematical theories by measures of complexity related to definability. In a recent paper, Gabriel Conant and I studied the model theory of projective planes and (m,n)-pseudoplanes. In this project, we will attempt to resolve some open questions raised in that paper concerning the combinatorial structure of (m,n)-pseudoplanes. Additionally, depending on the interests of the student, we may 1. study the model theory of planar ternary rings, a class of algebraic structures which coordinatize projective planes, or 2. explore other classes of incidence structures omitting certain configurations.
Prerequisites: Some experience with graph theory and/or abstract algebra. Experience with logic (especially first-order/predicate logic or model theory) would be welcome, but not necessary.
- Searching for Congruence Surfaces: Nick Miller
Faculty mentor: Nick Miller
Description: A congruence surface is a hyperbolic Riemann surface whose fundamental group is a congruence subgroup of an arithmetic group. In the non-compact setting, there are many examples of such surfaces, some of which have been shown by Schmutz to maximize the length of systole in their respective genus. However in the compact setting, it is unknown whether one can even produce a congruence surface for each possible genus.
In this project, we will first learn some basics of hyperbolic geometry with an emphasis on Riemann surfaces and then learn about arithmetic constructions of hyperbolic surfaces. We will then search for congruence surfaces in low genus, with the eventual goal of constructing infinitely many congruence surfaces, one for each genus.
Prerequisites: Some knowledge of elementary group theory and number theory would be useful but is not necessary. Some experience with SAGE and/or python would also be useful but again is not necessary.
- Cutting and Pasting of Manifolds and Group Actions: Carmen Rovi
Faculty mentor: Carmen Rovi
Prerequisites: Some familiarity with basic algebraic topology would be desirable.
- Homotopy Type Theory: Amr Sabry
Faculty mentor: Amr Sabry
Description: Homotopy Type Theory is a relatively 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 with a focus on computational interpretations. The exact topics will depend on the student's interest and background.
Prerequisites: Good background in abstract algebra and logic; excellent programming skills; experience with proof assistants and dependent types is a plus; knowledge of some topology is a plus.
- Simple Skein Theories in Finite Characteristic: Noah Snyder
Faculty mentor: Noah Snyder
Description: A string can be tied into a knot in many ways, but it can be hard to see whether you can turn one knot into another one. In order to tell knots apart we need to find invariants that distinguish. Several very interesting invariants like the Jones polynomial are defined using "skein relations", which let you cut out a small part of a knot (called a tangle) and replace it with a sum of simpler tangles. This procedure has a very algebraic flavor and you can think of each of these invariants as giving a kind of ``algebra" where you can multiply pictures by connecting up loose ends of string. Links aren't the only kind of objects you can study skein theoretically. Another great example is planar 3-valent graphs. For both knots and planar 3-valent graphs there are theorems classifying the "simplest" skein theories. In this classification a lot of fascinating examples come up related to things called quantum groups.
Usually the skein relations have coefficients that are complex numbers. In this project we will try to classify simple skein theories but where the coefficients instead live in a finite field like the integers modulo p.
Prerequisites: The main pre-requisite is a strong understanding of modular arithmetic, and enough ring and field theory to understand finite fields. Some familiarity with knot theory or graph theory is helpful but not required.
- Markov Chain Mixing Times: Graham White
Faculty mentor: Graham White
Description: It is a commonly-quoted fact that when shuffling a deck of cards, seven riffle shuffles are 'enough'. That is, after fewer than seven shuffles, many possible states of the deck are much less likely than they would be in a 'perfectly random' deck, while after more than seven shuffles the probabilities of most states are close to what they should be. More precisely, after seven shuffles the distance between the two probability distributions is small.
This sort of question may be asked for other Markov chains. If you have an interesting (and nice enough) random process, and you run it for long enough, it will approach a limiting distribution. How long is `long enough'? For instance, if one shuffles a deck of n cards by repeatedly swapping random pairs of cards, then about n * log(n) steps are required. If instead one swaps random pairs of adjacent cards, then this changes to n^3 * log(n) steps. Of course, both of these shuffling procedures are much slower than riffle shuffling, which only takes a number of steps proportional to log(n).
We will explore techniques which can be used to prove bounds of this sort, and investigate what bounds can be obtained for a variety of random processes. Many of these methods are combinatorial, for instance some involve carefully constructing a bijection between various sets of paths in the chain. Some possibilities for study could include random walks on graphs, groups, or combinatorial objects, depending on your areas of interest.
Prerequisites: Familiarity with probability will be helpful. There may be opportunities to steer the project toward graph theory, group theory, or combinatorics, so interest in any of these is a plus, but is not required.