### 2019 research project descriptions

- Geometry of Hyperbolic Percolation Clusters: Chris Connell
**Faculty mentor:**Chris Connell**Description:**On any infinite connected graph G we can play a popular game (at least among probabilists) called Bernoulli percolation whereby independently we remove each edge with probability 1-p. Percolation processes have been heavily studied in some settings such as Euclidean Lattices like Z^n and infinite trees, in part perhaps because of its connection to problems in physics and other areas.

Let p_c in [0,1] be critical connection parameter, namely the infimum of the values of p where there almost surely remains at least one infinite connected component of the graph. We are interested in studying "expected" geometry of connected components under percolation, especially near a threshold value like p=p_c, on graphs that exhibit hyperbolic behaviour such as infinite trees. (We expect to have no infinite components at p=p_c.) We are seeking to understand metric questions like the expected growth rates of balls in infinite components at supercritical values of p, the expected escape rates of certain paths and properties of certain metrics arising from the related "first passage percolation." One motivation for this project is the search for useful algebraic invariants in the case that G is the Cayley graph of a group for a given generating set.**Prerequisites:**An introductory knowledge of probability would be very helpful, but can be learned. Some knowledge of group theory could be an asset as well.- Pattern formation and high dimensional probability: Wai Tong (Louis) Fan
**Faculty mentor:**Wai Tong (Louis) Fan**Description:**Reaction-diffusion systems have attracted much attentions as prototypical models for pattern formations in nature such as stripes and spots. For example, reaction-diffusion equations are the main mathematical tools employed to analyze the Turing-instability (also known as diffusion-driven instability), a fundamental mechanism that leads to pattern formations out of homogeneous states. However, when different sources of random perturbations are present, it is no longer clear how to quantify different kinds of patterns and how do these patterns arise and evolve in time.

In this project, we will survey recent work on stochastic Turing patterns as well as probabilistic techniques in analyzing these random objects. This interdisciplinary research involves stochastic simulations of high-dimensional Markov chains, high-dimensional image data analysis, comparisons of spatial statistics and classification methods in machine learning.**Prerequisites:**Basic knowledge in Markov chains and random variables. Proactive in learning by doing. Computer programming skills would be a plus, but not required.- The topography of quantum states in polygons: Chris Judge
**Faculty mentor:**Chris Judge**Description:**Wave functions in quantum mechanics describe the probability that some observable quantity in the system takes on certain values.

This project concerns wave functions for a single particle that is constrained to a polygonal region in the plane.

The purpose of this project is to find wave functions whose level sets have interesting features.

For example, can one find figure-eights or cusps? For rectangles, constructions of wave functions can be accomplished by superposing products of sines and cosines.

For many other polygons, one needs computer software.

Much of the work will consist of combining thought experiments and computer software explorations with rigorous mathematical justification.**Prerequisites:**No prior experience with quantum physics is necessary though it might be helpful.

Familiarity with software such as Mathematica, Maple, Matlab, or Sage is preferable.- Cauchy's Integral Formula for Clifford Valued Functions: Matvei Libine
**Faculty mentor:**Matvei Libine**Description:**reu_libine_2019.pdf**Prerequisites:**Strong background in undergraduate complex analysis, a course on algebra

involving rings and modules (needed to understand the definitions of Clifford

algebras and modules), some familiarity with integrating differential forms

over manifolds and Stokes' Theorem (since we are looking for an integral

formula where the loop in (1) is replaced by a higher dimensional contour

of integration).- Extremal length of non-simple curves: Dylan Thurston
**Faculty mentor:**Dylan Thurston**Description:**We will look at the conformal (a.k.a. complex analytic) structure of surfaces, and new ways of finding interesting metrics on them, via*extremal length*of curves.

Fix a Riemann surface $S$; this is a closed 2-manifold with a complex structure. This is the same as a class of Riemannian metrics up to conformal rescaling (preserving the local angles but not lengths). Fix also a closed curve $C$ on $S$, considered up to homotopy. Then, for any metric $g$ in the correct conformal class on $S$, we can consider the*length*$l_g(C)$ of $C$, the shortest it can get in the homotopy class. To find the*extremal length*, we optimize over the metrics as well; we compute \[

\mathrm{EL}(C) = \sup_g \frac{l_g(C)^2}{\mathrm{Area}(g)},

\]

where we take the supremum over the conformal class of metrics. (We divide by the total area of the metric as a way of normalizing, so that scaling the whole metric uniformly does not change the value.)

To get an idea what this looks like, it's easy to see that for a metric realizing the supremum, there must be a shortest-length representative of $C$ passing through every point of the surface. (Otherwise you could locally scale down the metric near that point.)

When the curve $C$ is simple (does not intersect itself), the solution to the supremum problem is known: the metric is locally flat except at a finite number of points.

In this project, we will investigate what the supremum looks like for non-simple curves $C$. Only a few examples are known, and we will aim to extend our understanding. We will approach this from two viewpoints:- Theoretical: try to construct extremal lengths for some examples of surfaces and non-simple curves.
- Experimental: figure out a way to represent surfaces via combinatorial models such as elastic graphs, square-packings or triangulations in order to approximate the optimizing metric and thus compute extremal length numerically.

**Prerequisites:**Undergraduate real analysis is required. Complex analysis and programming experience will both be helpful, but not required.- Markov Chain Mixing Times: Graham White
**Faculty mentor:**Graham White**Description:**It is a commonly-quoted fact that when shuffling an ordinary deck of playing 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.