### 2020 research project descriptions

- Colleen Delaney / Julia Plavnik: Quantum link invariants and the classification of modular tensor categories
**Faculty mentor:**Colleen Delaney Julia Plavnik**Description:**Modular tensor categories are algebraic structures that encode topological quantum field theories in 3 spacetime dimensions and give theories of particle-like objects called anyons in exotic quantum states of matter. In the theory of anyons, certain spacetime trajectories of anyons carve out shapes like knots and links with some probability that is given by a number which depends only on the topology of the knot or link. Mathematically these are given by invariants of (framed) knots and links that can in principle be computed from the data of modular tensor category (MTC). Until recently it appeared that MTCs were determined by the values of just two link invariants: the Hopf link and the once-twisted unknot, also called the modular data. In this project we will investigate whether another invariant of MTCs coming from the Whitehead link is a powerful invariant for the small MTCs whose classification is already known.

**Prerequisites:**Prerequisites for a successful project include strong linear algebra skills and familiarity with the basics of knot theory. Exposure to abstract algebra or computer programming may be helpful but is not required.

- Louis Fan: Probabilistic techniques in evolutionary dynamics
**Faculty mentor:**Wai Tong (Louis) Fan**Description:**Through basic examples and research problems in probability, we will investigate the power the the limitation of random dynamical systems (including Markov chains and differential equations with noise) in the study of evolutionary dynamics. The latter is concerned with the mathematical equations that govern the evolution of complex systems arising in biology, economics, physics, social science and many research areas. Examples of these systems include population genetics of living systems, cancer evolution, human language formation, opinion propagation, tropical cyclone development, etc.

**Prerequisites:**Basic knowledge in Markov chains and ordinary differential equations. Proactive in learning by doing. Partial differential equations and computer programming skills would be a plus, but not required.

- Dylan Thurston: Polygonal fractal decompositions
**Faculty mentor:**Dylan Thurston**Description:**There are many fractal tiles that decompose into copies of themselves. Perhaps the best known is the dragon fractal (on the left below), but there are many others.

What is less well-understood is how to get good polygonal approximations to these tiles, suitable for making physical models. In many circumstances, there are canonical polygonal models. In this project, we will look for algorithms for cconstructing these models and characterize when they exists. We will also look at classification results, like the fractal Penrose tiles (on the right above): these come in two types, each decomposing into two smaller tiles of the two different types. How many other tile shapes are there with the same properties?**Prerequisites:**Linear algebra and basic topology. Familiarity with these self-similar systems is helpful but not necessary, as is computer programming experience.

- Jane Wang: Dynamics on dilation surfaces
**Faculty mentor:**Jane Wang**Description:**Suppose that we have a polygon with n pairs of parallel sides. Then, we can create a surface by gluing each side to its parallel partner with a translation and a dilation. The object that we have created is called a dilation surface, and is a surface that has locally Euclidean geometry except at finitely many cone points. The local geometry of dilation surfaces allows us to study the dynamics of the straight line flow on the surface: if we start at a point on a dilation surface and travel in a straight line in a given direction, what happens? For example, can we end up back where we started? Does our path equidistribute? There is also interesting global dynamics of the action of the group SL(2,R) on the space of dilation surfaces. In this project, we aim to study the dynamics on individual dilation surfaces as well as on the space of dilation surfaces. As dilation surfaces are a relatively new object of study, there are many open questions about them that we could try to tackle: Can we find dilation surfaces with many symmetries? Can we completely understand the dynamics of the straight line flow on any particular dilation surfaces? Can we understand what all dilation surfaces of a particular genus look like? These are just some examples of questions that we could try to answer. The ultimate direction of this project will depend on your background and interests.

**Prerequisites:**Basic group theory and linear algebra. Some experience with complex analysis and topology would be helpful, but are not required.

- Shouhong Wang: Backward-forward approach to the central manifold reductions
**Faculty mentor:**Shouhong Wang**Description:**In dynamical systems theory, a center manifold function (CMF) links the fast variables to the slow variables, and it plays a crucial role in studying nonlinear dynamics for many problems in mathematical physics. The computation of CMFs is often a difficult task. This project focuses on the backward-forward approach for finding leading-order approximations of CMFs.

**Prerequisites:**The prerequisites include undergraduate level Calculus 1-3, linear algebra and differential equations. More computational inclined student may wish to further explore numerical codes for computing CMFs using the backward-forward approach. In this case, working knowledge of Matlab will be needed.

- Colleen Delaney / Julia Plavnik : Quantum representations of 3D motion groups
**Faulty Mentors :**Colleen Delaney and Julia Plavnik**Description :**Motion groups are structures that organize the topologically distinct ways that submanifolds can move around inside of an ambient manifold. For example, the motion groups of points in the plane are given by braid groups. In physics, where point-like quasiparticles in effectively 2-dimensional quantum phases of matter can be modeled by points on a surface, quasiparticle spacetime trajectories are represented by elements of the braid group. This mathematical framework can be used to study ``topological quantum computation", the idea of building a computer where data is stored in quantum states of excitations and processed by controlling their motions. These physical and computational interpretations generalize beyond the braid group to physical excitations in 3-dimensional space that take the shape of knots and links, where the study of motion groups and their representation theory are on the cutting edge of research at the intersection of low-dimensional topology, theoretical physics, and quantum information science. In this project, students will investigate motion groups for some small knots and links, study properties of their representations, and interpret them in the context of quantum matter and computation.**Prerequisites**for a successful project include strong linear algebra skills and familiarity with the basics of knot theory. Exposure to abstract algebra or computer programming may be helpful but is not required.