Department of Mathematics, Duke University

The program consists of groups of 2-4 undergraduate students working together on a single project. Each team will be led by a faculty mentor assisted by a graduate student.
Participants will receive a $4,000 stipend, out of which they must arrange their own housing and travel. Funding and infrastructure support are primarily provided by the Department of Mathematics at Duke University. Participants may not accept other employment or take classes during the program.

The program runs from May 29 until July 20, 2018. The application deadline is February 15, 2018, but we will consider late applications until all slots are filled. Teams are expected to be finalized by March 16 but this is not guaranteed. We expect to notify applicants about the decision by email by the end of March.

There are 7 teams planned for summer 2018 in the numbered list below. Please indicate the number of the projects you choose when you apply; you may list up to four choices in ranked order of preference. For each of your choice of projects, please provide a short explanation of why you have chosen them and how you feel you could contribute to them. Please provide contact information for two faculty references (no letters required).

(1) Beyond curves and shapes of constant width, led by Professor Robert Bryant

It has been known for a long time that there are curves and shapes (other than the circle or the sphere) that have `constant width’, so that, no matter how they are turned, they will just fit between two parallel lines (or planes if we are in 3-D) separated by the width of the object. Famous examples show up in the Wankel engine and so-called Reuleaux triangles. (If you want to see these in action, search on YouTube for `curves of constant width’ and/or `shapes of constant width’.)

The mathematics of these objects is reasonably well understood (although there are still some surprises), but it turns out that one can go much beyond this. For example, if T is a triangle in the plane whose angles are rational multiples of a right angle, then, in addition to the inscribed circle, there are non-round convex figures C that can be inscribed in the triangle so that they touch all sides, no matter how the figure C is oriented. We don't know a lot about these curves of constant `triangular fit’, but the point of this project is to explore them and try to prove some theorems about them. There are similar phenomena in dimension 3, and we might be able to print out some of these objects and experiment with them, if time and interest permit.

We'll talk about what this has to do with Fourier series and other geometric concepts and see where it takes us.

(2) Designing circuits for a quantum computer, led by Professors Robert Calderbank and Henry Pfister

A quantum computer is one that exploits quantum mechanical principles in its operation, and if a quantum computer could be built, then it could break public key cryptosystems exponentially faster than classical computers.

The effectiveness of quantum computing derives from coherent quantum superposition or entanglement, which allows a large number of calculations to be performed simultaneously, and this coherence is lost as a quantum system interacts with its environment. In classical computing one can assemble computers that are much more reliable than any individual component by exploiting error-correcting codes. The aim of this project is to do the same for quantum computers.

Quantum error-correcting codes are designed to protect bits that participate in quantum computation. Logical operators acting on protected qubits need to be translated to physical operators (circuits) acting on quantum states. Circuit synthesis involves representing a physical operator as a binary symplectic matrix, then writing this symplectic matrix as a product of elementary symplectic matrices, each corresponding to an elementary circuit. Given a quantum error-correcting code and a model for quantum errors, this project will explore how to choose operators that maximize the probability of correct execution.

Helpful skills for this project may include linear algebra, probability, a first course in algebra and some ability to compute.

(3) Local affinity construction for dimension reduction methods, led by Professors Xiuyuan Cheng and Hau-Tieng Wu

In many applications, data coming as high dimensional vectors actually lie on or close to low dimensional manifolds that are embedded in the high dimensional space. The problem of dimension reduction aims at revealing such hidden geometrical structures for data visualization and further analysis, and it is important for unsupervised representation learning. A large class of dimension reduction methods are based upon the construction of an affinity graph or a kernel matrix, and the affinity is computed only for data points in local neighbourhoods. The construction of such local affinity has key effects on the performance of dimension reduction methods, as has been widely observed in experiments, yet the theoretical understanding remains limited. The project will investigate the effect of different ways of constructing the local affinity on the results of dimension reduction and representation learning, both in simulation and in mathematical analysis, starting from simple examples of synthetic data sampled on curves and surfaces. It will be helpful for at least one student researcher in this group to be able to code.

(4) Modelling of random processes by recurrent neural networks, led by Professor Xiuyuan Cheng

Recurrent neural networks (RNN) have been widely used in the modelling of sequential data such as speech and natural language data. This project will investigate the modelling power of RNN on simulated sequential data generated by a stochastic process with known equations. The approximation accuracy and the training of the RNN will be studied firstly in simulation and then from an analytical point of view - will a neural network be able to predict the next step of a random process given previous steps, or learn to generate typical trajectories of the process after training? What architecture of neural networks would be more effective for the modelling of a random process, and how does this depend on the parameters of the process itself? Time permitting, we will also attempt to construct processes which would be hard for a neural network to model. It will be helpful for at least one student researcher in this group to be able to code.

(5) Epidemics on random graphs, led by Professors Richard Durrett and Matthew Junge

Our processes will take place on graphs generated by the configuration model in which the degree distribution is specified and then connections are made at random. In an epidemic, sites can be in one of three states: Susceptible, Infected or Removed (i.e., immune to further infection). There are three basic epidemic models SI, SIR, and SIS. In all three versions, susceptibles become infected at a rate λ times the number of infected neighbors. In the SI model recovery is not possible. In the other two versions infecteds become removed (or susceptible) at rate 1. We will investigate (i) how the duration and severity of epidemics depends on λ, and (ii) how the behavior of SIR and SIS epidemics are changed when susceptible individuals can sever their connection to infected individuals and become neighbors of another individual chosen at random from the graph. It will be useful to have some participants who can program but our aim will be to prove theorems.

(6) Stochastic properties of dynamical systems, led by Professor Sayan Mukherjee

In this project, students will study the stochastic complexity of certain dynamical systems. They will use mainly simulations as well as some basic probability theory and ergodic theory to characterize how complex certain dynamical systems are. In addition to classical systems like the logistic map, some systems inferred from real data on microbial communities and gene regulatory systems will be explored.

(7) Topological quantum edge states, led by Professor Alexander Watson

Certain two-dimensional materials which are insulators in their bulk (i.e. far from the edge, or boundary, of the material) are known to be conducting at their edge. This phenomenon has been shown in experiments to be robust to localized defects of the edge, and can be theoretically explained in terms of topological invariants associated to the crystal structure (arrangement of atoms) of the material. In this project we will study such states analytically and computationally.

If you have any questions about DOmath2018, please email either Professor Lenny Ng (ng@math.duke.edu) or Professor Heekyoung Hahn (hahn@math.duke.edu).