**Faculty: Yingda Cheng (Mathematics)**

*Design of high order discontinuous Galerkin schemes for partial differential equations*

The discontinuous Galerkin (DG) schemes are a class of finite element methods developed in recent years. They use completely discontinuous basis functions, and have gained wide popularity in solving differential equations arising from various fields of science and engineering. In this project we will consider the design of DG schemes for PDEs with higher order spatial derivatives. By reviewing currently available solvers, we wish to design new schemes with provable stability and convergence results. The prerequisites for the project include basic knowledge about differential equations and programming, such as MATLAB or C.**Faculty: Mark Iwen (Mathematics)**

*Data compression of genomic information*

The students will work on computationally efficient metric space embedding methods for the data compression of genomic information, in particular they will investigate the embedding of the Levenshtein edit distance within Euclidean space. Students should be familiar with (or willing to learn) basic computer science concepts, probability, and first semester linear algebra. Programming experience is a big plus, as students will probably want to test their algorithms on a real gene sequence data.**Faculty: Aaron Levin (Mathematics)**

*Topic in number theory*

The research project will focus on a topic in number theory (to be determined). Number theory studies properties of the integers and their generalizations (e.g., rational numbers and algebraic integers). This includes the study of prime numbers and the study of integral and rational solutions to polynomial equations (Diophantine equations). The background material required for the project is primarily a familiarity with algebra (groups, rings, fields), but any previous experience with number theory (at any level) will be useful.**Faculty: Jeff Schenker (Mathematics)**

*Random walks with forward bias and their application to certain problems from field biology*

We will investigate random walks with forward bias and their application to certain problems from field biology. Knowledge of calculus and an introductory course in mathematical analysis should be sufficient background. Some prior knowledge of probability theory would be useful, but is not absolutely necessary. Some experience with Matlab would be beneficial, as we may want to program some simulations.**Faculty: Emil Valdez (Mathematics and Statistics, Actuarial Science Director)**

*Pension schemes and annuity plans*

Our research scholars from China will do actuarial research on topics related to advanced risk management, actuarial techniques and pension schemes. The students will focus on a small project related to pension schemes and annuity plans. These are increasingly becoming important in China especially with the continuing trend of a growing number of elderly in the population. One aspect is to examine the pension landscape and regulation aspects in the United States, Canada and possibly Mexico to develop and create an improved model that may be applied in China. Another possible aspect is to examine the rural social pension scheme in China as modernization and urbanization continue to evolve. A prior exposure to financial mathematics or investments and a little of probability and statistics would be helpful.**Faculty: Qiliang Wu and Keith Promislow (Mathematics)**

*Structures in amphiphilic mixtures (Click here for further information)*

**Faculty: Yimin Xiao (Statistics)**

*Characterize scaling properties and long range dependence of multivariate time series and Gaussian random fields*

Characterize scaling properties and long range dependence of multivariate time series and Gaussian random fields. This problem is not only of interest in theory of stochastic processes, but also for multivariate space-time modeling in statistics.**Faculty: Dapeng Zhan (Mathematics)**

*Approximation of random walk hitting distribution to the harmonic measure*

Wilson’s algorithm uses loop-erased random walk to generate a uniform spanning tree – an unbiased sample of all possible spanning trees. The algorithm initializes the maze with an arbitrary starting cell. Then, a new cell is added to the maze, initiating a random walk. The random walk continues until it reconnects with the existing maze. However, if the random walk intersects itself, the resulting loop is erased before the random walk continues. In this project, we will try to generalize this important Wilson's algorithm. Some knowledge of random walk will be helpful. Prerequisite: Some knowledge of the basics of probability theory, random walk, Brownian motion.

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