## Proposed research projects for Summer 2012

Supervisor Name
Research Project
Omer Angel and
Lior Silberman
Title: Value distribution of eigenfunctions on random graphs

Description: We wish to explore the value distributions of eigenfunction of random matrices, especially those associated to random regular graphs. We hope to obtain both experimental and analytical results.

The experimental part will involve numerical linear algebra; experience with Octave, Matlab or the like would be helpful.
Richard Anstee
The problem area is Extremal Hypergraph Theory although I prefer to use the language of matrix theory. The following is a typical problem. Let F be a given kxt (0,1)-matrix and let A be an mxn (0,1)-matrix with no repeated columns and no submatrix F. There is a conjecture of Anstee, Frankl, Furedi and Pach that claims that there is a constant c(F) so that n < c(F) x m^k. One can establish that n < m^{2k-1-e} where e is a function of k,t and is small. The goal would be to improve this result. The work will focus on specific F in order to gain insight. A related problem is where we forbid any row and column permutation of F (so called problem of Forbidden Configurations).

Interested students could contact my previous USRA students: Connor Meehan and Ronnie Chen.
Bud Homsy
Title: Studies of the attraction of floating particles

Background:

Particles at an interface get attracted to each other and form aggregates and clusters. This is commonly referred to as the “Cheerios effect”, owing to the tendency of pieces of cereal floating on milk to stick together.  It is known that the attractive force is due to the combination of surface tension and the curvature of the meniscus separating the particles.  In spite of this basic fact, prevailing theories of particle attraction can handle only a few particles and often make restrictive assumptions.

Our interest in this subject derives from an interest in understanding how the “Cheerios effect” influences coating flows.  In coating applications, (such as coating of memory devices and other recording media, producing photographic film, etc.), a thin layer of fluid is deposited by a flow onto a solid substrate.  It has been found recently that particles lying at the interface of the liquid being coated can modify the coating properties to a significant degree. This project will aim to understanding the forces present at an interface due to the presence of particles in simple flow topologies.

Project Description:

This project has both experimental and computational components.  The overall objective is to establish the quantitative relationship between the particle configuration, the interparticle spacing, and the force of attraction.  On the experimental side, simple table-top experiments with well characterized particles and fluids in controlled conditions will be carried out.  Fundamental data on the position, velocity and acceleration of particles as they come together will provide important tests of theories.  On the computational side, the partial differential equations governing interface dynamics will be solved numerically using available Open Source software like Surface Evolver for groups of multiple particles.

Kalle Karu
There are several problems that can be chosen according to student's background knowledge and interests. The general area is the geometry of polytopes and fans.  Topics include formulas for lattice point counting, face enumerations, subdivisions of fans, tropical geometry.

The necessary background is linear algebra, elementary combinatorics, and preferably some algebra.
Leah Keshet
Math Biology
Fok-Shuen Leung
Title: Mathematics modules in Science One

Description: Science One is a highly selective first-year program taught by eight faculty members from the Mathematics, Biology, Chemistry, and Physics & Astronomy departments. These subjects are taught in parallel with occasional crossover topics. The goal of this project is to rewrite the mathematics curriculum to be completely integrated with other subjects. The student would help design modules with broad themes -- for example, "Decay", "The Normal Distribution", "Evolutionary Processes" -- but specific mathematical content. A background in Science One is preferable but not necessary.
Akos Magyar and
Malabika Pramanik
Title: Problems in density Ramsey theory.

Description: Ramsey theory is a beautiful area at the interface of analysis, combinatorics and number theory whose aim is to show that large but otherwise arbitrary sets of integers or integer points necessarily contain highly regular structures. Its basic results are both natural and easy to formulate, however their proofs can be surprisingly difficult and varied. A famous example is Roth's theorem that states that if a set contains a "positive proportion" of the integers then it must contain three equally spaced points. Another is van der Courput's theorem which establishes the same for the set of primes.

There has been remarkable progress over the past decade in understanding the underlying principles behind the different approaches in Ramsey theory, and by now a number of problems can be formulated whose solution does not require excessive formal knowledge. The aim of this project is to introduce the students to the basic analytic techniques and possibly try them on some open questions in this area.
Brian Marcus and
Andrew Rechnitzer
APPROXIMATING ENTROPY OF MARKOV RANDOM FIELDS

The entropy of a stationary random process quantifies the degree of randomness of the process. Entropy is important in information theory, where it governs optimal rates of data compression and data transmission, in statistical physics, where it represents disorder in physical systems, and dynamical systems where it quantifies the complexity of a system.

In the case of a one-dimensional stationary Markov chain, there is a simple, exact formula for entropy in terms of the transition probabilities of the chain. In two dimensions, the Markov property generalizes to the notion of a stationary Markov random field (MRF), which can be viewed as a random process of arrays on the integer lattice.  While MRF`s can be very concrete, there is no known formula for the entropy. However, there have been many methods developed for approximating entropy of MRF`s.  These methods use linear algebra, probability and combinatorics.

In this project, the student will first learn the basics of entropy in theory and practice and the methods for computing approximations.  Then the student will write computer programs to compare the performance of the methods.

Background in computer programming, preferably C++, and linear algebra is required. Background in probability and combinatorics would be helpful.
Anthony Peirce
1) NUMERICAL SOLUTION OF A SEMI-INFINITE HYDRAULIC FRACTURE PROPAGATING IN AN ELASTIC MEDIUM:
Hydraulic Fractures (HF) are created in the oil and gas industry to enhance the production of hydrocarbons. This has been the subject of some controversy recently due to the potential negative impacts of this process also known as "Fracking". Our goal is to better understand the mechanics of these fractures that are generated in brittle rock formations by the injection of a high pressure viscous fluid.
This project aims to investigate various numerical schemes to solve the problem of a semi-infinite Hydraulic Fracture propagating in an elastic medium at a constant velocity V. This problem is important from a theoretical point of view as it forms the fundamental solution for many tip asymptotic solutions for finite fractures, which can be obtained via an appropriate re-scaling of the variables. The project will involve partial differential equations, complex analysis, and numerical analysis using the MATLAB programming language, which the USRA will learn during the course of the project. A strong background in PDEs (Math257/316, Math 300, and Math 405 is a recommendation).

2) DEVELOPMENT OF AN EFFICIENT CUDA IMPLEMENTATION OF AN EXTENDED FINITE ELEMENT (XFEM) ALGORITHM FOR MODELING HYDRAULIC FRACTURES:
This project involves the development of an object-oriented set of CUDA routines for the efficient implementation of an existing Extended Finite Element Algorithm Developed at UBC (in the MATLAB language) for modeling propagating Hydraulic Fractures. We have recently acquired a GPU server comprising 2 GPU boards with 480 GPU processors and 6 GB of RAM each as well as 12 Xeon Multicore CPUs with 96 GB of shared RAM. The project will be ideal for a combined Honors student in Computer Science and Mathematics. Being a development of C++, CUDA is a language developed to exploit the multi-threading capabilities of the CGU architecture processors. The USRA will learn both the numerics behind the XFEM algorithm as well as the CUDA programming language. (A strong background in numerics Math405 - or equivalent courses in CS and C++ programming are recommended).

Projects from last year: 2011