Research Experience for Early Graduate Students (REG)

May 26– July 31, 2009
Program Director:  Loek Helminck

Where: NC State University, Raleigh, NC .
Stipend and support: $4000 for ten weeks. Housing and a partial meal allowance provided for students who are not from NCSU. Travel funds up to $300 per participant provided as needed.

Topics for 2009 REU+ program: http://www.math.ncsu.edu/REU/#abstracts


Program Approach and Philosophy: This is a summer research program for early graduate students. Projects are similar in nature and scope to those of undergraduate summer research, in that these will be small problems with a good chance of success in the short term. These are not expected to turn into PhD thesis, though in some cases they could.

The students’ involvement in the project will mostly consist of 10 weeks in the summer, though NC State students will hear about the area and related problems in seminars during the spring semester. Unlike thesis work, these projects will be done collaboratively, as much of mathematics is done. At the end of the summer all teams will write up their results and present a short talks. Some students may present their results at SIAM, AMS or other meetings as well.

Why you should consider an REG:

• You get to do research, which is the main goal of graduate school, without waiting until after you pass the qualifiers.
• Get to know faculty better, by working with them on unsolved problems.
• Learn that mathematics is full of unanswered questions, and that you can help answer them.
• Learn to work with as part of a team. Partition tasks effectively, taking advantage of each individual's unique skills, be accountable to a group and integrate results.
• Practice communicating about mathematics both through both informal discussion and formal presentations and papers.
• Experience that Mathematics is a vibrant living discipline through which we can solve significant problems for government, industry, other scientists, and ultimately society at large.
• In short: Be part of the research mathematics community.

Participant background, requirements and selection: Participants are expected to meet the following criteria:

• must be a citizen or permanent resident of the United States or its possessions,
• must be a full-time mathematics graduate student as of September 2009,
  be committed to devote their full time to the program and not engage in any other course work or employment during the program,
• must be an early graduate student. Usually this means that the student has not yet passed qualifying exams. [Students who pass qualifiers early may still be eligible].

  Participants will be selected on the basis of demonstrated mathematical creativity, motivation, and good work habits as well as meeting the above requirements, as determined from the application materials and recommendation letters.

Week 1: Opening reception with the Chancellor, REG, REU, REU+ students and faculty

Week 1: Introduction to projects and mentors. Project teams are determined and begin to work on the projects.

Weeks 2 - 8: Work on projects. Progress reports are due each Friday.

In addition there will be a writing workshop and seminars on:

• mathematics related to the student projects
• research ethics
• how to give poster presentations
• how to give research talk

  Extra-curricular activities include Tea Time, organized game and movie nights, a trip to see the AAA Durham Bulls play a baseball game, as well as excursions to free local summer classical music concerts. The North Carolina Smoky Mountains and North Carolina beaches are within 3 hours drive from Raleigh.

Week 9: Students working on applied projects attend the initial presentations of projects at the Industrial Mathematical and Statistical Modeling Workshop for Graduate Students (IMSM). Students working on theoretical projects will give intial presentations to other students and faculty to obtain feedback.

Week 10: Students working on applied projects attend the final presentations of results of IMSM projects. All students will complete their final reports and do poster presentations of their work.
Team projects will follow the calendar above. Students participating in individual projects may follow a more flexible calendar. For example, they may begin working on a project in Spring 2009.

Notification Policy

All applicants will be notified by email about the completeness of their application a couple of days after the deadline date. Unless previously notified, a final notification that the search is closed will be emailed after all positions have been filled and confirmed (this could take a month).  If you have any questions about the status of your application, especially if you are trying to make a decision on accepting another summer position, please email the program director who will happy to give you a prompt response.

Research Projects of the NC State REG

The following lists currently available projects. Additional projects are added as the year progresses. NCSU Faculty and students can also iinitiate a project by talking to the Graduate Program Director. With approval of the graduate program director, these individual projects can be set up in a flexible manner, for example, with modified start or end dates.

SAMPLE Projects in Dynamical Systems, Optimization and Control:
North Carolina State University has a very active mathematical sciences group that works with several aspects of control systems. In addition to the faculty mentioned as advisors of specific projects below, the Control group includes Kazi Ito (stochastic systems), Hien Tran (delay differential systems), and Ralph Smith (multiscale and PDE systems). Additional projects in the group include composite material modeling and design, control of Fluids, physiological control systems, control and estimation in electromagnetics, acoustics, and biomedicine (HIV, biotissue and PBPK models). All of these projects are in collaboration with non-mathematical scientists and/or clinicians, thereby providing a rich multidisciplinary environment for graduate student projects.

1. Dynamic Social Networks. (H. T. Banks and N. G. Medhin). Multidisciplinary research studying latent variable modeling with tools from control theory and optimization, dynamical systems, and random processes. Student projects will include carrying out modeling, studying identification/observability issues, asymptotic behavior of the model, forecasting, and the analysis of alternative models.
   
2. Theory, Computation, and Application of Constrained Dynamical Systems. (S. L. Campbell). Student project areas include (1) the numerical solution of constrained optimal control problems in the context of aerospace applications from Boeing and (2) failure detection and identification.
   
3. Heavy Traffic methods for Time-Varying Wireless Systems (R. Buche). This is a collaborative research with NEC Laboratories America, Inc. in Princeton, NJ and Lucent Technologies, Bell Labs in Holmdel, NJ, two corporations which specifically encourage graduate student involvement.

SAMPLE Projects in Bio-Mathematics:
North Carolina State University has a strong and growing group who work in Bio-mathematics with projects that are very attractive for graduate students and have been an effective way of attracting undergraduates into graduate study in mathematics and enhancing diversity. In addition to the advisors listed below, the biomathematics group includes H.T. Banks, Hien Tran, and Alun Lloyd.

1. Modeling blood ?ow and pressure in arteries and veins. (M. Olufsen and M. Haider). Students involved with this project will be able to participate in viscoelastic model development and numerical implementation, as well as test and validation of improved models.
   
2. Biofilms for control of pathogenic surface bacteria. (S. Lubkin). Students will work with Dr. Lubkin and Dr. F. Breidt of the USDA on modeling several aspects of this biocontrol problem (depending on their interests and expertise): surface feature modeling, acid production, bacterial growth competition, deposition, surface diffusion, and other aspects as they emerge.
   
3. Modeling hormonal control of the menstrual cycle. (J. F. Selgrade). Students will do mathematical modeling and analysis, parameter identification, and computer simulation.

SAMPLE Projects in Differential Equations:
In addition to the project below the differential equations group at NCSU includes A. Chertock, H. T. Banks, S. Campbell, E. Chukwu, J. Franke, P. Gremaud, K. Ito, N. Medhin, X. Lin, S.Lubkin, S. Schecter, J. Selgrade, R. Smith, H. Tran, and S. Tsynkov.

1. Thin Film Flow: Modeling, Experiments and Simulations. (Michael Shearer). The flow of thin liquid films is of interest in chemical engineering and in biology, where lubrication by natural fluids is a crucial component of locomotion.
   
2. Systems of reaction-diffusion equations and viscous conservation laws. (S. Schecter and X-B. Lin).
   
3. Financial Mathematics: Modeling, Data, and Computation. (Advisors: M. Kang and T. Pang). Specific topics include volatility risk; free boundary value problems for American options; portfolio optimization using Merton’s portfolio optimization model; and variance reduction in Monte Carlo simulations.
   
4. Is periodic forcing beneficial or deleterious. (J. Franke). Population dynamics from one generation to the next can be modeled using a discrete dynamical system. One important question concerns the effects of periodic forcing on these models. Is the periodic forcing beneficial or deleterious to the species? There are several types of periodic forcing that can be studied and the model that is to be forced could include stage structure as well as dispersion between different patches.

SAMPLE Projects in Numerical Analysis:
North Carolina State University has a strong and active group in Numerical Analysis (NA) which has considerable experience in directing students and guiding their development. In addition to faculty mentioned as advisors in the sampling of projects described below, the NA group includes A. Chertock (PDE and Boundary Value Problems), S. Campbell (Simulation), M. Chu (Linear Algebra), K. Ito (Control), J. Scroggs (PDEs), and S. Tsynkov (PDEs).

1. Numerical methods for free boundary and moving interface problems. (Z. Li).
   
2. Search Engines: Updating Google’s PageRank. (I. Ipsen and C. Meyer). This research involves a rich blend of mathematical ideas from probability theory and Markov chains; networks; and graph theory, linear algebra, and numerical analysis.
   
3. Determinants for Lattice Simulations. (I. Ipsen). This project studies the quantum simulation of nuclear matter on a lattice and, in particular, the contribution of nucleon-nucleon-hole loops at non-zero nucleon density. The calculations involve computing determinants of large sparse interaction matrices.
   
4. Simulation and Design. (C. T. Kelley). There are several interdisciplinary projects in simulation and design. The applications include modeling and remediation of groundwater contamination, simulation and tuning of nanoscale semiconductor devices , and physical chemistry.
   
5. Numerical methods for optimization. (C. T. Kelley). In this project we look at numerical methods for optimization with applications to groundwater remediation, medicine, and electronics. Any student with good programming skills and a strong background in calculus and linear algebra could contribute to this project. The opportunities are especially good for students with good physics and/or chemistry backgrounds.
   
6. Control of Populations Using Impulsive Culling. (R. E. White). The project involves efficient solution to PDEs, optimization methods and KKT equations. This will be accessible to students with some numerical PDEs exposure and the numerical skills will be transferable to other PDEs and control problems.
   
7. Stochastic Initial Value Problems. (A. Chertock). Application of particle methods to the numerical simulation of statistical initial-value problems including the randomly perturbed KdV equation and random solutions to Buger's equation.

SAMPLE Projects in Symbolic Computation:
The Symbolic Computation Group at NCSU consists of H. Hong, E. Kaltofen, M.F. Singer, A. Szanto, A. Helminck and I. Kogan. This group has several activities that would help integrate beginning graduate students into active research. Each of the mentioned faculty has proposed projects suitable for beginning graduate students but we will only describe some of these:

1. LinBox. (E. Kaltofen). LinBox [1] is the name of an open source library of C++, Maple, and GAP functions [URL: www.linalg.org] that can manipulate very large sparse or structured matrices with exact integral or modular entries. The summer REG student would participate in the project by investigating a given function, say the one for matrix determinant, studying the newest advances, and attempting an implementation in the library.
   
2. Symmetries of differential equations. (I. Kogan). Participation in this project will give a beginning graduate student or an advanced undergraduate a practical, research-oriented experience in differential geometry, differential equations variational calculus, and symbolic computation.
   
3. Symbolic-numeric solution of over-constrained systems of nonlinear equations. (A. Szanto). In this non-traditional project the students are led through the experience of discovering new concepts via making intelligent conjectures based on computational experiments.
   
4. Differential/difference equations. (M.F. Singer). Algorithms for solving systems of partial or ordinary differential or difference equations in terms of sets of special functions where the differential and difference operators act on the same set of variables. A beginning graduate student or advanced undergraduate would participate by developing and implementing algorithms for restricted classes of equations and be part of an international research team.

SAMPLE Projects in Algebra and Analysis:
In addition to the projects below the Algebra and Analysis group at NCSU includes A. Fauntleroy, H. Hong, N. Jing, E. Kaltofen, I. Kogan, D. Labate, T. Lada, K. Misra, M. Putcha, N. Reading, J. Rodriguez, M. Singer, E. Stitzinger, and A. Szanto.

1. Structure of symmetric spaces. (A.G. Helminck). Symmetric spaces are complicated analytic and geometric structures which occur naturally in many areas of mathematics and physics. Frequently algebraic or combinatorial characterizations can be given for what seem to be purely analytic or geometric properties. These characterizations enable one to design algorithms for the analytic and geometric properties. In this project we will look at some of these algebraic or combinatorial characterizations.
   
2. Specialized characters and combinatorial identities. (K. Misra). Lie algebras are vector spaces with a multiplication operation called "bracket" satisfying certain properties. In this project students will be introduced to certain affine Lie algebras and investigate their relations to known combinatorial identities. Prerequisites: MA405 and MA407 or equivalent (knowledge of some programming language is preferred).
   
3. Cohomology of the Lie pseudo-algebra W(d). (B. Bakalov). The project is to compute the cohomology of one of the most important Lie pseudo-algebras, W(d). The students involved in this project will acquire basic knowledge of Lie algebras, their representations, and cohomology.
   
4. Multiscale geometric analysis of multidimensional data. (P. Gremaud and D. Labate). From the new FBI fingerprint database to the JPEG-2000 standard for image compression, efficient representation of multidimensional data plays an increasingly important role in a variety of applications. Harmonic analysis and wavelet theory are used.
   
5. Geometry and Cominatorics. (N. Reading) Possible project areas include: the combinatorics of finite partially ordered sets; the combinatorics of Coxeter groups; the combinatorics and geometry of simplicial hyperplane arrangements. The projects would likely involve computer experimentation in Maple.

SAMPLE Projects in Mathematical Physics:
In addition to the project below the Mathematical Physics group at NCSU includes R. Fulp, A. Kheyfets, I. Kogan, T. Lada and L. Norris.

1. Lyapunov - Malkin Theorem and Nonholonomic Integrators. (D. Zenkov). The goal of this project is to obtain a discrete version of the Lyapunov–Malkin theorem and to use it in the stability analysis of discrete mechanical systems.
   
2. Understanding superspace. (D. Zenkov). A number of research projects are available which have as their goals the development of various mathematical structures on superspace. Some of these projects are analysis oriented, others focus on representations of certain superalgebras, and others on geometric structures on superspaces.