Sample Research Projects of the NC State REG

 

SAMPLE Projects in Dynamical Systems, Optimization and Control:

North CarolinaState 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 (delaydifferential 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). This project is part of a multidisciplinary research effort by a group of social scientists, statisticians, and applied mathematicians who are studying latent variable modeling with tools from control theory and optimization, dynamical systems, and random processes. The generalized network models result in a high dimensional linear time varying system with stochastic, parameter dependent structural matrices [1, 2, 3]. The models can be nonlinear delay systems with the network depending on the history of the observed and latent variables. Student projects will include carrying out modeling, studying identification/observability issues, asymptotic behavior of the model, forecasting, and the analysis of alternative models.

    2. [1] Banks, D., and Carley, K., Models for network evolution., J. MathematicalSociology, 21: 173-196, 1996

    [2] Molenaar, P. C. M. (1994) Dynamic Latent Variable Models in Developmental Psychology Latent Variable Analysis, Alexander von Eye, Clifford C. Clogg, editors, SAGE Publications 1994.

    [3] Skrondal, A.,Rabe-Hesketh, S., Generalized Latent Variable Modeling, Chap-man&Hall/CRC 2004.

  2. Theory, Computation, and Application of Constrained Dynamical Systems. (S. L. Campbell). This topic contains ideal research projects for entering and first and second year graduate students and has a collaborative aspect so the students involvedwill get an early exposure to interdisciplinary research. On most of the topics, the PI has advanced Ph.D. students who will assist the PI in providing mentoring. 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 in an active approach for designing finite horizon test signals, of minimum size, that guarantee failure detection in the presence of uncertainty.
  3. Heavy Traffic methods for Time-Varying Wireless Systems (R. Buche). Future wireless systems will have demanding capacity requirements, particularly in the forward-link, as applications such as internet browsing, video conferencing, multimedia, etc. become commonplace. This project investigates the application of heavy traffic methods for the associated allocation problem. The heavy traffic method [2, 1, 3] is applied under the assumption the queueing system is operating at near capacity. Stochastic control techniques can be used with a limit model and the policies obtained often work well when “far" from the heavy traffic assumption. 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.

    4. [1] H.J. Kushner, P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Second Edition Springer, New York, 2001.

    [2] R. Buche and H. J. Kushner, “Control of Mobile Communications With Time-Varying Channels in Heavy Traffic", IEEE Transactions on Automatic Control, Vol. 47, No.6, June 2002, pp.992-1003.

    [3] A. L. Stolyar, “Max Weight scheduling in a generalized switch: State space collapse and workload minimization in heavy traffic", The Annals of Applied Probability, Vol. 14, No. 1, February 2004.

 

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 flow and pressure in arteries and veins. (M. Olufsen and M. Haider). While three dimensional fluid dynamic models for the cardiovascular system exist [1], they are computationally expensive and time consuming to solve. In clinical applications such as surgery planning, there is a need for simpler models that can be solved within minutes. An example is the one-dimensional model developed by Olufsen [2, 3], which has been further developed and validated by Steele [4] in a study of cardiovascular dynamics during rest and exercise. The models [4] and [3] were developed for steady flow and do not take into account losses due to pulsatility resulting from the nature of blood flow in viscoelastic veins and arteries [5]. 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. [1] C.A. Taylor, T.J. Hughes and C.K. Zarins, Finite Element Modeling of Three-dimensional Pulsatile Flow in the Abdominal Aorta: Relevance to Atherosclerosis, Ann. Biomed. Eng. 26(6):975-87, 1998.

    [2] M.S. Olufsen, Structured Tree Outflow Condition for Blood Flow in Larger Systemic Arteries, Am. J. Physiol. 276(45):H257-H268, 1999.

    [3] M.S. Olufsen, C.S. Peskin, W.Y. Kim, E.M. Pedersen, A. Nadim and J. Larsen, Numerical Simulation and Experimental Validation of Blood Flow in Arteries with Structured-tree Outflow Conditions, Ann. Biomed. Eng. 28:1281-1299, 2000.

    [4] B.N. Steele, J. Wan, J.P. Ku, T.J.R. Hughes and C.A. Taylor, In Vivo Validation of a One-Dimensional Finite Element Method for Predicting Blood Flow in Cardiovascular Bypass Grafts, IEEE Transactions in Biomedical Engineering, 50:649656, 2003.

    [5] Y.C. Fung, Biomechanics: Mechanical Properties of Living Tissue, SpringerVerlag, 2nd edition, 1993.

  2. Biofilms for control of pathogenic surface bacteria. (S. Lubkin). One promising method of controlling bacterial growth on raw or minimally processed foods is inoculation of the food surface with benign lactic acid bacteria (LAB) [1], which have been shown to suppress the growth of numerous pathogenic bacteria [3]. Students will work with Dr. Lubkin and Dr. F. Breidt of the USDA [2] 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. The mathematics primarily involves applications of nonlinear ordinary and partial differential equations as well as statistics.

    3. [1] F. Breidt and H. P. Fleming, Using Lactic Acid Bacteria to Improve the Safety of Minimally Processed Fruits andVegetables, Food Technol., 51:44-51, 1997.

    [2] D.P. Dougherty, F. Breidt, Jr., R.F. McFeeters and S.R. Lubkin, Energy-based Dynamic Model for Variable Temperature Batch Fermentation by Lactococcus Lactis, Appl. Environ. Microbiol., 68 (5):2468-2478, 2002

    [3] M. Vescovo, S. Torriani, C. Orsi, F. Macchiarlol and G. Scolari, Application of Antimicrobial-Producing Lactic Acid Bacteria to Control Pathogens in Ready-To-Use Vegetables, J. Appl. Bacteriol., 81:113-119, 1996.

  3. Modeling hormonal control of the menstrual cycle. (J. F. Selgrade). Complex endocrine signaling between the ovaries and the pituitary gland is the key ingredient for regulating and maintaining the menstrual cycle in adult women. Selgrade [3, 4] and collaborators have been developing a mathematical model describing the concentrations of five hormones important for this system. The model is a highly nonlinear system of delay differential equations which hasbeen validated with normal cyclical data [2]. As a woman approaches menopause, production of ovarian hormones decreases which subsequently affects cycle length and on other cycle variations; hence the model needs to be modified to include aging and perimenopausal women. Students will do mathematical modeling and analysis, parameter identification, and computer simulation.

    4.  

    [1] Harris Clark, L., Schlosser, P.M., and Selgrade, J.F. [2003] Multiple stable periodic solutions in a model for hormonal control of the menstrual cycle, Bulletin of Math. Biology 65, 157-173.

    [2] McLachlan, R.I., Cohen, N.L., Dahl, K.D., Bremner, W.J., and Soules, M.R. [1990] Serum inhibin levels during the periovulatory interval in normal women: Relationships with sex steroid and gonadotropin levels. Clin. Endocrinol. 32, 39-48.

    [3] Schlosser, P.M., and Selgrade, J.F. [2000] A model of gonadotropin regulation during the menstrual cycle in women: Qualitative features, Environmental Health Perspectives 108(suppl 5), 873-881.

    [4] Selgrade, J.F., and Schlosser, P.M. [1999]Amodel for the production of ovarian hormones during the menstrual cycle, Fields Institute Communications 21, 429–

    446.

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, where such flows are used in coatings of solid surfaces. They are also increasingly recognized as being important in biology, where lubrication by natural fluids is a crucial component of locomotion. In this project, we adopt a more fundamental approach, regarding these flows more from the point of view of the mathematics and physics of fluid mechanics at the microscopic and even nanometer scale. Experiments with various configurations are being pursued in the lab of Karen Daniels (NCSU Physics). The most challenging (but still mathematically tractable) of these involves flow induced by surfactant. The model of such flow is a coupled system of nonlinear PDE, which at its core is a hyperbolic conservation law (for the film thickness) and a porous medium equation (for the surfactant distribution). Simulations in one dimension show that wave-like solutions take various forms, only some of which are understood analytically. Moreover, experiments reveal a rich variety of multidimensional patterns associated with instabilities of the one-dimensional solutions. Karen Daniels’ PhD student will set up an experiment in Spring 2007 to study flow down an inclined plane, with surfactant.
  2. Financial Mathematics: Modeling, Data, and Computation. (Advisors: M. Kang and T. Pang). This project covers several topics related to financial mathematics with the following common approach: 1) review and compare the existing models; 2) improve on them or propose a new class of models; 3) perform tests on simulated and real data; and 4) develop the computational tools needed to use the new models. Regular meetings will enable the team of advisors and students to assess the progress and discuss the various tasks so that the students will be efficiently advised and guided in their first steps in mathematical research. Specific topics include volatility risk (one of the prime sources of risk in financial markets [6]); free boundary value problems for American options; portfolio optimization usingMerton’s portfolio optimization model [3, 7, 8]; and variance reduction in Monte Carlo simulations [1, 2, 5]. In addition, the research project will include the problems faced by a privately owned company [4].

    3. [1] P. Del Moral. Feynman-Kac formulae, genealogical and interacting particle systems with applications. Springer Verlag, 2004.

    [2] P. Del Moral and J. Garnier. Genealogical particle analysis of rare events. Preprint, Université Paul Sabatier, Toulouse, 2004.

    [3] W. H. Fleming and T. Pang, An Application of Stochastic Control Theory to Financial Economics, SIAM J. of Control and Optimization, Vol. 43, No.2, 502531, 2004

    [4] W. H. Fleming and T. Pang,Astochastic control model of investment, production and consumption, submitted, 2003.

    [5] Paul Glasserman. Monte Carlo Methods in Financial Engineering. Springer Verlag, 2003.

    [6] P. Hagan, D. Kumar, A. Lesniewski and D. Woodward. Managing Smile Risk, Wilmott Magazine, September 2002, 84-108.

    [7] R. C. Merton, Continuous Time Finance, revised edition. Blackwell, Cambridge, MA, 1992.

    [8] T. Pang, Portfolio optimization models on infinite time horizon, Journal of Optimization Theory and Applications, Vol. 22, No 3, 119-143, 2004.

 

 
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 boundaryand moving interface problems. (Z. Li). Free boundary and moving interface problems have many applications in computational fluid mechanics, mathematical biology, and material sciences. Lack of irregularities in the solution and topological changes such as splitting and merging of moving interfaces pose challenges. This project is to develop state-of-the-art of numerical methods that combine the advantages of the level set method and the immersed interface method and includes (1) development of efficient hybrid finite-element level-set methods for the evolution of interfaces that are driven by both local geometry and global stresses; (2) adaptive mesh refinement techniques for immersed interface method and the level set method; (3) efficient numerical methods for Maxwell equations involving interface and far field boundary conditions; and (4) development of software packages for interface problems.

  2. Search Engines: Updating Google’s PageRank. (I. Ipsen and C. Meyer). Google is arguably the most popular internet search engine. Its efficiency derives, in part, from its PageRank algorithm. The link structure of the web graph can be represented by an irreducible aperiodic Markov matrix P. Computing the PageRank amounts to computing the stationary vector π of P. The dimension of P is on the order of billions. PageRank is often called ’the world’s largest matrix computation’ [6] with an executiontime measured in days. The world wide web is a dynamic network in which pages and links are constantly added and deleted. Google is on record as saying that it has no more effective way todeal with the global updating of PageRank other than starting from scratch every three or four weeks. This updating dilemma is the focus of this project. The strategy is to investigate the feasibility of updating based on the aggregation/disaggregation (state lumping) theory for Markov chains [3, 4, 5]. This research involves a rich blend of mathematical ideas fromprobability theory and Markov chains; networks [1, 2]; and graph theory, linear algebra, and numerical analysis.

    3. [1] A.L. Barabasi. Linked: The New Science of Networks. Plume, 2003.

    [2] A.L. Barabasi, R. Albert, and H. Jeong. Scale-free characteristics of random networks: The topology of the world-wide web. PhysicaA, 281:69–77, 2000.

    [3] I.C.F. Ipsen and S. Kirkland. Convergence analysis of an improved PageRank algorithm. SIAM J. Matrix Anal. Appl., 2004. Submitted.

    [4] A.N. Langville and C.D. Meyer. Deeper inside PageRank. Internet Mathematics, 2004. To appear.

    [5] C.D. Meyer. Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems. SIAM Review, 31(2):240–272, 1989.

    [6] C. Moler. The world’s largest matrix computation. MatLab News and Notes, pages 12–13, October 2002.

  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 M. Zone determinant expansion computes the determinant of a complex matrix to specified accuracy [4, 5] and is not limited to Hermitian positive-definite matrices. Other types of methods [1, 2, 3, 6, 7] can be faster but they apply only to Hermitian positive-definite matrices. This project studies a hybrid method that combines the generality of the zone determinant expansion with the speed of existing methods can be successfully pursued by a beginning graduate student.

    4. [1] Z. Bai and G.H. Golub. Bounds for the trace of the inverse and the determinant of symmetric positive definite matrices. Ann. Numer. Math., 4(1-4):29–38, 1997.

    [2] S. Duane, A.D. Kennedy, B.J. Pendleton, and D. Roweth. Hybrid Monte Carlo. Phys. Lett.B, 195(2):216–22, 1987.

    [3] S. Gottlieb, W. Liu, D.Toussaint, R.L. Renken, and R.L. Sugar. Hybrid-moleculardynamics algorithms for the numerical simulation of quantum chromodynamics. Phys. Rev.D, 35(8):2531–42, 1987.

    [4] I.C.F. Ipsen and D.J. Lee. Determinant approximations. BIT, 2003. Submitted.

    [5] D.J. Lee and I.C.F. Ipsen. Zone determinant expansions for nuclear lattice simulations. Phys. Rev.C, 68:064003, 2003.

    [6] A. Reusken. Approximation of the determinant of large sparse symmetric positive definite matrices. SIAM J. Matrix Anal. Appl., 23(3):799–818, 2002.

    [7] R.T. Scalettar, D.J. Scalapino, and R.L. Sugar. New algorithm for the numerical simulation of fermions. Phys. Rev.B, 34(11):7911–23, 1986.

  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 [2], simulation and tuning of nanoscale semiconductor devices [6], and physical chemistry [5]. The graduate students who work on these projects work closely with scientists and engineers to design algorithms for optimization and the solutions of linear and nonlinear equations, implement those algorithms in tools for the applications, and analyze the convergence and accuracy of the methods. Parallel computing plays a major role in most of these projects. One example is resonant tunneling diodes which are nanoscale devices [3, 7, 8]. Very fine grids must be used to capture the physics. For problems this large, distributed memory parallel computers and matrix-free [1, 4] linear and nonlinear solution are necessary.

    5. [1] P. N. Brown and Y. Saad. Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Stat. Comp., 11:450–481, 1990.

    [2] K. R. Fowler, C. T. Kelley, C. T. Miller, C. E. Kees, Robert W. Darwin, J. P. Reese,

    M. W. Farthing, and Mark S. C. Reed. Solution of a well-field design problem with implicit filtering. Optimization and Engineering, 5:207–234, 2004.

    [3] W. R. Frensley. Wigner-function model of a resonant-tunneling semiconductor device. Phys. Rev.B, 36:1570–1580, 1987.

    [4] C. T. Kelley. Iterative Methods for Linear and Nonlinear Equations. Number 16 in Frontiers in Applied Mathematics. SIAM, Philadelphia, 1995.

    [5]C.T.KelleyandB. Montgomery Pettitt.Afast algorithmforthe Ornstein-Zernike equations. J. Comp. Phys., 197:491–591, 2004.

    [6] M. S. Lasater, C. T. Kelley, P. Zhao, and D. L. Woolard. Numerical tools for the study of instabilities within the positive-differential-resistance regions of tunnelling devices. In Proceedings of 2003 3nd IEEE Conference on Nanotechnology, San Francisco, CA, August 12–14, 2003, pages 390–393. IEEE, 2003.

    [7] E. Wigner. On the quantum correction for thermodynamic equilbrium. Phys. Rev., 40:749–759, 1932.

    [8] P. Zhao, H. L. Cui, and D. L. Woolard. Dynamical instabilities and I-V characteristics in resonant tunneling through double barrier quantum well systems. Phys. Rev.B, 63:75302, 2001.

  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.
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. It offers a year-long introductory course in Symbolic Computation as well as special topics courses reflecting the particular research interests of the faculty. There are several international research projects involving faculty and students, and there is a lively weekly informal lunch and a weekly seminar. 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. For example, it can solve large sparse diophantine linear systems and compute the Smith form of large sparse matrices such as those arising in the analysis of combinatorial mathematical structures, e.g., simplicial complexes, and in mathematical physics. LinBox strives to be the symbolic computation equivalent of the numerical symbolic linear algebra libraries such as LAPack, NAG, or MatLab. The summer REG, REU or REU+ 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. [1] J.-G. Dumas, T. Gautier, M. Giesbrecht, P. Giorgi, B. Hovinen, E. Kaltofen, B. D. Saunders, W. J. Turner, andG. Villard. LinBox: Ageneric library for exact linear algebra.In ArjehM. Cohen, Xiao-ShanGao, and Nobuki Takayama, editors, Proc. First Internat. Congress Math. Software ICMS 2002, Beijing, China, pages 40–50, Singapore, 2002. World Scientific.

  2. Symmetries of differential equations. (I. Kogan). The majority of the differential equations and variational problems arising in science and engineering possessnatural symmetry properties and taking these into account yield theoretical and computational advances. Kogan is developing and implementing (in collaboration with researchers at Utah State University and INRIA) symbolic differential and variational calculus in various geometric moving frames based on Cartan’s moving frame method and advances in symbolic computation [1, 2, 3, 4]. Students participating in the project will study computational and theoretical problems in the symmetry-conservation laws correspondence for reduced Euler-Lagrange equations using moving frame techniques. Participation in this project will give an beginning graduate student or advanced undergraduate a practical, research-oriented experience in differential geometry, differential equations variational calculus, and symbolic computation.

    3. [1] I. Anderson, The Vessiot handbook, Technical Report, Utah Sate University, 2000, http://www.math.usu.edu/∼fg_mp/

    [2] M. Fels and P. Olver Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math., 55, (1997) 99–136.

    [3] I. Kogan and P. Olver, Invariant Euler-Lagrange Equations and The Invariant Variational Bicomplex, Acta Appl. Math. 76 (2003), 137–193.

    [4] I. Kogan, Two algorithms for a moving frame construction, Canad. J. Math. 55, (2003), no. 2, 266 – 291.

  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. Problems in symbolic-numeric computation are especially well suited for such discoveries, since properties of many known symbolic algorithms are yet to be explored in the numerical setting. However, the topics will not be constrained to symbolic-numeric computation, but will include other areas of computational mathematics.
  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 have been known for several years and are now readily available on Computer Algebra systems like Maple and Mathematica [1, 2, 3, 4]. We propose to extend this work to finding closed form solutions of differential/difference equations 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.

    5. [1] P. Hendriks, M.F. Singer Solving Difference Equations in Finite Terms, Journal of Symbolic Computation, 27/3, 1999, 239-259.

    [2] G. Labahn, Z. Li, Hyperexponential Solutions of Finite-rank Ideals in Orthogonal Ore rings, Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation, J. Gutierrez, ed., ACM Press, 2004, 213-220.

    [3] M. van der Put, M.F. Singer Galois Theory of Difference Equations, Lecture Notes in Mathematics„ vol. 1666, Springer-Verlag,1997.

    [4] M. van der Put, M.F. Singer, Galois Theory of Linear Differential Equations, Grundlehren der mathematischen Wissenschaften, Volume 328, Springer, 2003.

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. Surprisingly, algebraic or combinatorial characterizations can frequently be given for what seem to be purely analytic or geometric properties. A long term project is to build a comprehensive computer algebra package for all computations related to symmetric spaces. There are computational issues such as what data structures are most computationally efficient as well as many algebraic or analytical open problems about these symmetric spaces. For example, symmetric spaces have only been totally classified for real and complex symmetric spaces, and this required a case by case study. For other fields, like finite and the padic numbers, there are partial classifications given by Helminck and his students. There are many more cases to analyze. These would make excellent REG projects with some suitable for REU+ and REU students as well.
  2. Specialized characters and combinatorial identities. (K. Misra). Lie algebras are vector spaces with a multiplication operation called "bracket" satisfying certain properties. Representation of a
    Lie algebra is a way to represent the elements in the Lie algebra in terms of linear transformations (hence matrices) on a vector space which is called the representation space. A "root" or "weight" can be thought of as the generalized eigenvalues and "root spaces", "weight spaces" are the corresponding eigenspaces. A character of a representation codes the dimensions of the weight spaces of a representation. It is known that certain specializations of the characters of the representation of affine Lie algebras give rise to certain combinatorial identities called partition identities. 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).

  1. Cohomology of the Lie pseudo-algebra W(d). (B. Bakalov). Two-dimensional conformal field theory (CFT) is important in physics as a model of two-dimensional quantum field theory, and it plays a crucial role in string theory, statistical thermodynamics and condensed matter physics. In [1, 2, 3] we introduced and studied generalizations of conformal algebras (introduced in [4]) called Lie pseudo-algebras, which are expected to be relevant for CFT in a dimension higher than two. The project is to compute the cohomology of one of the most important Lie pseudo-algebras, the one called W(d) in [1]. The students involved in this project will acquire basic knowledge of Lie algebras, their representations, and cohomology.

    2. [1] B. Bakalov, A. D’Andrea and V.G. Kac, Theory of finite pseudoalgebras, Advances in Math. 162 (2001), 1–140.

    [2] B. Bakalov, A. D’Andrea and V.G. Kac, Irreducible modules over finite simple Lie pseudoalgebras I. Primitive pseudoalgebras of type W and S, to appear.

    [3] B. Bakalov, V.G. Kac and A.A. Voronov, Cohomology of conformal algebras, Comm. Math. Phys. 200, (1999) 561–598.

    [4] V.G. Kac, Vertex algebras for beginners, 2nd ed., University Lecture Series, vol. 10, Amer. Math. Soc., Providence,RI, 1998.

    [5] D.B. Fuchs, Cohomology of infinite-dimensional Lie algebras, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986.

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 andto use it in the stability analysis of discrete mechanical systems. Consider a system of differential equations x˙= Ax+X(x, y), y˙= Y(x, y), where x ∈ Rm , y ∈ Rn, and A is a constant m×m matrix and X(x, y) and Y(x,y) are nonlinear vector fields with X(0,y) = 0 and Y(x,y) = 0. The Lyapunov–Malkin theorem (see [3, 4]) states that the equilibrium (x, y) = (0, 0) of the above system is stable and asymptotically stable with respect to x if the spectrum of A belongs to the left half plane. The proposed project is to obtain a similar result for certain discrete dynamical systems. This stability result can be later applied to the stability analysis of equilibria and of relative equilibria of nonholonomic integrators studied in [1] and [2] and will introduce the student to modern methods to study dynamical systems.

    2. [1] Cortés J. and S. Martínez [2001], Nonholonomic Integrators, Nonlinearity 14, 1365–1392.

    [2] Fedorov, Yu.N. and D.V. Zenkov [2004], Discrete Nonholonomic LL Systems on Lie Groups, preprint.

    [3] Lyapunov, A.M. [1992], The General Problem of Stability of Motion, Int. J. Control 55, 531–773; translated into EnglishbyA.T. Fuller.

    [4] Zenkov, D.V., A.M. Bloch, and J.E. Marsden [1998], The Energy-Momentum Method for Stability of Nonholonomic Systems. Dynamics and Stability of Systems 13, 123–165.