Michael C. Reed

Professor of Mathematics

Department of Mathematics
Duke University
P.O. Box 90320
Durham, NC, 27708-0320
Phone: 919-660-2808
Fax: 919-660-2821
reed@math.duke.edu

Education

B.S., Yale University, 1963
Ph.D., Stanford University, 1969

Employment

Instructor, Lecturer, Princeton University, 1968-71
Assistant Professor, Princeton University, 1971-74
Professor, Duke University, 1974-
Chairman, Department of Mathematics, 1982-85, 1986-89
Bishop-MacDermott Proffesor of Mathematics, 2003-               
 
Math 49S: Applications of Mathematics to Physiology and Medicine [HTML] [PDF]

Ph.D. Students

Guy Arthur Battle III (1977), The Applicaton of Banach Algebra Techniques to the Equilibrium Statistical Mechanics of Continuous, Infinite Systems.

E.J. Holder, JR (1980), On the Existence, Scattering, Blow Up, and Decay of Solutions to Systems of Nonlinear Schrodinger Equations.

Michael B. Oberguggenberger (1981), Propagation of Singularities for Semilinear Mixed Hyperbolic Systems in Two Variables.

Thomas C. Messer (1984), The Propagation and Creation of Singularities of Solutions of Quasilinear, Strictly Hyperbolic Systems in One Space Dimension.

Lucio Micheli (1984), The Propagation of Singularities for Non-strictly Hyperbolic Semi-linear Sysetems in One Space Dimension.

Karen Foster Israel (1984), Monotone Behavior for Equilibria of Dynamical Systems.

Harold E. Layton (1986), A Mathematical Model of the Urine Concentrating Mechanism.

Insun Shin (1990), Diffusion with Periodic Obstacles and Applications to Intracellular Diffusion.

Jeanne Nailor (1991), Behavior of Equilibria in Quasi-Thermodynamic Chemical reaction Networks with Mass-Action Kinetics.

Freda Locklear (1991), A Numerical Study of Propagation of Singularities for Semilinear Hyperbolic Systems.

Danielle Carr (1993), Reaction-Hyperbolic Systems in One Space Dimension.

Elizabeth Brooks (1995), Probabilistic Methods for Hyperbolic Partial Differential Equations

Kirsten Travers (1998), Strongly Singluar Solutions of Hyperbolic Partial Differential Equations

Research Interests

Functional Anaylsis, Mathematical Physics, Non-linear Partial Differential Equations, Non-linear Harmonic Analysis, Applications of Mathematics to Physiology and Medicine.

Publications

Functional Analysis and Mathematical Physics

  1. The damped self-interaction, Comm. Math. Phys.  11 (1969), 346-357.
  2. On self-adjointness in finite tensor product spaces , J. Funct. Anal. 5 (1970), 94-124.
  3. A Garding domain for quantum fields, Comm. Math. Phys., 14 (1969), 336-346.
  4. The GNS Construction--a pedagogical example, in Lectures on Elementary Particles and Quantum Field Theory, MIT Press, 1970, 415-436.
  5. Torus invariances for the Clifford Algebra, I, Trans. Amer. Math. Soc. 154 (1971), 177-183.
  6. Torus invariances for the Clifford Algebra, II, J. Funct. Anal. 8 (1981).
  7. (with R. Herman), Covariant representations of tensor product algebras, Pacific Jour. 40 (1972).
  8. (with B. Simon), A spectral mapping theorem for tensor products of unbounded operators, Bull. Amer. Math. Soc. 78 (1972), 330-333.
  9. (with B. Simon), Tensor products of closed operators on Banach spaces, Journ. Func. Anal. 13 (1973), 107-124.
  10. (with J. Rauch), Two examples illustrating the differences between classical and quantum mechanics, Comm. Math. Phys. 29 (1973), 105-111.
  11. Functional analysis and probability theory, in Constructive Quantum Field Theory, ed. A. Wightman, Springer Lecture Notes in Physics 25 (1973).
  12. (with L. Rosen), Support properties of the free measure for Boson fields, Comm. Math. Phys. 36 ( 1974), 123-132.
  13. (with B. Simon), The scattering of classical waves from inhomogeneous media, Math. Z. 155 (1977), 163-180.

Non-linear Partial Differential Equations

  1. Higher order estimates and smoothness for non-linear wave equations, Proc. A.M.S. 51 (1975), 79-85.
  2. Construction of the scattering operator for nonlinear wave equations, Indiana Journ. 25 (1976), 1017-1027.
  3. (with P. Blanchard), Theorie de la diffusion pour l'equation de Sinus-Gordon a trois dimensions, Ann. Inst. H. Poincare 26 (1977), 163-180.
  4. Unsolved problems in the theory of non-linear wave equations and Propagation of singularities in Many Degrees of Freedom in Particle Physics and Field Theory, (ed. L. Streit), Plenum Press, 1978.
  5. (with R. Illner), Decay of solutions of the Carleman Model, Meth. Math. in Applied Sci.3 (1981), 121-127.
  6. Geometry and discrete velocity approximations to the Boltzmann equation, in Differential Equations, editors I. Knowles and R. Lewis, North-Holland, Amsterdam, 1984, pp. 477-484.
  7. (with R. Illner), Decay to equilibrium for the Carleman model in a box, SIAM Journal of Appl. Math. 44 (1984), 1067-1075.
  8. (with J. Blum and S. Venakides), Approximate Travelling Waves in Linear Reaction Hyperbolic Equations, SIAM J. Applied Math. 50 (1990), 167-180.

Non-linear Harmonic Analysis

  1. Propagation of singularities in wave equations of Klein-Gordon type in Non-linear Partial Differential Equations and Applications, Springer Lecture Notes in Mathematics, 648, Edit. J. Chadam, 1978.
  2. Propagation of singularities in non-linear wave equations in one dimension, Commun. Part. Diff. Eq. 3 (1978), 155-199.
  3. (with J. Berning), Reflection of singularities of one-dimensional semilinear wave equations at boundaries, J. of Mathematical Anal. and Appl. 72 (1979), 635-653.
  4. (with J. Rauch), Propagation of singularities for semilinear hyperbolic equations in one space dimension, Annal. Math. 111 (1980), 531-552.
  5. (with J. Rauch), Jump discontinuities of semilinear strictly hyperbolic systems in two variables: creation and propagation, Comm. Math. Phys. 81 (1981), 203-227.
  6. (with M. Beals), Propagation of singularities for hyperbolic pseudodifferential operators with non-smooth coefficients, Comm. Pure Applied Math. 35 (1982), 169-184.
  7. (with J. Rauch), Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension, Duke Math. J. 49 (2), (1982), 397-475.
  8. (with J. Rauch), A general regularity theorem for hyperbolic waves in one space dimension, Bull. Amer. Math. Soc. 6 (3), (1982), 445-446.
  9. (with J. Rauch), Propagation of singularities in non-strictly hyperbolic semilinear systems: Examples, Comm. Pure Applied Math. 35 (1982), 555-565.
  10. (with J. Rauch), Singularities produced by the nonlinear interaction of three progressing waves: Examples, Comm. Part Diff. Eq. 7 (9), (1982), 1117-1133.
  11. Geometric structures in the singularity theory of hyperbolic equations, in Advances in Nonlinear Waves, I, editor, E. Debnath, Cambridge University Press, 1984, 273-280.
  12. (with M. Beals), Microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems, Trans. Amer. Math. Soc. 285 (1984), 159-184.
  13. (with J. Rauch), Ray-like solutions of semilinear wave equations, Contemporary Mathematics 27, Amer. Math. Soc., Providence, R.I., 1984.
  14. (with J. Rauch), Striated solutions of semilinear two-speed wave equations in dimension n > 2, Indiana Journal of Mathematics, Indiana University Math. Journal 34 (1985), 337-353.
  15. (with J. Rauch) Discontinuous progressing waves for semilinear systems, Comm. P.D.E. 10(1985), 1033-1077.
  16. (with J. Rauch), Classical, conormal, semilinear waves, Seminaire Goulaouic-Meyer-

    17. Schwartz, 1985-1986, n V.
  17. (with J. Rauch) Nonlinear superposition and absorption of delta waves in one space dimension, J. Functional Analysis, 73, No. 1 (1987), 152-178.
  18. (with J. Rauch), Ultrasingularities in nonlinear waves, in Non-linear Semigroups, Partial Differential Equations, and Attractors, Springer Lecture Notes in Mathematics, 1248, edited T. Gill, W. Zachary, pp.134-141.
  19. (with J. Rauch), On the absorption of singularities in dissipative nonlinear equations, in Differential Equations and Mathematical Physics, editors I. W. Knowles Y. Saito, Springer-Verlag, 1987. pp. 403-407.
  20. (with J. Rauch), Bounded, stratified, and striated solutions of hyperbolic equations, Nonlinear Partial Differential Equations and their Applications, Volume IX, editors H. Brezis and J. L. Lions, Research Notes in Mathematics # 181, Wiley, New York, 1988.
  21. (with J. Rauch) Propagation of Equality and Classicality for Conormal Solutions of Semilinear System, Comm. PDE 13 (10) (1988), 1297-1335.

Mathematical Biology

  1. (with J. Blum) A model for fast axonal transport, J. Cell Motility 5 (1985), 507-527.
  2. (with J. Blum), Theoretical Analysis of Radioactivity Profiles During Fast Axonal Transport: Effects of Deposition and Transport, Cell Motility and the Cytoskeleton 6 (1986), 620-627.
  3. (with J. Blum), A reaction-hyperbolic system in Physiology,, in Non-linear Semigroups, Partial Differential Equations, and Attractors, Springer Lecture Notes in Mathematics, 1248, edited T. Gill, W. Zachary, pp. 127-133, 1987.
  4. (with J. Blum), The Transport of Organelles in Axons,Math. Biosci., 90 (1988), 233-245.
  5. (with J. Blum), A Model for Slow Axonal Transport and Its Application to Neurofilamentous Neuropathies, J. Cell Motility and the Cytoskeleton, 12 (1989),53-65.
  6. (with J. Blum, G. Lawler, I. Shin) The Effect of Cytoskeletal Geometry on Intracellular Diffusion, Biophys. J., 56 (1989), 995-1005.
  7. (with J. Blum), A Model for the Computation and Encoding of Azimuthal Information by the Lateral Superior Olive, J. Acoust. Soc. Amer. 88(1990),pp. 1442-1453.
  8. (with J. Blum), "Information Processing in the Auditory Brainstem" in Engineering Principles of Physiologic Function, (D. Schneck ed.),New York Univ. Press, New York, 1990.
  9. (with J. Blum), Further Studies of a Model for Azimuthal Encoding: LSO Neuron Response Curves and Developmental Processes, J. Acoust. Soc. Amer.90(4)(1992), pp. 1968-1978.
  10. (with J. Blum and D. Carr). Theoretical Analysis of Lipid Transport in Sciatic Nerve, Biochimica et Biophysica Acta 1125(1992), pp. 313-320.
  11. (with J. Blum) Models of Axonal Transport: Applications to Understanding Certain Neuropathies in Handbook of Neurotoxicology I: Basic Principles and Current Concepts, Dekker, N.Y., 1994.
  12. Reed, M. & J.J. Blum, 1994. Mathematical Questions in Axonal Transport, in Lectures on Mathematics in the Life Sciences, Vol. 24, American Math. Soc., Providence, pp. 1-21.
  13. (with J. Blum) A Computational Model for Signal processing by the Dorsal Cochlear Nucleus: I. Responses to Pure Tones, J. Acoust. Soc. Amer. 97(1995), 425-438.
  14. (with J. Blum and J. Davies) A Computational Model for Signal processing by the Dorsal Cochlear Nucleus: II. Responses to Broadband and Notch Noise, J. Acoust. Soc. Amer.,98(1995), 181-191.
  15. (with L. Durbeck) Delay Lines and Auditory Processing, Comments in Theoretical Biology, 3(1995), 441-461.
  16. (with J. Blum) Envelope coding in the Auditory Brainstem in Proc. Conference on Biomedical Simulation, P. Cellier, Ed., 1997.
  17. (with J. Blum) Effects of Wide-band Inhibitors in the Dorsal Cochlear Nucleus I: Model Calculations of the Responses to Pure Tones, J. Acoust. Soc. Amer., 102(1997), 2238-2244.
  18. (with J. Blum) Effects of Wide-band Inhibitors in the Dorsal Cochlear Nucleus II: Model Calculations of the Responses to complex sounds, J. Acoust. Soc. Amer., 103(1998), 2000-2009.
  19. (with J. Blum) Model Calculations of Steady State Responses to Binaural Stimuli in the Dorsal Nucleus of the Lateral Lemniscus, Hearing Research, 136 (1999), 13-28.
  20. (with J. Blum, Jo Ann Janovick and M. Conn) A Mathematical Model for Quantitation of Gonadotropin-Releasing Hormone Stimulation of Luteinizing Hormone Secretion for Gonadotropes, Amer. J. Physiol., accepted.

Textbooks

  1. (with B. Simon), Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, Academic Press (1972), (Russian Translation 1976).
  2. (with B. Simon), Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-adjointness, Academic Press, 1975. (Russian Translation, 1978).
  3. (with B. Simon), Methods of Modern Mathematical Physics, Vol. III: Scattering Theory, Academic Press, 1979, (Russian Translation, 1982).
  4. (with B. Simon), Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators, Academic Press, 1978. (Russian Translation 1982).
  5. Abstract non-linear wave equations, Springer Lecture Notes in Mathematics 507 (1976).
  6. Fundamental Ideas of Classical Analysis, J. Wiley and Sons, New York, 1997.
  7. (with L. Blake), Duke Laboratory Calculus, J. Wiley and Sons, New York (Math. Dept. copyright), 1998.

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Last modified: 28 June 1999