Mark H Holmes
Professor, Mathematical Sciences
My research interests are time dependent. The reason is that I like mathematics, and I like to use mathematics to understand the world we live in (i.e., I am an applied mathematician). Here are two specific research topics of current interest: Mechanoreception. This is the study of how a living organism senses and then recognizes mechanical stimuli. The mechanoreceptors that I'm presently working on are in the auditory system and in skin. For example, the mechanisms through which skin transduces tactile stimuli from a mechanical to neural signal are not understood. Therefore, the goal of my research is to construct a mathematical model that describes this process. This project involves the study of the nonlinear deformation of skin, the analysis of the dynamics of the tactile receptor, and the study of the nonlinear diffusion associated with the electro-chemical processes in the nerve. Sleep-Wake Cycle. The goal of this research project is to derive, and then analyze, a physiologically based model of the human sleep-wake cycle. The approach is to use the known properties of the neurotransmitters associated with wake and sleep, and the regions of the brain in which they function, to derive the model. This approach also incorporates, or accounts for, the states of sleep (REM and NREM) and the mechanisms that regulate it (homeostatic drive and circadian synchronization).
Books Introduction to the Foundations of Applied Mathematics, Springer, 2009. Web page Introduction to Numerical Methods for Differential Equations, Springer, 2006. Web page Introduction to Perturbation Methods, Springer-Verlag, New York, 1995. Table of Contents Answers to Selected Exercises Errata List: 2nd Printing, 1st Printing Mixture Theories for the Mechanics of Biological Tissues, RPI Web Book, 1995. Info Page
Project Web-Sites Project CSUMS RTG (coming soon) Project Links
A Few Awards Guggenheim Fellow Y.C. Fung Young Investigator Award (ASME) 2000 Premier Award for Excellence in Engineering Education Courseware 2001 ASME Curriculum Innovation Award 2002 Award for Innovative Excellence in Teaching, Learning and Technology 2002 Best Paper Award, 13th International Conference on College Teaching and Learning 2007 ICTCM Award for Excellence and Innovation with the Use of Technology in Collegiate Mathematics 2007 Rensselaer Trustee's Outstanding Teacher Award
Larger Grants (as Project Director) "GAANN: Graduate Assistance in Areas of National Need," with M. Cheney, I. Herron, G. Kovacic, F. Li, D. Schwendeman, and V. Roytburd, $1,330,000, Department of Education. "RTG: Research Training Group in Large-Scale Nonlinear Systems," with G. Kovacic, P. Kramer, F. Li, Y. Lvov, and V. Roytburd, $1,272,000, National Science Foundation. Summary for first three years of grant. "CSUMS: Computational Science Training in the Mathematical Sciences at Rensselaer," with I. Herron, G. Kovacic, P. Kramer, and V. Roytburd, $1,251,000, National Science Foundation. Summary for first three years of grant. "Initiative for Vertical Integration of Research and Education in Applied Mathematics," with J. Flaherty, G. Kovacic, J. McLaughlin, and D. Schwendeman, $3,830,000, National Science Foundation. Summary for last two years of grant. "Mathematics and its Applications in Engineering and Science: Building the Links," with W. Boyce, R. Spilker, K. Conner, and J. Wilson, $4,016,000, National Science Foundation.
Ph.D. University of California, Los Angeles, 1978
- Development and analysis of mathematical models for physiological systems, including modeling the biological tissues found in joints (such as the knee).
- A Model for the Nonlinear Mechanism Responsible for Cochlear Amplification, Math Biosci Eng. 2014 Dec;11(6):1357-73 (with K. Fessel)
- Connections Between Cubic Splines and Quadrature Rules, American Mathematical Monthly, Vol 121, Num 8, Oct 2014, pp. 723-726
- Asymmetric Random Walks and Drift-Diffusion, EPL (Europhysics Letters), v102(3), 2013, 30005