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 a few of the research topics for my more recent PhD students (in reverse chronological order):
Northern Lights. The problem for the aurora involves solving an electron transport equation in the upper atmosphere that is coupled to the kinetic model for light emission due to scattering. The transport equation was solved using an eigenvalue decomposition method that allows for the accurate resolution of the rather extreme boundary layer near the Earth's surface. When coupled to the light emissions model, it was possible to predict accurately the (red, green, and blue) light seen in an aurora.
Model and Analysis for the Onset and Progression of Parkinson's Disease. Parkinson's disease (PD) is associated with abnormally synchronized oscillations in the beta (~20 Hz) frequency band in the basal ganglia. Developing a quantitative understanding of PD involves one of the major challenges in theoretical neurobiology, which is to be able to characterize the dynamic interactions in a complex and distributed network model of neural circuits. Using a mean field firing-rate model it was shown how the interactions between the neuronal groups change from a steady-state response (healthy) to a limit cycle behavior as the disease progresses.
Nonlinear Amplification in the Cochlea. A fundamental open question in understanding how we hear concerns the role of a nonlinear feedback mechanism known as the cochlear amplifier. In this project, a nonlinear three-dimensional continuum model for the amplification of a wave in the cochlea was analyzed. This involved using a nonlinear WKB approximation, and a hybrid numerical scheme, to show that the model is capable of reproducing some of the more well-known affects of the amplifier.
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).
Introduction to the Foundations of Applied Mathematics
Introduction to Perturbation Methods
Introduction to Differential Equations
Introduction to Scientific Computing and Data Analysis
Introduction to Numerical Methods in Differential Equations
Applied Math Days 2019
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
2007 ICTCM Award for Excellence and Innovation with the Use of Technology in Collegiate Mathematics
2007 Rensselaer Trustee's Outstanding Teacher Award
"RTG: Research Training in Applied Mathematics," with G. Kovacic, P. Kramer, F. Li, Y. Lvov, and D. Schwendeman, $2,099,878, National Science Foundation. Summary. [active]
"GAANN: Graduate Assistance in Areas of National Need ," with I. Herron, G. Kovacic, F. Li, and D. Schwendeman, $1,330,000, Department of Education. [inactive]
"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. [inactive]
"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. [inactive]
"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. [inactive]
"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. [inactive]
"A model and analysis for the nonlinear amplification of waves in the cochlea," with K. Fessel. Mathematical Biosciences
"Invariance properties for the error function used for multilinear regression," with M. Caiola. PLOS ONE
"Numerical solution of the electron transport equation in the upper atmosphere," with M. Woods and W. Sailor. Journal of Computational Physics
"Model and Analysis for the Onset of Parkinsonian Firing Patterns in a Simplified Basal Ganglia," with M. Caiola. International Journal of Neural Systems
"Conservative numerical methods for nonlinear oscillators," accepted for publication American Journal of Physics
"Empirical Analysis of Phase-Amplitude Coupling Approaches," with M. Caiola, T. Wichmann, and A. Devergnas, accepted for publication PLOS ONE
Numerical Computing Fall '19
Art and Science of Mathemtics Fall '19
Numerical Linear Algebra with Applications Spring '19
Foundations of Applied Mathematics Fall '18
Calculus II Fall '16
Introduction to Differential Equations Spring '16
Mathematics in Medicine and Biology Fall '15
Perturbation Methods Spring '12
Intro to Topology Fall '10
Intro to Math Research Spring '09
Art and Science of Mathemtics II Spring '09
Intro to the Numerical Solutions of Differential Equations Spring '05
Mac Resource Page Recommended software and recreational reading for Mac users.
Reason for Nobel Prizes?
Reason for above observation?
Office: Amos Eaton 322
E-mail: holmes (@rpi.edu)
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