Professor, Mathematical Sciences
Dr. Kramer received his B.A. in Physics from Princeton University, and earned his Ph.D. in Applied and Computational Mathematics at Princeton University under the supervision of Professor A.J. Majda. Dr. Kramer took a three-year Courant Instructor and NSF postdoctoral research fellow position at the Courant Institute at New York University, before joining the faculty in the Department of Mathematical Sciences at Rensselaer as an assistant professor in August, 2000.
Dr. Kramer's research focuses on the application of ideas and techniques from probability theory and differential equations to model and analyze complex systems which evolve in time and involve too many variables to represent explicitly in a practical computational model. The effects of the unresolved variables on the quantities of interest are treated in a statistical fashion. Current areas of research include intracellular transport, the statistical dynamics of suspensions of swimming microorganisms, and metastable dynamics in sleep-wake networks. This work is in collaboration with scientists and engineers at Arizona State University, Ohio State University, Pennsylvania State University, Tulane, as well as at Rensselaer.
Dr. Kramer supervises the preparation of undergraduates at Rensselaer for the annual Mathematical and Interdisciplinary Contest in Modeling, and supports the organization of the Graduate Student Mathematical Modeling Camp (2013-present) as well as the endeavors of the Mathematical Problems in Industry Workshop (2000-present). Dr. Kramer is also an active member of the NSF-funded Research Training Grant (RTG) program in the department, which supports the research and education of several graduate students and postdoctoral research fellows.
Ph.D., Princeton University, 1997
- Stochastic modeling in microbiology
- Stochastic dynamics on networks
- Spatial ecology
- Nonlinear wave turbulence
- K. A. Newhall, M. S. Shkarayev, P. R. Kramer, G. Kovacic, and D. Cai, "Synchrony in stochastically driven neuronal networks with complex topologies," Physical Review E 91 (2015): 052806.
- J. C. Latorre, P. R. Kramer, and G. A. Pavliotis, "Numerical Methods for Computing Effective Transport Properties of Flashing Brownian Motors," Journal of Computational Physics 257A (2014): 57-82.
- O. Kurbanmuradov, K. Sabelfeld, and P. R. Kramer, "Randomized Spectral and Fourier-Wavelet Methods for Multidimensional Gaussian Random Vector Fields," Journal of Computational Physics 245 (2013): 218-234.
- K. A. Newhall, E. P. Atkins, P. R. Kramer, G. Kovacic, and I. R. Gabitov, "Random Polarization Dynamics in a Resonant Optical Medium," Optics Letters 38 (6), (2013): 893-895.
- S. A. McKinley, A. Athreya, J. Fricks, and P. R. Kramer, "Asymptotic Analysis of Microtubule-Based Transport by Multiple Identical Molecular Motors," Journal of Theoretical Biology 305 (2012): 54-69.
- S. R. Keating, P. R. Kramer, and K. S. Smith, "Homogenization and Mixing Measures for a Replenishing Passive Scalar Field,"Phys. Fluids. 22, (2010): 075105.
- P. R. Kramer, C. S. Peskin, and P. J. Atzberger, "On the foundations of the stochastic immersed boundary method," Computer Methods in Applied Mechanics and Engineering, 197(25-28), 2008: 2232-2249
- A. J. Majda and P. R. Kramer, "Simplified models for turbulent diffusion: Theory, numerical modelling and physical phenomena," Physics Reports, 314 (4-5), 1999: 237-574