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 transport within biological cells, the swimming dynamics of irregular colonies of flagellated cells, statistical modeling in ecology, and the statistical inference of neuronal network topology from firing data. This work is in collaboration with scientists and engineers at Arizona 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 (2013-17) 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
- F. Olmez, P. R. Kramer, J. Fricks, D. R. Schmidt, and J. Best, "Penalized KS method to fit data sets with power law distribution over a bounded subinterval," Journal of Statistical Computation and Simulation 91 (8), 1524-1563.
- J. J. Klobusicky, J. Fricks, and P. R. Kramer, ''Effective behavior of cooperative and nonidentical molecular motors," Research in the Mathematical Sciences 7 (2020): 29.
- M.-V. Ciocanel, J. Fricks, P. R. Kramer, and S. A. McKinley, "Renewal reward perspective on linear switching diffusion systems," Bulletin of Mathematical Biology 82 (2020): 126.
- Y. Qian, P. R. Kramer, and P. T. Underhill, "Stochastic Kinetic Theory for Collective Behavior of Hydrodynamically Interacting Active Particles,'' Physical Review Fluids 2 (2017): 043104.
- 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.
- 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.
- 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 (2008): 2232-2249
- A. J. Majda and P. R. Kramer, "Simplified models for turbulent diffusion: Theory, numerical modelling and physical phenomena," Physics Reports, 314 (1999): 237-574