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Mathematical Sciences

Mathematical Sciences
Peter Kramer

Ph.D. Princeton University
Applied and Computational Mathematics

Research Areas 

My overarching research interest is applying stochastic modeling and analysis to characterize the behavior of complex systems, particularly linking phenomena at a formally macroscopic or collective level to the dynamics prescribed at an individual or formally microscopic level. 

Statistical Mechanics of Swimming Microorganisms

One of the most vigorous fields of recent inquiry regarding the connection of collective behavior to the rules of individuals is in the quantitative modeling of animal swarming or flocking.   In fact for terrestial-scale organisms, for which much of this work is focused, the individual mobility and interaction rules are not yet precisely understood, and various hypotheses are under active exploration.  Swimming microorganisms, on the other hand, mainly because of their rather limited cognitive powers, have comparatively very well understood mechanisms for motion as well as physical and biological rules for interaction.  A growing range of experiments on suspensions of swimming microorganisms moreover reveals a range of remarkable collective behavior, such as patterned flows and changes to effective fluid properties such as viscosity.   From a mathematical point of view, therefore, because of the sound theoretical foundation of individual dynamics and interactions and the range of well-controlled experiments on populations, swimming microorganisms serve as an excellent model system for developing analytical techniques for connecting collective dynamics to the individual dynamical rules.  Moreover, beyond this theoretical motivation, the understanding of suspensions of swimming microorganisms has strong potential for applications to driving microfluidic systems.   Together with Patrick Underhill (Chemical Engineering), recent graduate student Kajetan Sikorski, and current graduate student Yuzhou Qian (Chemical Engineering), we have been working on incorporating both statistical correlations and stochastic effects into a theory for predicting the effective behavior of a swimming microorganisms in a suspension, given the well-defined physical equations for individual motion and hydrodynamic interaction.  The stochastic aspect here arises predominantly not from thermal fluctuations, but from coarse-graining of the detailed dynamics of the swimming process.  An important part of this project is clarifying, on quantitative physical grounds, the source and impact of these stochastic components.

Intracellular transport

The fundamental actors in transporting organelles within cells are families of protein molecules, predominantly dynein and kinesin, which act as molecular-level engines, converting chemical fuel (ATP) into some sort of useful work, such as directed transport.  Mathematical models for these molecular motors are typically stochastic, as thermal fluctuations play a strong role in influencing their dynamics, both by the noisy buffeting by the solvent molecules as well as the random diffusive transport of ATP to the proper binding location to activate the next chemomechanical cycle for progress.  A great deal of mathematical and physical study of the mechanisms and performance of natural and synthetic molecular motors has taken place over the last 20 years.  With Juan Latorre (Freie Universitaet Berlin) and Grigorios Pavliotis (Imperial College), we contributed to this body of work by applying homogenization theory and various techniques from stochastic analysis to map out how the transport effectiveness of a fundamental ``flashing ratchet" model depends on the governing parameters, and to evaluate the regimes in which simplified Markov chain models adequately represented the effective transport of the more detailed model.   Over the last five years, a rapidly growing amount of theoretical and experimental work has been focusing on how the dynamics of a cargo is governed by multiple molecular motors bound to it simultaneously, as is expected to be relevant in vivo.   Within a working group at the Statistics and Applied Mathematical Sciences Institute (SAMSI) we critically examined this literature, and with John Fricks (Penn State, statistics), Scott McKinley (Univ. Florida) and Avanti Athreya (Johns Hopkins), we formulated a new modeling framework for molecular motors cooperatively bound to a cargo which both accounted for stochastic spatial fluctuations in a more physically consistent manner than most of the existing models and allowed for analytical techniques to connect single-motor properties to those of the motor-cargo complex.   We are now, together with experimentalist William Hancock (Penn State, bioengineering), working on developing our basic model to incorporate more realistic features, with the broad aim of linking detailed dynamical understanding of molecular motors at an individual level to larger scale experimental observations.    During discussions of this effort at the Zentrum fuer Interdisziplinaere Forschung in Bielefeld, Germany, some of the other research fellows in residence, namely Leonid Koralov (Maryland), Yuri Makhnovskii (Topchiev Institute), and Leonid Bogachev (Leeds) have begun collaborating with me on developing some of the more technically challenging mathematical aspects needed to capture the relevant physics.

Stochastic Dynamics in Neuronal and Other Networks

A grand challenge in theoretical neuroscience is to explain how higher-level brain functions, such as memory and object recognition, can be explained in terms of the known biophysics of the constituent neurons and their connections.  The overwhelming complexity of the brain naturally frustrates a direct mathematical attack on this problem, so mathematical neuroscience has in recent years focused on more tractable analogue questions, whose resolution can bring insight into these broader conceptual issues.  I became involved in mathematical neuroscience through co-advising former graduate student Katherine Newhall (now at UNC Chapel Hill) with colleague Gregor Kovacic and collaborator David Cai (Shanghai Jiao-Tong), where we were concerned with a project of characterizing the presence of strongly synchronous behavior in a neuronal network model in terms of the individual neuron dynamics and the network model describing their connections.  We employed the basic stochastically driven integrate-and-fire model for the neurons, and both simple and, with Maxim Shkarayev (Iowa State) complex scale-free network model architectures.   Randomness enters into these models both to describe generic unstructured input signals (typically via canonical Poisson point processes) and to generate complex networks, for which only certain statistics are prescribed.  Katherine Newhall, Peter Mucha (UNC), Amanda Traud (NC State), and I are exploring generalizations of these ideas to more general dynamical network models with cascades.  During a recent visit to the Mathematical Biosciences Institute (MBI), Janet Best (Ohio State) and I formed a research group, together with Fatih Olmez, Jung-Eun Kim (UNIST Korea), and Deena Schmidt (UN Reno), to pursue a related question of how the architecture of a metastable neuronal network model for a sleep-wake system affects the distribution of times spent in the sleep or wake states.   Undergraduate research student Anthony Trubiano has been assisting this program through examination of how to effectively characterize sleep and wake states, and transitions between them, in the neuronal network model. 

Stochastic network modeling in epidemiology

Kristin Bennett have been working with graduate student Lei Yao to develop a mathematical model and statistical technique for inferring unobserved transmission events from data of treated tubercolosis patients in New York State.   The objective is to apply individual-based stochastic dynamical models for the spread of tubercolosis with computational methods from network tomography and phylogenetics to integrate temporal information from the data into Bennett's clustering methods.

Statistical Aspects of Multiscale Computing

During a visit to the Statistical and Applied Mathematical Sciences Institute (SAMSI) for their program on ``Stochastic Dynamics," a cross-disciplinary project emerged between computational scientist Sorin Mitran (University of North Carolina), myself, and statisticians M. Susie Bayarri (Universitat de Valencia),  James Berger (Duke), Murali Haran (Penn State, ), and Hans Rudolf Kuensch (ETH Zurich) from a contemporaneous program.   We have been exploring, with undergraduate student Ben Walker, how to strengthen, using systematic techniques from the statistics community, the representation and interaction of the microscale and mesoscale elements of a novel multiscale simulation method (time-parallel continuum-kinetic-molecular) recently introduced by Mitran.

Stochastic modeling in ecology

Brad Lister (biology) and I have co-advised a number of undergraduate students on theoretical ecology modeling projects as part of the ``Computational Science Training in the Mathematical Sciences'' program, and out of this experience, we have been developing mathematical models for ecosystems which incorporate more biophysical realism.  In particular, we are working on an size-structured model for daphnia with explicit coupling to nutrient concentrations, applying a novel probabilistic model for computing the transfer of individuals between size compartments.  We are also developing quantitative frameworks for how foraging decisions are affected by partial information accumulated over time concerning predators.  Graduate student Karen Cumings has begun working with us on understanding recently discovered mechanisms for how plants interact with each other.  Multidisciplinary graduate student Christine Goodrich is in the process of formulating a research project concerning a quantitative study of how human activites affect the services which the environment provides to human living.  

Nonlinear Wave Turbulence

With graduate student Michael Schwarz, Gregor Kovacic and I have been investigating the application of turbulence and kinetic theories to the dispersionless form of the one-dimensional Majda-McLaughlin-Tabak model.   With recent graduate student Warren Towne (now at Lincoln Laboratories), Yuri L'vov (Rensselaer) and I have been developing an approach to simulating weakly nonlinear statistical systems which mitigates artifacts of periodic boundary conditions commonly employed.

Some of the above projects are supported by NSF grants DMS-1211665 and DMS-1344962 (RTG). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

Contact Information

Peter Kramer
Department of Mathematical Sciences
Rensselaer Polytechnic Institute
110 8th Street
Troy, New York 12180

Phone: (518) 276-6896
Fax: (518) 276-4824



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