Research
I am interested in applied and computational algebraic topology, and am attending the IMA Thematic Year on Scientific and Engineering Applications of Algebraic Topology. Here is my research statement. Below are a list of publications, some of which are still in preprint form.Publications


Automatic recognition and tagging of topologically different regimes in dynamical systems. with Marian Gidea and Mikael VejdemoJohansson.
Full Manuscript, 2014 (preprint). Complex systems are often modeled as (typically chaotic) highdimensional deterministic dynamical systems, or as random dynamical systems with lower dimensional deterministic components. From the point of view of forecastabiliy, a highdimensional chaotic system is equivalent to a stochastic system. Since a detailed description of the dynamics is virtually unattainable, a more sensible goal is to recognize and mark the transition of a system between qualitatively different regimes of behavior. In practice, one is interested in developing techniques for detection of such transitions from sparse observations, possibly contaminated by noise. In this paper we develop a framework to accurately tag different regimes of complex systems based on topological features. In particular, our framework works with a high degree of success in picking out a cyclically orbiting regime from a stationary equilibrium regime in highdimensional stochastic dynamical systems. 



Critical Transitions in a Model of a Genetic Regulatory Network. with Marian Gidea.
Full Manuscript, 2013. We consider a model for substratedepletion oscillations in genetic systems, based on a stochastic differential equation with a slowly evolving external signal. We show the existence of critical transitions in the system. We apply two methods to numerically test the synthetic time series generated by the system for early indicators of critical transitions: a detrended fluctuation analysis method, and a novel method based on topological data analysis (persistence diagrams). 



NonGlobal Parameter Estimation Using Local Ensemble Kalman Filtering (to appear in Monthly Weather Review). with Thomas Bellsky and Lewis Mitchell.
Full Manuscript, 2013. We study parameter estimation for nonglobal parameters in a lowdimensional chaotic model using the local ensemble transform Kalman filter (LETKF). By modifying existing techniques for estimating global parameters using observational data, we present a methodology whereby spatiallyvarying parameters can be estimated using observations only within a localized region of space. Taking a lowdimensional nonlinear chaotic conceptual model for atmospheric dynamics as our numerical testbed, we show that the LETKF accurately estimates parameters which vary in both space and time, as well as parameters representing physics absent from the model. 



Predicting HighCodimension Critical Transitions In Dynamical Systems Using Active Learning. with Tomas Gedeon and Kelly Spendlove.
Full Manuscript, 2013. Complex dynamical systems, from those appearing in physiology and ecology to Earth systems modeling, often experience critical transitions in their behavior due to potentially minute changes in their parameters. While the focus of much recent work, predicting such bifurcations is still notoriously diffcult. We propose an active learning approach to the classification of parameter space of dynamical systems for which the codimension of bifurcations is high. Using elementary notions regarding the dynamics, in combination with the nearest neighbor algorithm and Conley index theory to classify the dynamics at a predened scale, we are able to predict with high accuracy the boundaries between regions in parameter space that produce critical transitions 



Using Machine Learning to Predict Catastrophes in Dynamical Systems. with Tomas Gedeon and John Sheppard.
Full Manuscript, 2012. Nonlinear dynamical systems, which include models of the Earthâs climate, financial markets and complex ecosystems, often undergo abrupt transitions that lead to radically different behavior. The ability to predict such qualitative and potentially disruptive changes is an important problem with farreaching implications. Even with robust mathematical models, predicting such critical transitions prior to their occurence is extremely difficult. In this work we propose a machine learning method to study the parameter space of a complex system, where the dynamics is coarsely characterized using topological invariants. We show that by using a nearest neighbor algorithm to sample the parameter space in a specific manner, we are able to predict with high accuracy the locations of critical transitions in parameter space. 



Modeling Complexity of Physiological Time Series InSilico. with Tomas Gedeon and John Sheppard.
Full Manuscript, 2010. A freerunning physiological system produces time series with complexity which has been correlated to the robustness and health of the system. The essential tool to study the link between the structure of the system and the complexity of the series it produces is a mathematical model that is capable of reproducing the statistical signatures of a physiological time series. We construct a model based on the neural structure of the hippocampus that reproduces detrended fluctuations and multiscale entropy complexity signatures of physiological time series. We study the dependence of these signatures on the length of the series and on the initial data. 



Miscellaneous (nonmath) Publications 

The Logic of Simpson's Paradox. with P. Bandyopadhyay, D. Nelson, M. Greenwood, and G. Brittan.
Full Manuscript, 2011. Our research on Simpson's paradox suggests that the "to proceed question", needs to be divorced from what makes Simpson's paradox "paradoxical". 

