I am an NSF Graduate Research Fellow in the Department of Mathematics at Rutgers University. My research concerns applied algebraic topological approaches to dynamical systems. My advisor is Konstantin Mischaikow. I spent half of 2018 as an NSF GROW Fellow at VU Amsterdam working with Rob Vandervorst. I was a long term visitor at the Mathematical Biosciences Institute over the fall of 2016 and an NSF EAPSI Fellow at Kyoto University with Hiroshi Kokubu over the summer of 2013. Here is a brief CV.
Broadly speaking, I am interested in the development of mathematical theory and computational tools for the analysis of nonlinear or high-dimensional data generated by dynamical systems. I typically work within the realms of applied topology or computational dynamics, in particular on computational Conley theory, connection and transition matrix theory, and topological data analysis. I'm also interested in the applications of these tools to sytems biology and general data analysis. For more details, see my research statement.
With Shaun Harker and Konstantin Mischaikow.
Abstract: The connection matrix is a powerful algebraic topological tool from Conley index theory that captures relationships between isolated invariant sets. Conley index theory is a topological generalization of Morse theory in which the connection matrix subsumes the role of the Morse boundary operator. Over the last few decades, the ideas of Conley have been cast into a purely computational form. In this paper we introduce a computational, categorical framework for the connection matrix theory. This contribution transforms the computational Conley theory into a computational homological theory for dynamical systems. More specifically, within this paper we have two goals:
1) We cast the connection matrix theory into appropriate categorical, homotopy-theoretic language. We identify objects of the appropriate categories which correspond to connection matrices and may be computed within the computational Conley theory paradigm by using the technique of reductions.
2) We describe an algorithm for this computation based on algebraic-discrete Morse theory.
Here is an arxiv preprint (2018).
With Bree Cummins and Tomas Gedeon.
Abstract: We present a theoretical framework for inferring dynamical interactions between weakly or moderately coupled variables in systems where deterministic dynamics plays a dominating role. The variables in such a system can be arranged into an interaction graph, which is a set of nodes connected by directed edges wherever one variable directly drives another. In a system of ordinary differential equations, a variable x directly drives y if it appears nontrivially on the right-hand side of the equation for the derivative of y. Ideally, given time series measurements of the variables in a system, we would like to recover the interaction graph. We introduce a comprehensive theory showing that the transitive closure of the interaction graph is the best outcome that can be obtained from state space reconstructions in a purely deterministic system. Our work depends on extensions of Takens' theorem and the results of Sauer et al. [J. Stat. Phys., 65 (1991), pp. 579--616] that characterize the properties of time-delay reconstructions of invariant manifolds and attractors. Along with the theory, we discuss practical implementations of our results. One method for empirical recovery of the interaction graph is presented by Sugihara et al. [Science, 338 (2012), pp. 496--500], called convergent cross-mapping. We show that the continuity detection algorithm of Pecora et al. [Phys. Rev. E, 52 (1995), pp. 3420--3439] is a viable alternative to convergent cross-mapping that is more consistent with the underlying theory. We examine two examples of dynamical systems for which we can recover the transitive closure of the interaction graph using the continuity detection technique. The strongly connected components of the recovered graph represent distinct dynamical subsystems coupled through one-way driving relationships that may correspond to causal relationships in the underlying physical scenario.
Here is an official version in SIAM Journal on Applied Dynamical Systems.
With Jesse Berwald and Tomas Gedeon.
Abstract: Complex dynamical systems, from those appearing in physiology and ecology to Earth system modelling, often experience critical transitions in their behaviour due to potentially minute changes in their parameters. While the focus of much recent work, predicting such bifurcations is still notoriously difficult. 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-neighbour algorithm and Conley index theory to classify the dynamics at a predefined scale, we are able to predict with high accuracy the boundaries between regions in parameter space that produce critical transitions.
Here is an official version in Mathematical and Computer Modelling of Dynamical Systems.
Available here is a version of the talk given at the CRM 2019 Workshop on Data Driven Dynamics, the SUNY Albany Algebra/Topology Seminar, the Fall 2018 IAS-Penn-Rutgers joint workshop Identifying Order in Complex Systems, and the UPenn Applied Topology seminar.
Available here is a version of a talk given at the ICMC Summer Meeting on Differential Equations 2019, Workshop on Applied Topology 2019 and the Fall 2018 AMS Eastern Sectional Meeting.
Available here is the talk I gave at the MBI Visitor Seminar in December 2016.
I wrote a User's Guide for the Conley-Morse database during my time as an NSF EAPSI Fellow at Kyoto University. It is available here. The intended audience is potential users of the Conley-Morse database who have had little exposure to dynamics or even mathematics.
I am a TA at Large for Math 252. Students should consult the Sakai site for more information.
In Fall 2018 I was a TA for Math 151: Calculus I for the Mathematical and Physical Sciences Sections 25-27.
In Fall 2017 I was a TA for Math 244: Differential Equations for Engineering and Physics at Rutgers.
Over summer 2017 I was the instructor for the introductory algebra course (Math 351) at Rutgers University. The textbook we used was baby Hungerford. Here are some lecture notes I wrote for the course, which were popular among the students.
In Spring 2018 I was TA for both Calculus II and Linear Algebra for Business Analytics at VU University in Amsterdam.
Hill Center - Busch Campus
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Piscataway, NJ 08854-8019, USA
kelly.spendlove at rutgers.edu