Self Photo
Research Fellow, Michigan Society of Fellows
Department of Mathematics and Center for the Study of Complex Systems
University of Michigan

I am an applied mathematician and general scientist with a specialization in dynamical systems, networks, and algorithms. I use methods from these and other areas of mathematics to study problems in a range of fields, including biology, ecology, sociology, linguistics, engineering, and philosophy. I am especially interested in problems that are both mathematically interesting and of strong fundamental or practical significance in some clear and direct sense. (I find this constraint limiting enough that drawing from a broad range of fields is helpful.) For each problem, I try to arrive at a new solution by building up from fundamental principles.

Algorithms from nonlinear systems
Project Schematic 1

Following Lax's study of the Korteweg-de Vries equation, dynamical flows were identified that converge to many of the basic decompositions in linear algebra:  QR, LU, Cholesky, and LZ, for example.  Such flows provide alternate ways of computing these decompositions that are sometimes faster than the usual algorithmic routines, especially where approximate answers are sufficient.  Motivated by these and related observations, we are studying whether dynamical systems could also be used to produce fast approximate solutions for harder tasks in computer science (e.g. NP-complete problems).

We are also interested in dynamical systems-based approaches for identifying analytically tractable algorithms for self-organization, primarily for engineering applications.

Fundamental hierarchy of human concepts and knowledge
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Oasis requires the concept of desert. Danger requires the concept of harm. Humans have an elaborate hierarchical network of such conceptual dependencies. Although aspects of this hierachy are understood, there is no comprehensive, large-scale characterization of its structure. Perhaps the most developed contributions toward such a characterization have been theoretical efforts within philosophy and cognitive linguistics. In contrast to this work, we take an approach driven by empirical analysis of English language lexicons and crowd-sourced data from native speakers. We are using insights from this work to assemble a general framework for human knowledge. Aspects of this framework may be necessary for reliable performance of deep learning methods on tasks like translation of informal communication and automated medical diagnosis.

Mathematical medicine
Project Schematic 3

Modern Western medicine excels at treating pathogenic diseases and focal lesions. It is less effective at treating a broad range of system-wide dysfunctions of the endocrine, immune, and nervous systems. These include various metabolic disorders, neurodegenerative diseases, and chronic pain syndromes. Perhaps explanatory of this difference in effectiveness, the first category is often easily diagnosed using non-dynamical means (single blood tests, cell morphology, cell culturing, radiological imaging, etc.), while the second is harder to evaluate with static methods. An alternative diagnostic methodology for the second category may be to monitor the dynamics of factors measurable in fluid samples from patients and then correlate diagnoses with features of the time series. We are in initial stages of combining available kinetic models for endocrine, immune, and nervous subsystems to obtain a unified model for clinical evaluation of this sort of approach.

(All in preparation for publication.)
Equivalence of rapid epidemic dispersal and small-world structure
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We have discovered that a classic model for the spatial spread of a susceptible-infectious epidemic has its transition between spreading at bounded and unbounded speeds at the same "point" as the transition from large-world to small-world structure in an influential spatial network model. We believe this provides a first mathematical foundation for the popular belief that both transitions were simultaneous in history—i.e., that epidemic spreading transitioned from local wave-like propagation to rapid global dispersal just as the network of physical contacts between individuals transitioned from a large to a small world. For additional information, see the preprint of this work.

Minimal model of phase synchronization and desynchronization
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We have constructed and analyzed a new elementary family of models for synchronization and desynchronization. These models offer advantages over the Kuramoto and Peskin models for engineering applications that seek a distributed method to establish asynchrony among users sharing a common resource – for example, multiple access protocols, certain scheduling problems, and distributed tasks like efficient spatial foraging.

A quality factor for scientific theories
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We have derived a quality factor that abstractly quantifies how well a scientific theory predicts outcomes. Methodologically, our work follows an approach similar to that for Arrow’s impossibility theorem or Shannon’s derivation of information entropy: we list the properties that we would like such a quality factor to have and then prove that there is only one such quality factor.

A simple philosophy for mathematics
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Infinities are encoded in mathematics from its foundations. Yet human minds can only remember and manipulate a finite number of digits. How then can we say these infinities are meaningful to a human mind? In this project, we give a brief philosophical account of mathematical practice that delimits mathematics from other non-mathematical human activities and clarifies the meaningfulness of the abstract objects that it wields.

Optimal damping of collisions
Project Schematic 8

The problem of designing materials that minimize damage to vulnerable objects during collisions is common in engineering: packing materials, airbags, head restraints, automotive crumple zones, running shoes, and sports helmets all attempt to reduce the impulse transferred to protected objects in the event of an impact.  We provide a new analysis of this problem that yields a nonlinear second-degree differential equation for the optimal solution. This equation turns out to be exactly solvable.

Characterizing test accuracy
Project Schematic 9

Testing and sampling procedures are used extensively in science and education. For several practical cases, it remains unclear what effect factors like collaborative studying (e.g., study groups), multiple test versions, and various forms of measurement error have on the accuracy of test scores. We use probabilistic rather than statistical approaches to address these questions, finding several new fundamental relationships.