Projects

Existing philosophical literature offers many "local" analyses of concepts, that is, definitions of composite concepts in terms of constituent concepts without broader assertions of which concepts are fundamental.  For example, we might define "certainty" using some notion of algorithmic verification, define "moral wrong" using an idea of unjustified harm, and define "free will" as a conceptual blend of a branch in a possible world and the feeling of making a conscious choice. This project ambitiously endeavors to assemble these sorts of local notions into a single "global" account – a formal language of conscious experience within which most or all major philosophical concepts can be defined.

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, this project examines whether dynamical systems could also be used to produce fast approximate solutions for hard tasks in computer science (e.g., NP-complete problems).

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.  This project proposes a new analysis of this problem.

Atrial fibrillation is the most common arrhythmia encountered in the clinic, affecting 1.5-2% of the global population.  It confers a five-fold increased risk of stroke, a three-fold increased risk of heart failure, and a 2.4-fold increased risk of dementia.  High-energy shocks have been used to successfully terminate cardiac arrhythmias for over 100 years.  However, using these shocks to manage atrial fibrillation―for example, with implantable cardioverter defibrillators―is traumatic, painful, and poorly tolerated.  This project seeks to develop a painless, low-energy method of cardiac defibrillation.

Human disease takes an astonishing array of different forms. The "root causes" of these diseases, however, must either be present at birth, arrive from the environment, or emerge randomly within the patient. The last of these possibilities is particularly mysterious. It appears, for example, that some cases of cancer and prion disease can emerge spontaneously, but what other random events, or combinations of random events, within the body lead to disease? Answering this question is essential for designing a rigorous method of differential diagnosis. It may also be helpful for narrowing down the possible mechanisms of unexplained medical phenomena like sudden infant death syndrome, IgA nephropathy, and inclusion body myositis. And on a broader scale, it may steer us toward the best strategies for population-level prevention and early detection of disease. Motivated by these and related benefits, this project seeks to offer a proposal for a high-level classification of the root causes of disease.

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 hierarchy 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 this project, we seek to built a hierarchy of conceptual dependencies between human concepts using computational analysis of English-language lexicons. Such a hierarchy could be useful for tasks like translation of verbal and written communication and automated medical diagnosis.

While Fisher and Kolmogorov offered a first analysis of spatial contagions driven by diffusive mobility, many spatial contagions in nature are not driven by diffusion but by local influence and recurrent (i.e., out-and-back) mobility. For example, infectious diseases in range-bound animal populations, electrical conduction in the heart, and riot behavior in metropolitan areas each spread by nondiffusive mechanisms. It is quite intuitive that spatial networks that allow epidemics to spread at an accelerating pace must have a small-world structure—and vice versa—that spatial networks over which epidemics spread at constant speed should be "large-world" in their connectivity. However proving this using the standard definition of "small-world" turns out to be nontrivial. In this project, we give an elementary, two-part argument of one direction of this equivalence in a classically important deterministic case.

In this project, we seek to derive a quality factor that quantifies how well an idealized scientific theory predicts outcomes. Methodologically, this 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 pursue a proof that there is only one such quality factor.

Testing and sampling procedures are used extensively in science and education. Under certain realistic assumptions, it remains unclear what mathematical 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. This project uses probabilistic approaches to characterize these relationships.