Some Mathematical Problems in Shape Formation
In mathematics, we formulate conjectures and discover principles attesting to the coherence and harmony in nature. What is specic to mathematics is that we achieve this through rigorous deduction. Therefore, mathematics may provide a firm ground to our empirical understanding of the physical phenomena. It is important, however, to remember that mathematicians (like myself) engage in pure mathematics, without having any concrete "application" in mind but in a free pursuit driven by intellectual curiosity. This curiosity is rewarded by surprising discoveries of the elegant and consistent structures in the abstract objects we study. Sometimes, it also leads to discoveries regarding the practical applications for what began as pure mathematics. Therefore, there is no clear line separating pure and applied mathematics.
Recently, there has been sustained interest in the growth-induced morphogenesis, particularly of the low-dimensional structures such as fillaments, laminae and their assemblies, arising routinely in biological systems and their artificial mimics. The physical basis for morphogenesis can be presented in terms of a simple principle: differential growth in a body leads to residual strains that generically result in changes of the body's shape. Eventually, the growth patterns are expected to be, in turn, regulated by these strains, so that this principle might well be the basis for the physical self-organization of the biological tissues.
While such questions lie at the interface of biology, chemistry and physics, fundamentally they have a deeply geometric and analytical character. Indeed, they may be seen as a variation on a classical theme in differential geometry - that of embedding a shape with a given metric in a space of possibly different dimension. The goal now, in addition to stating the conditions when it might be done (or not), is to: 1) constructively determine the resulting shapes in terms of an appropriate mathematical theory, and: 2) investigate the separation of scales which arise, naturally, in slender structures and discover the constraints associated with the prescription of a metric.
Our abstract results may be applied to the model energy of the nematic liquid crystal elastomers, which are rubber-like, cross-linked, polymeric solids, having both positional elasticity due to solid response of the polymer chains, and the orientation elasticity due to the separately deforming director. Another particular choice of the metric G is consistent with the experiment fabricating programmed at disks of gels having a non-constant monomer concentration which induces a "differential shrinking factor". In this experiment, the disk is then activated in a temperature raised above a critical threshold, whereas the gel shrinks locally with a factor proportional to its concentration.
Similar questions can, naturally, also be posed on networks, via a discrete differential geometry approach; however, this area is much less developed than the well established field of classical differential geometry and the calculus of variations. Our forthcoming Math Research Center semester-long program at the University of Pittsburgh, will concentrate exactly on such topics. Following our previous successful programs (namely, the 2013 Semester on Game Theory and PDEs, and the 2014 Semester on Convex Integration and Analysis), we will host the "Fall 2014 Theme Semester on Discrete Networks: Geometry, Dynamics and Applications". This program, which is organized by B. Doiron, B. Ermentrout, J. Rubin and M. Lewicka, will be devoted to studying topics pertaining to: discrete networks, structural rigidity and morphogenesis of discrete structures, dynamics on networks, and dynamic network problems in neuroscience. It will take place in the period of September-December 2014.
Stay tuned for more information coming soon!
The role of ongoing spatio-temporal activity on shaping responses to inputs in biological networks
PI: Bard Ermentrout
Sponsor: Benter Foundation
Biological systems such as networks of neurons are able to produce a wide range of intrinsic patterns such as waves and synchronous oscillations. On the other hand, these systems must constantly respond to external stimuli. The goal of this project is to find conditions generating complex spatio-temporal activity in these networks networks and then use various mathematical tools to examine how these dynamics affect responses to extrinsic activity such as noise or patterned inputs.
Model-Based Decisions in Sepsis
CoPI: David Swigon,
PI: Gilles Clermont, UPMC
Computational modeling has been an integral part of knowledge discovery in many fields, but not clinical medicine, owing mostly to the complexity of the human as a system and the absence of adequate data sets where models could be validated. Yet, computational models will prove central to the design of smart and individualized treatment strategies of complex diseases. We are proposing to develop such practical models for the specific case of sepsis, a condition which has proven particularly difficult to treat and where clinical trials of new drugs have failed repeatedly. Combining the expertise of a transdisciplinary group of modelers pioneering infection research and the vast data set generated from a prospective, randomized trial controlled trial of how to best support patients in the first few hours of sepsis, we will construct and validate predictive computational models of sepsis which could form the basis for smarter clinical trials and personalized therapies.