Charles Collins, Ph.D. - US grants

This Focused Research Group is composed of pure
mathematicians, computational mathematicians, and
neuroscientists. They develop implementations of discrete
conformal mapping for multidisciplinary use, both within
mathematics itself where complex analysis is being reinvigorated
by new discrete techniques, and in the larger scientific context
with visualization and analysis of scientific data. The Riemann
Mapping Theorem guarantees unique conformal maps between any pair
of conformal 2-discs (or conformal 2-spheres); the conformal
geometry preserved by such maps carries valuable mathematical
structure. Such surfaces arise naturally in many scientific
contexts as piecewise flat (from data) or smoothly embedded (from
theory) surfaces in 3-space. Recently the new computational
technique of circle packing has allowed computational
approximations to these conformal maps. Implementing such
approximations for large scientific datasets faces both
theoretical and computational challenges. The investigator and
his colleagues work on three related topics: theoretical
superstructure of the circle packing technique, refinement and
parallelization of the circle packing algorithm for use on large
datasets, and the application of these conformal maps to
visualization and analysis of scientific data. The main
application focuses on conformal flattening of human brain
cortical surfaces. The investigators use uniqueness of conformal
maps to install surface-based coordinate systems on these
surfaces; these coordinate systems allow localization of
activation foci in Positron Emission Tomography (PET) and
functional Magnetic Resonance Imaging (fMRI) brain scans.
Conformal flattening has wider applicability as a visualization
and graph embedding technique, and these connections inform the
research.
This Focused Research Group develops algorithms to bring a
classical mathematics theorem (the Riemann Mapping Theorem, 1854)
to bear on problems of visualization of data. The Riemann Mapping
Theorem guarantees the existence of unique conformal
(angle-preserving) maps between surfaces, but does say how to
compute these maps. Modern computers and new algorithms have
changed all that, because our new computational ability can
breathe life into classical existence theorems of mathematics,
turning theory into computational tools. This project develops
algorithms to implement the computation of conformal maps on
complex spatial surfaces. The main application is the flat
mapping of human brain cortical surfaces. The brain surface is
highly convoluted and folded in space, and most of the brain
surface is folded up and hidden from view. If one flattens the
surface, one can simultaneously see down into all the folds. The
mathematically unique conformal maps produced by the algorithms
allow surface-based coordinate systems to be computed on the
brain surface so that surface positions can be precisely
determined. Moreover, if one puts foci of functional activation
onto the flattened surface, one can then visualize and measure
the relationship between brain function and brain anatomy. These
new surface-mapping techniques and their application to the brain
surface permit biomedical researchers and clinicians to rapidly
and accurately map and compare the locations of physiological and
pathological "events" in the brains of research subjects and of
patients with a variety of neurological and psychiatric
disorders. The project is supported by the Computational
Mathematics, Applied Mathematics, and Geometric Analysis programs
and the Office of Multidisciplinary Activities in MPS and by the
Computational Neuroscience program in BIO.