FRG: Collaborative Research: Geometric and Topological Methods for Analyzing Shapes

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As we interact with our environment we constantly assess, measure, and compare the objects within it. Our perception of such objects, either with our eyes, or through the results of scanners, is determined by their shapes, which are themselves characterized by the geometry of their surfaces. We are currently experiencing an explosion of discrete geometric data on shapes of objects obtained from scanners, cameras, imaging systems, sensors, satellites, and even cell phones. There is an urgent need for this geometric data to be processed automatically, for shape matching, shape comparison, and shape recognition. This need arises in areas such as facial recognition, identifying and classifying fossilized bones, distinguishing fractures in bones, diagnosing tumors and anomalies in organs, and measuring changes over time in brain images. Creating a mathematical theory and developing algorithms to recognize and to align such geometric shapes are therefore major research challenges that have far-reaching implications. Deep mathematical theories in geometry and analysis that were developed over the past centuries are now finding applications in this field of shape matching. This project explores fundamental issues in this exciting area, which is on the cusp of seeing major advances. It does so by using the theory of conformal, harmonic, and isometric mappings to align surfaces. While these theories have been extensively studied in a mathematical context, their adaptation to computational algorithms is still under development. This project develops a cohesive and comprehensive theoretical framework for this emerging discipline along with concrete connections to scientific applications. It plans to implement and make publicly available a collection of software that will offer new tools and will open new lines of inquiry to scientists in biology, medicine, anthropology and other fields where the analysis of shape plays a central role.The surfaces in our three-dimensional world can be described mathematically as two-dimensional Riemannian manifolds. Study of the geometric structures on such surfaces is a central topic in mathematical areas such as topology and differential geometry. It leads to classical theories of conformal geometry, moduli spaces, harmonic and conformal maps, and Riemann surfaces. These fields are now being applied to study surfaces of bones, brain cortices, proteins and other bio-molecules. When viewing an object with a laser, or radar, or CAT scan, we obtain a discrete representation of such a surface. Classical theories are inadequate for processing this real world data. This project will develop discrete counterparts of conformal and harmonic maps of surfaces, explore their existence, uniqueness, and diffeomorphism properties, and establish the convergence of the discrete theory to the classical smooth theory. It will also create and implement algorithms that incorporate this theory to create usable software for scientists and other practitioners. In this way, this project will bridge the gap between the mathematical theories of geometry and topology and the application of such ideas to algorithmic analysis of shape data.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.