"The central goal of this proposal is to settle the algorithmicfoundations of geometry understanding in dimensions higher than 3. Wecoin the term geometry understanding to encompass a collectionof tasks including the computer representation and the approximationof geometric structures, and the inference of geometric or topologicalproperties of sampled shapes.The need to understand geometric structures is ubiquitous in scienceand has become an essential part of scientific computing anddata analysis.Geometry understanding is by no means limited tothree dimensions. Many applications in physics, biology,andengineering require a keen understanding of the geometry of a varietyof higher dimensional spaces tocapture concise information from the underlying often highlynonlinear structure ofdata. Our approach is complementary to manifold learningtechniques and aims at developing an effective theory for geometric andtopological data analysis.To reach these objectives, the guiding principle will be to foster asymbiotic relationship between theory and practice, and to addressfundamental research issues along three parallel advancingfronts. We will simultaneously develop mathematical approachesproviding theoretical guarantees, effective algorithms that areamenable to theoretical analysis and rigorous experimental validation,and perennial software development. We will undertake thedevelopment of a high-quality open source software platform toimplement the most important geometric data structures and algorithmsat the heart of geometry understanding in higher dimensions. Theplatform will be a unique vehicle towards researchers from otherfields and will serve as a basis for groundbreaking advances inscientific computing and data analysis."