We describe a result by Niyogi, Smale and Weinberger that learns the homology groups of a submanifold of Euclidean space with high confidence from random samples. Then, we briefly explain a recent result, partially based on the work above, on the complexity of computing the homology of real projective varieties.
Felipe Cucker is Chair Professor of Mathematics. His research covers a variety of subjects including semi-algebraic geometry, computer algebra, complexity, emergence in decentralized systems (in particular, emergence of languages and flocking), learning theory, and foundational aspects of numerical analysis. He serves on the editorial board of several journals and is Managing Editor of the journal Foundations of Computational Mathematics, published by the society of the same name. He has published 4 books including one ("Manifold Mirrors", Cambridge University Press, 2013) exploring the relations between art and mathematics.