In a very large monograph of he 70s Almgren provided a deep analysis of the singular set of area minimizing surfaces in codimension higher than 1. I will explain how a more modern approach reduces the proof to a manageable size and allows to go beyond his groundbreaking theorem.
 

Mathematics plays a central role in many recent technological advances. The speaker will describe his experience at a math institute that promotes connections between math and other disciplines. The impact of these interdisciplinary interactions will be demonstrated in three examples: Compression of very large datasets for medical imaging; machine learning as a tool for finding new materials for batteries; and mathematical modeling and computer simulation that enable predictive policing.

Minimal surfaces are among the most natural objects in Differential Geometry, and are fundamental tools in the solution of several important problems in mathematics. In these two lectures we will discuss the variational theory of minimal surfaces  and describe recent applications to geometry and topology, as well as mention some future directions in the field. 

In particular we will discuss our joint work with Andre Neves on the min-max theory for the area functional. This includes the solution of the Willmore conjecture and the construction of infinitely many minimal hypersurfaces in manifolds with positive Ricci curvature. We will also discuss joint work with Agol and Neves on the Freedman-He-Wang conjecture about links. 

I shall introduce local and integral diffusion processes, free boundary problems with and without memory, and discuss applications to American options and economics.

In many imaging problems such as X-ray crystallography, detectors can only record the intensity or magnitude of a diffracted wave as opposed to measuring its phase.  Phase retrieval concerns the recovery of an image from such phaseless information.  Although this problem is in general combinatorially hard, it is of great importance because it arises in many applications ranging from astronomical imaging to speech analysis. This talk discusses novel acquisition strategies and novel convex and non-convex algorithms which are provably exact, thereby allowing perfect phase recovery from a minimal number of noiseless and intensity-only measurements. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. This may be of special contemporary interest because phase retrieval is at the center of spectacular current research efforts collectively known under the name of coherent diffraction imaging aimed, among other things, at determining the 3D structure of large protein complexes.