Quantum cohomology is a deformation of the classical cohomology algebra of an algebraic variety X that takes into account enumerative geometry of rational curves in X. This has many remarkable properties for a general X, but becomes particularly structured and deep for special X. One of the most interesting class of varieties in this respect are the so-called equivariant symplectic resolutions. These include, for example, cotangent bundles to compact homogeneous varieties, as well as Hilbert schemes of points and more general instanton moduli spaces. A general vision for a connection between quantum cohomology of sympletic resolutions and quantum integrable systems recently emerged in supersymmetric gauge theories, in particular in the work of Nekrasov and Shatashvili. In my lecture, which will be based on joint work with Davesh Maulik, I will explain a construction of certain solutions of Yang-Baxter equation associated to symplectic resolutions as above. The associated quantum integrable system will be identified with the quantum cohomology of X. If time permits, we will also explore K-theoretic generalization of this theory.

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.