Abstract: When numbers are added in the usual way, "carries" occur along
the way. Making math sense of the carries leads to all sorts of
corners, in particular to the mathematics of shuffling cards. I will
show that it takes seven ordinary riffle shuffles to mix up 52 cards and
explain connections to fractals and other lovely mathematical objects.
This is a talk for a general audience, no specialist knowledge needed.

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.