Bulk sand, if cyclically sheared gently enough, will spontaneously crystallize, in some ways like water crystallizing into ice. For water the standard model is Classical Nucleation Theory. We discuss the difficulties in such an approach to the crystallization of sand.

Symmetric tensors are multi-dimensional arrays invariant to permutation of indices. Symmetric tensors arise in data science when applying the method of moments, as higher-dimensional analogs of the sample covariance matrix. In tasks from demixing Gaussian mixture models to blind source separation, it is informative to decompose a symmetric tensor as a sum of symmetric outer products of vectors. This talk presents a novel algorithm for computing low-rank symmetric tensor decompositions, based on a modified tensor power method. Numerical experiments demonstrate that our algorithm significantly outperforms state-of-the-art methods, per standard performance metrics. We provide supporting theoretical guarantees, through connections to optimization theory, algebraic geometry and dynamical systems. We also extend the algorithm to compute a certain generalization of symmetric tensor decompositions. By applying the method of moments, this enables estimation of a union of linear subspaces from noisy point samples, i.e., robust subspace clustering. Applications to motion segmentation and image segmentation are discussed. Finally, parallels to numerical linear algebra motivate a plethora of interesting open problems.

The McKay correspondence takes many guises but at its core connects the geometry of minimal resolutions for quotient singularities C^n / G to the representation theory of the group G. When G is an abelian subgroup of SL(3), Craw-Ishii showed that every minimal resolution can be realised as a moduli space of stable quiver representations associated to G, although the chamber structure for the stability parameter and associated wall-crossing behaviour is poorly understood. I will describe my recent work giving explicit representation-theoretic descriptions of the walls and wall-crossing behaviour for the chamber corresponding to a particular minimal resolution called the G-Hilbert scheme. Time permitting, I will also discuss ongoing work with Yukari Ito (IPMU) and Tom Ducat (Bristol) to better understand the geometry, chambers, and corresponding representation theory for other minimal resolutions.

Let Pn denote real projective n-space, Sn an n-sphere, and Bn an n-ball. 1) P3 is split along an embedded P2 and an S2 boundary is attached. Capping with a B3 results in S3. 2) P3 is split along an embedded S2. Two components are formed and an S2 boundary is attached to each. Capping both with a B3 results in an S3 and a P3.