Monday, November 16, 2020
11/16/2020 - 2:30pm
Abstract: We propose a latent variable model to discover faithful low-dimensional representations of high-dimensional data. The model computes a low-dimensional embedding that aims to preserve neighborhood relationships encoded by a sparse graph. The model both leverages and extends current leading approaches to this problem. Like t-distributed Stochastic Neighborhood Embedding, the model can produce two- and three-dimensional embeddings for visualization, but it can also learn higher-dimensional embeddings for other uses. Like LargeVis and Uniform Manifold Approximation and Projection, the model produces embeddings by balancing two goals—pulling nearby examples closer together and pushing distant examples further apart. Unlike these approaches, however, the latent variables in our model provide additional structure that can be exploited for learning. We derive an Expectation–Maximization procedure with closed-form updates that monotonically improve the model’s likelihood: In this procedure, embeddings are iteratively adapted by solving sparse, diagonally dominant systems of linear equations that arise from a discrete graph Laplacian. For large problems, we also develop an approximate coarse-graining procedure that avoids the need for negative sampling of nonadjacent nodes in the graph. We demonstrate the model’s effectiveness on datasets of images and text.
11/16/2020 - 4:00pm
Anosov representations are the higher rank analogues of convex cocompact
representations into rank one Lie groups. However, very little is known about the class of groups
which admit Anosov representations. In this talk, we discuss characterizations of groups
which admit Anosov representations into SL(3,R), projective Anosov representations into
SL(4,R) and Borel Anosov representations into SL(4,R). In general, we provide restrictions
on the cohomological dimension of groups admitting Anosov representations into SL(d,R).
Joint work with Konstantinos Tsouvalas.
11/16/2020 - 4:30pm to 6:00pm
Recent work of Aganagic proposes the construction of a homological knot invariant categorifying the Reshetikhin-Turaev invariants of miniscule representations of type ADE Lie algebras, using the geometry and physics of coherent sheaves on a space which one can alternately describe as a resolved slice in the affine Grassmannian, a space of G-monopoles with specified singularities, or as the Coulomb branch of the corresponding 3d quiver gauge theories. We give a mathematically rigorous construction of this invariant, and in fact extend it to an invariant of annular knots, using the theory of line operators in the quiver gauge theory and their relationship to non-commutative resolutions of these varieties (generalizing Bezrukavnikov's non-commutative Springer resolution).
https://yale.zoom.us/j/99433355937 (password was emailed by Ivan on 9/11, also available from Ivan by email)