Monday, December 2, 2019
Time  Items 

All day 

4:00pm 
12/02/2019  4:00pm The art of using quantum field theory to derive mathematical results often lies in a mysterious transition between infinite dimensional geometry and finite dimensional geometry. In this talk we describe a general mathematical framework to study the quantum geometry of \sigmamodels when they are effectively localized to small fluctuations around constant maps. We illustrate how to turn the physics idea of exact semiclassical approximation into a geometric setup in this framework, using GaussManin connection. This leads to a theory of “counting constant maps” in a nontrivial way. We explain this program by a concrete example of topological quantum mechanics and show how “counting constant loops” leads to a simple proof of the algebraic index theorem. Location:
LOM 214
12/02/2019  4:00pm Abstract: Symmetric tensors are multidimensional arrays invariant to permutation of indices. Symmetric tensors arise in machine learning and statistics when applying the method of moments, as higherdimensional 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 lowrank symmetric tensor decompositions, based on a modified tensor power method. Numerical experiments demonstrate that our algorithm significantly outperforms stateoftheart 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. Location:
DL 220
