## Geometry Seminar

**Date:** September 7, 2018

**Time:** 4:00PM - 5:00PM

**Location:** BLOC 628

**Speaker:** Rafael Oliveria, U. Toronto

**Title:** *Scaling algorithms, applications and the null-cone problem*

**Abstract:** Scaling problems have a rich and diverse history, and thereby have found numerous applications in several fields of science and engineering. For instance, the matrix scaling problem has had applications ranging from theoretical computer science to telephone forecasting, economics, statistics, optimization, among many other fields. Recently, a generalization of matrix scaling known as operator scaling has found applications in non-commutative algebra, invariant theory, combinatorics and algebraic complexity; and a further generalization (tensor scaling) has found more applications in quantum information theory, geometric complexity theory and invariant theory. In this talk, we will describe in detail the scaling problems mentioned above, showing how alternate minimization algorithms naturally arise in this setting, and we shall present a general (3-step) framework to rigorously analyze such algorithms. We will also present a more general perspective on scaling algorithms, connecting it to the null-cone problem in invariant theory. This framework is based on concepts from invariant theory, which we will define. No prior background on Invariant Theory will be needed. Talk based on joint works with Peter Buergisser, Ankit Garg, Leonid Gurvits, Michael Walter and Avi Wigderson.