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This lecture will provide an introduction to the Weingarten calculus of polynomial Haar integrals on compact matrix groups, starting from first principles. We will focus on Weingarten calculus for the unitary group $U(N),$ and compare on contrast the standard method developed by Collins and Sniady, which built on work of Weingarten, with an emerging new approach which builds instead on work of Samuel. After a careful development of the $U(N)$ theory, we will move on to $SU(N)$ and discuss the relationship between polynomial integrals and Latin squares discovered by Kumar and Landsberg.

The design of efficient and reliable algorithms for computing the eigenvalues and eigenvectors of a matrix (i.e. solving the eigenvalue problem) is critically important in both science and engineering. Despite significant advancements in various practical aspects, fundamental theoretical questions about the eigenvalue problem remain poorly understood.
In this talk, I will present joint work with Jess Banks, Archit Kulkarni, and Nikhil Srivastava, in which we provide nearly optimal rigorous guarantees, on all inputs, for the shifted QR algorithm. Similar results were established by Wilkinson in 1968 and Dekker and Traub in 1971 for Hermitian matrices; however, despite sustained interest and several attempts, the non-Hermitian case had remained elusive since then.

We will discuss the notion of stability in Geometric Invariant Theory and applications to tensors, complexity and statistics. Some examples are the G-stable rank for tensors, the complexity of noncommutative rational identity testing, Brascamp-Lieb inequalities, and Maximum Likely Estimates for matrix and tensor normal models.

Tensors and their ranks play a central role in mathematics, physics and computer science: from constructing fast matrix multiplication algorithms, to understanding entanglement in quantum physics, to the study of combinatorial structures in discrete mathematics. Despite tremendous interest, much is still unknown.

We will give a brief introduction to tensors and their applications, building on Strassen's pioneering perspective developed in his quest to understand the complexity of matrix multiplication. We will then focus on the study of the asymptotic behaviour of tensors via asymptotic spectrum duality and via techniques from representation theory (Schur-Weyl duality and moment polytopes). We will survey recent results in this direction (in particular explicit computation of moment polytopes), and discuss open problems.

Many optimization problems arising in statistics and theoretical computer science are non-convex, but under the appropriate Riemannian geometry are *geodesically* convex --- a generalization of Euclidean convexity to Riemannian manifolds. After giving a few such examples (including scaling problems), I’ll introduce the notion of geodesic convexity and highlight its key mathematical properties. I will then discuss the query complexity of solving geodesically convex problems, surveying the current landscape of algorithms (upper bounds). The talk will conclude with known lower bounds and their connections to the computational complexity of scaling problems. This will be a whiteboard talk.