The Rayleigh-Ritz method approximates the eigenvalues of a large
Hermitian matrix $B$ with Ritz values, which are the eigenvalues of $B$'s
restriction to a smaller trial subspace $S$. We can view the $k$-th
Ritz value as a real-valued function ("Ritz energy landscape") on the
manifold of all possible s-dimensional trial subspaces, the
Grassmannian $Gr_s(C^n)$ (or $Gr_s(R^n)$ for the real symmetric case).
Motivated by questions from quantum chemistry and spectral
optimization, this talk explores the topology of the Ritz energy
landscape. A Morse function is called "perfect" if it describes the
topology of its domain in the most efficient way possible, meaning the
number of its critical points of each type exactly matches the
corresponding Betti number of the space. We demonstrate that for a
matrix $B$ with distinct eigenvalues, the Ritz landscape is indeed
perfect. While the function itself is not everywhere smooth and its
critical points are not isolated --- and not even Morse-Bott --- its
critical structure is nevertheless well-defined and ultimately
reflects the topology of the Grassmannian in a minimal, perfect way.
To be more precise, we show that the filtration of the Grassmannian by
the sublevel sets of the $k$-th Ritz value is homologically perfect.
The proof proceeds by introducing a suitable perturbation which
ensures that points of non-smoothness are not critical (by a theorem
of Zelenko and the presenter) and that the remaining smooth critical
points are isolated.
Based on a joint work with Mark Goresky (IAS).
An important special case of Schubert calculus for the Grassmannian
concerns flags osculating the rational normal curve, which is equivalent
to the Bethe Ansatz in the Gaudin model for $\mathfrak{gl}_d$. The most natural
family of these problems are indexed by partitions $\lambda$. Liao and
Rybnikov recently studied a subgroup of the monodromy group for the
Bethe Ansatz equivalent to the action of the cactus group on standard
Young tableau of shape $\lambda$. When $\lambda$ is a hook or is symmetric,
they showed that it was not 2-transitive, but was otherwise giant
(contains the alternating group).
I will describe this background and then give some geometric arguments
which refine their work on hooks and symmetric partitions, and then
[resent some computational evidence that Harris's principle holds in
that the monodromy is a large as possible. This is based on joint work with Leonid Rybnikov (UdeM).
The study of the closest point(s) on a statistical model from a given distribution in the probability simplex with respect to a fixed Wasserstein metric yields a polyhedral norm-distance optimization problem.
There are two components to the complexity of computing the Wasserstein distance between a data point and a model.
One is the combinatorial complexity, governed by the combinatorics of the Lipschitz polytope of the finite metric to be used.
Another is the algebraic complexity, which is governed by the polar degrees of the Zariski closure of the model.
In this talk, I will discuss the formulas for the polar degrees of rational normal scrolls and graphical models whose underlying graphs are star trees.
If time permits, I will discuss our efforts to compute polar degrees for graphical models with four binary random variables, including the path on four vertices and the four-cycle, as well as for small, no-three-way-interaction models.
