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Thunderbird
This is the departmental seminar on algebraic topology and related topics.
We usually have a semester theme, for which speakers from the department are invited to contribute talks, and external guest speakers who will give talks on current research in topology. The semester themes for Fall 2022 are "sheaves and logic" and "applied topology". The seminar website is here.
Hours and Venue: Tuesdays, 3:30pm - 4:30pm in MATH 350
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 (or 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).
Footnote (*): the term "restriction" is appropriate in the equivalent formulation of eigenvalues of quadratic forms. For an operator B, we apply it to vectors in S and then project the result orthogonally back onto S; this is called a "compression" of B.
Ritz energy landscape is perfect (almost) Morse