A fundamental problem in topology is to determine when two spaces are distinct. For surfaces, classical invariants suffice, but for three-dimensional manifolds the question is considerably harder. The Witten-Reshetikhin-Turaev invariants, originating in quantum field theory, provide one family of tools. For torus bundles, i.e. 3-manifolds built by cutting a torus, applying a mapping class in SL(2,Z), and regluing, the skein algebra embeds into a non-commutative torus at roots of unity, yielding a finite-dimensional representation that makes these invariants concretely computable. Exploiting this structure, we develop a classical dynamic programming algorithm and a quantum algorithm that encodes the computation into logarithmically many qubits, achieving an exponential reduction in space. We further identify a natural coefficient-counting problem that is #P-complete yet admits efficient quantum approximation.
The first half of the talk will be expository, requiring only linear algebra and modular arithmetic. The second half will outline the algorithmic results.
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
Tue, Feb. 17 5pm (Math 350)
Math Club
Edouard Heitzmann
X
The legal standard set in Rucho v Common Cause puts the onus on the plaintiffs in a Voting Rights Act (VRA) case to demonstrate that a map is a racial rather than a partisan gerrymander in order for it to be struck down as unlawful. This means plaintiffs have to 'do the homework' of the defendants for them: they must produce a map that is at least as politically gerrymandered as the defendants drew, while achieving better racial representation outcomes. This standard was successfully met by a group of mathematicians in the El Paso redistricting case, winning in federal court before the Supreme Court shadow docket put a hold on the decision, allowing the racial gerrymander to go through. The mathematicians who accomplished this used Monte Carlo Markov Chain (MCMC) methods to draw a so-called 'ensemble' of millions of alternative congressional maps for Texas, and used statistical methods to show that the enacted map was an outlier for racial representation among the maps that had similar partisan outcomes. In this talk I will go over these mathematical details, as well as describe the computational tools currently used in this field. Note: If you have a working Jupyter Notebook installation, you are encouraged to bring it to the talk, as you will have the opportunity to follow along the computational demos included in the talk.
The Mathematics of Redistricting—what they did in Texas
How to tell shapes apart using quantum computers