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Motivated by questions from quantum chemistry and spectral optimization, this talk explores the topology of the Ritz energy landscape. Ritz values are the eigenvalues of a larger matrix (or operator) B compressed 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(n,s). We show that the k-th Ritz value (for any k) is a perfect Morse function, once the definition of "perfection" is suitably adjusted. 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. While the k-th eigenvalue is Lipschitz rather than smooth and while its critical points are not isolated --- and not even Morse-Bott --- its critical point count is nevertheless well-defined and reflects the topology of the Grassmannian in a minimal, perfect way. 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 critical points that remain are isolated. We also show that the notion of perfection we introduce ("homological perfection") is closed under such perturbations. Based on joint work with Mark Goresky (IAS).

Over the past several decades, the importance of mathematics in life science research has been growing, prompting high-profile calls to reform the mathematics education of biology researchers. These reports advocate for contexutalizedmathematics courses that focus concepts actually used in life science research. In this talk, I will present how UCLA has developed a course centered on modeling and dynamical systems to directly address this need. By taking a modeling-first approach, the course uses important examples from biology as the primary motivation to study mathematics. Students first learn to write models of biological systems such as epidemiological models, predator-prey models, and cellular protein production. Taking a geometrical approach, students learn to determine the long-term behaviors of systems using phase plane trajectories and stability of equilibrium points. This naturally leads to the idea of qualitative change, or bifurcation, enabling students to study important medical phenomena such as the abnormal breathing pattern called Cheyne-Stokes respiration. The second part of the course takes a more analytical approach culminating with the Hartman-Grobman theorem (or principle of linearization). I will conclude by discussing the adoption of this course by other universities. This work is supported by the NSF grant DUE-2225258.

Many data-generating processes arising in physics and engineering possess intrinsic geometric structures, such as symmetries, conservation laws, variational principles, and differential equation constraints. Incorporating these structures into machine learning models is crucial for achieving physically meaningful and reliable predictions. In this talk, I will introduce a structure-preserving kernel-based learning framework for recovering unknown functions while respecting the underlying geometric properties of the problem. The proposed method admits a closed-form solution, achieves strong numerical performance, and often outperforms existing approaches. In the manifold setting, the learned solution is globally defined and independent of the choice of local coordinates. I will also present theoretical guarantees, including convergence results under both fixed and adaptive regularization schemes, and discuss applications to structured learning problems arising in scientific computing.

We present FLOP (Fast Learning of Order and Parents), a score-based causal discovery algorithm for linear models. It pairs fast parent selection with iterative Cholesky-based score updates, cutting run-times over prior algorithms. This makes it feasible to fully embrace discrete search, enabling iterated local search with principled order initialization to find graphs with scores at or close to the global optimum. The resulting structures are highly accurate across benchmarks, with near-perfect recovery in standard settings. This performance calls for revisiting discrete search over graphs as a reasonable approach to causal discovery.