21.07.2026 16:15 Gregory Berkolaiko : Topology of the Ritz energy landscape
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).
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22.07.2026 13:00 Steve Bennoun (UCLA): Contextualizing First-Year Mathematics for Life Science Students Through Modeling and Dynamical Systems
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.
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22.07.2026 14:15 Jianyu HU (NTU Singapore): Learning from Structured Data with Structure-Preserving Kernels
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.
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23.07.2026 14:00 Leonard Henckel (University College Dublin, IRL): Embracing Discrete Search: A Reasonable Approach to Causal Structure Learning
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.
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23.07.2026 16:30 Kotaro Komatsu (University of Tsukuba): Introducing students to explorative aspects of proving in mathematical activity
Proving is a fundamental activity in mathematics, and its teaching has been widely discussed in mathematics education research. A central theme in this body of research concerns the transition from making or evaluating a conjecture to proving it, with proof construction often viewed as the ultimate goal of mathematical activity. However, proving also involves ongoing processes that extend beyond proof construction, including the revision and generalisation of proved statements. In this talk, I discuss these relatively understudied, explorative aspects of proving through two illustrative cases: one relates to Lakatos-style mathematical activity involving proofs and refutations in a secondary school context, and the other focuses on proof by mathematical induction at the undergraduate level. I also discuss implications for task design aimed at introducing explorative proving to students.
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Invited by Prof. Stefan Ufer
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27.07.2026 15:00 Esmée Theewis: Large deviation principles for stochastic evolution equations
In this talk, I will discuss new results on large deviations for stochastic evolution equations. Starting with the variational setting, I will present a large deviation principle (LDP) for a class of SPDEs that includes many new examples with gradient noise and unbounded spatial domains. We will then move to non-variational settings and explore LDPs for the stochastic 3D primitive equations and reaction-diffusion equations. While our method is based on the well-known weak convergence approach, main novel ingredients come from the theory of critical spaces and maximal regularity techniques. Based on joint work with Antonio Agresti and Mark Veraar.
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27.07.2026 16:15 Alejandro Morera: Mean-Field Limits for Stochastic Hilbert-Space-Valued Particle Systems on Digraph Measures: Convergence, Regularity, and Applications to Machine Learning Dynamics on Large Networks
This talk develops mean-field limits for heterogeneous stochastic interacting particle systems with states in separable Hilbert spaces and interaction structures described by digraph measures. The finite system is formulated in mild form, and its well-posedness is established using infinite-dimensional stochastic-analysis and semigroup methods.
The main result extends the digraph-measure mean-field framework to Hilbert-space-valued dynamics. The limit is a label-dependent family of path-space laws rather than a single exchangeable McKean--Vlasov law. We show existence and uniqueness of this limiting process, weak convergence in probability of empirical path measures to its averaged law, and Lipschitz regularity of in the label variable with respect to the Wasserstein distance. Under analytic-semigroup assumptions, the convergence further improves to stronger fractional-domain path topologies.
Finally, the theory is applied to two large heterogeneous learning systems, namely, an RKHS-valued diffusion-KLMS model and a delayed recurrent-neural-network model formulated on an infinite-dimensional Sobolev phase space. These examples provide continuum descriptions of distributed kernel learning and recurrent neural dynamics on large networks.
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03.08.2026 11:00 Junhyung Park (ETH Zürich, CH): Causal spaces: A mathematical axiomatisation of causality
Causal reasoning is usually formalized through structural causal models (SCMs) or potential outcomes. These frameworks have been enormously successful for modeling, identification, and inference, but they are not primarily designed as axiomatic foundations analogous to probability spaces in probability theory. This tutorial introduces causal spaces, a measure-theoretic framework in which interventions are represented by primitive causal kernels satisfying two minimal axioms: doing nothing leaves everything unchanged, and intervened coordinates take their prescribed values. The tutorial will explain the motivation for causal spaces, present the basic definition and semantics, and work through examples linking back to familiar causal models. It will then present some further development of basic causal space theory, such as causal effects, sources, identifiability, and counterfactual spaces, and touch upon advanced topics such as targeted interventions and continuous-time stochastic processes that are more difficult to express in existing frameworks.
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