Abstract: The classical Rokhlin Lemma asserts that for an aperiodic measure-preserving transformation $T$ of a probability space, one can find a "tower" of sets on which $T$ acts by translation and which covers almost all of the space. This result is a basic tool in ergodic theory and plays a crucial role in the proofs of several fundamental theorems (e.g., Dye's and Ornstein's). In the 1970s, Ornstein and Weiss extended the Rokhlin Lemma to free actions of discrete amenable groups, thus allowing many of these results to be generalized beyond the $\mathbb{Z}$-action setting.
It is natural to ask for analogous statements in topological dynamics, where an amenable group acts by homeomorphisms on a compact metrizable space. Indeed, such results proved to be extremely useful in certain topics, such as the study of C$^*$-crossed products and the theory of mean dimension. However, it turns out that the situation is significantly more subtle than in the measurable case. On the one hand, there are several different ways to formulate a topological analogue of the Ornstein–Weiss theorem, useful for different problems. On the other hand, unlike in the measurable setting, genuine difficulties arise, stemming both from the algebraic structure of the group and from the topology of the space.
In this talk, I will begin by reviewing the classical measurable results and introducing the corresponding topological formulations. I will then briefly discuss some applications, and conclude by outlining the conditions under which these topological Rokhlin-type properties can be established.
Abstract: The invariant cycle theorem was proved by Clemens and Schmid in 70's. It compares the cohomology of the total space of a family of complex varieties with that of a general smooth fiber, asserting that the restriction map surjects onto the invariant part when working with rational coefficients. A natural question is whether the theorem still holds over integral coefficients. Positive cases are known, for example, for semistable families of K3 surfaces, which is due to Friedman in 80's. In a joint work with Arapura and Greer, we construct a counterexample over integral coefficients, given by a family of elliptic surfaces with p_g=q=1. Our construction generalizes the Shioda–Inose construction for rational double covers of K3 surfaces.
Abstract: We introduce a multi-type interacting particle system on graphs to model heterogeneous agent-based dynamics. Within this framework, we develop algorithms that jointly learn the interaction kernels, the latent type assignments, and the underlying graph structure. The approach has three stages: (i) a low-rank matrix sensing step that recovers a shared interaction embedding, (ii) a clustering step that identifies the discrete types, and (iii) a post-processing step to factorize the graph and kernels. Under the assumption of the restricted isometry property (RIP), we obtain theoretical guarantees on sample complexity and convergence for a wide range of model parameters. Building on the same low-rank matrix sensing framework, I will then discuss quantum superoperator learning, encompassing both quantum channels and Lindbladian generators. We propose an efficient randomized measurement design and use accelerated alternating least squares to estimate the low-rank superoperator. The resulting performance guarantees follow from RIP conditions, which are known to hold for Pauli measurement ensembles.
Abstract: Deep learning has achieved groundbreaking performance in various application domains. Alongside its practical success, there has been a growing effort to explore the theoretical foundations of deep learning models. This talk will focus on the statistical foundations underlying deep neural network (DNN) models. From a statistical perspective, deep learning models can be largely viewed as a nonparametric function or distribution estimation problem, where the underlying function or distribution is parameterized by a DNN. In supervised settings, deep neural networks, including feedforward DNNs, are used for regression and classification tasks. For distribution estimation, deep generative models, where the generators or scores are modeled using DNNs, are the state-of-the-art deep learning models. Statistical theory provides insights into understanding why deep neural networks often outperform classical nonparametric models, and why and how these models perform exceptionally well in practice. Key insights include their ability to adapt to various intrinsic structures of the high-dimensional data, such as a lower-dimensional manifold structure, therefore circumventing the curse of dimensionality.
Abstract: Malaria, a parasitic disease spread to humans via effective bite by an infectious adult female Anopheles mosquito, continues to exude a major burden in endemic areas (causing in excess of 600,000 deaths annually, mostly in children under the age of five). Much progress was made over the last two or three decades in the fight against malaria, largely due to the heavy and large-scale use of chemical insecticides (particularly in the form of long-lasting insecticidal nets and indoor residual spraying) to kill the malaria mosquito, promoting a renewed quest for malaria eradication. Unfortunately, such heady use has also resulted in widespread Anopheles resistance to all the main chemical insecticides used in vector control, posing challenges to the eradication objective. New anti-malaria vaccines have been approved recently and are being deployed in a number of countries in sub-Saharan Africa. In this talk, I will present a new mathematical model, in the form of a system of delayed-differential equations, for assessing the population-level impact of the approved R21/Matrix-M vaccines in curtailing the disease burden in the targeted (vaccinated) population.
Abstract: I will discuss some classes of mathematical structures that can be reconstructed from their automorphism groups, starting with sets and ending with symplectic manifolds.
Abstract: The proximal Galerkin (PG) method is a finite element method for solving variational problems with inequality constraints. It has several advantages, including constraint-preserving approximations and mesh independence. In this seminar, I will present an a priori error analysis of the PG method, providing a general framework to establish convergence and error estimates. As applications, we demonstrate optimal convergence rates for both the obstacle and Signorini problems with various finite element subspaces. We will also discuss extensions of the method and its analysis to non-symmetric VIs, a posteriori error estimates, and non-conforming discretizations. Numerical results are provided to illustrate our findings.
Abstract: A Brownian particle subject to a random, divergence-free drift will have enhanced diffusion. The correlation structure of the drift determines the strength of the diffusion and there is a critical threshold, bordering the diffusive and superdiffusive regimes. Physicists have long expected logarithmic-type superdiffusivity at the threshold and algebraic superdiffusion above it. I will discuss recent [arXiv: 2601.22142, arXiv:2404.01115] and ongoing work with Scott Armstrong and Tuomo Kuusi in which we address these problems using techniques from the theory of stochastic homogenization.
Abstract:Â We classify all possible configurations of vectors in three-dimensional space with the property that any two of the vectors form a rational angle (measured in degrees). As a corollary, we find all tetrahedra whose six dihedral angles are all rational. While these questions (and their answers) are of an elementary nature, their resolution will take us on a tour through cyclotomic number fields, computational algebraic geometry, and an amazing fact about the geometry of tetrahedra discovered by physicists in the 1960s. Joint work with Sasha Kolpakov, Bjorn Poonen, and Michael Rubinstein.
Abstract: When Deligne gave his second proof of the Weil conjectures ("Weil II") in terms of etale local systems on algebraic varieties over finite fields, he observed that the Langlands correspondence for GL(n) over a global function field (now a theorem of L. Lafforgue) would imply that under mild conditions, any etale local system on a curve always has a "geometric origin" in the sense that it appears in the relative cohomology of some family of varieties over the curve. Roughly speaking, this means that the Frobenius action on this local system counts points on some family of varieties parametrized by the curve.
Over a higher-dimensional base, such a statement remains unknown, but Deligne conjectured some concrete corollaries which have subsequently all been proven. We state some of these corollaries and how they follow from work of Deligne, Drinfeld, the speaker, et al.
Abstract: Flow Matching, a promising approach in generative modeling, has recently gained significant attention. Despite its empirical success, the mathematical understanding of its statistical power remains limited, largely due to the strong dependence of existing theoretical bounds on the Lipschitz constant of the vector field driving the underlying ODE. In this talk, we investigate the assumptions under which this dependence can be controlled. We characterize classes of target distributions that admit vector fields with controlled Lipschitz constants, and exhibit examples where such control is impossible under arbitrary noise schedules. Building on these insights, we establish convergence rates in Wasserstein-1 distance between the learned and target distributions under stable noise schedules, improving upon previous results in high-dimensional settings.
Abstract: The FFT conference returns again, February 27 - March 1, with 16 invited talks over this three-day period. For schedule and abstracts, please see:
Abstract: In the 2000s, it was discovered by Fargues and Fontaine that the Galois theory of a mixed-characteristic local field can be described by a certain "curve" (i.e., a one-dimensional regular scheme which is "proper" in a suitable sense); this description is central to the proposed geometrization of local Langlands by Fargues-Scholze. We give a historical account of how the work of Fargues-Scholze emerged naturally from prior results, such as the field of norms construction of Fontaine-Wintenberger (which nowadays is viewed in terms of the "tilting correspondence" in the terminology of Scholze) and a result of the speaker which is retroactively equivalent to the classification of vector bundles on the Fargues-Fontaine curve.
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