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Abstract: This talk provides a discussion of the concept of redundacy for Hilbert frames. Frames are (over)complete sequences of vectors that satisfy Parseval-like inequalities.

The simplest example is a spanning set of m vectors in R^n. Its “redundacy” is the ratio m/n, while its “excess” is m-n. The talk explore ways to extend this notion of redundancy to settings where there is no underlying operator algebra structure. For Weyl-Heisenberg frames, the orthogonal projection onto the space of coefficients belongs to a finite W*-algebra. In this setting, the redundancy measure reduces to more tractable quantities, such as the Beurling-Landau density. Along the way, we construct a C^*-algebra that contains all Weyl–Heisenberg frame and Gram operators and admits a natural l2-localization.



Measure functions for frames
https://www.sciencedirect.com/science/article/pii/S0022123607002947?via%3Dihub

Density, Overcompleteness, and Localization of Frames. I. Theory
https://link.springer.com/article/10.1007/s00041-006-6022-0

Density, Overcompleteness, and Localization of Frames. II. Gabor Systems
https://link.springer.com/article/10.1007/s00041-005-5035-4

Redundancy for localized frames
https://link.springer.com/article/10.1007/s11856-011-0118-1

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Abstract: Quaternionic modular forms on G_2 carry a surprisingly rich arithmetic structure. For example, they have a theory of Fourier expansions where the Fourier coefficients are indexed by totally real cubic rings. For quaternionic modular forms on G_2 associated via functoriality with certain modular forms on PGL_2, Gross conjectured in 2000 that their Fourier coefficients encode L-values of cubic twists of the modular form (echoing Waldspurger's work on Fourier coefficients of half-integral weight modular forms). This talk will report on recent work proving Gross's conjecture when the modular forms are dihedral, giving the first examples for which it is known. Everything in the talk is joint work with Petar Bakic, Alex Horawa, and Siyan Daniel Li-Huerta.

View Abstract

Abstract: This talk provides a discussion of the concept of redundacy for Hilbert frames. Frames are (over)complete sequences of vectors that satisfy Parseval-like inequalities.
The simplest example is a spanning set of m vectors in R^n. Its “redundacy” is the ratio m/n, while its “excess” is m-n. The talk explore ways to extend this notion of redundancy to settings where there is no underlying operator algebra structure. For Weyl-Heisenberg frames, the orthogonal projection onto the space of coefficients belongs to a finite W*-algebra. In this setting, the redundancy measure reduces to more tractable quantities, such as the Beurling-Landau density. Along the way, we construct a C^*-algebra that contains all Weyl–Heisenberg frame and Gram operators and admits a natural l2-localization.

Measure functions for frames https://www.sciencedirect.com/science/article/pii/S0022123607002947?via%3Dihub
Density, Overcompleteness, and Localization of Frames. I. Theory https://link.springer.com/article/10.1007/s00041-006-6022-0
Density, Overcompleteness, and Localization of Frames. II. Gabor Systems https://link.springer.com/article/10.1007/s00041-005-5035-4
Redundancy for localized frames https://link.springer.com/article/10.1007/s11856-011-0118-1

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Abstract: I will explain why the étale fundamental group of a smooth projective variety is finitely presented, following the recent work in arXiv:2102.13424.

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Abstract: Advanced manufacturing enables rapid fabrication of complex materials and structures, allowing exploration of large design spaces in materials composition, microstructure, and processing conditions. However, the high dimensionality of these spaces, together with manufacturing uncertainties and defects, poses major challenges for traditional computational science and engineering methods. Addressing these challenges requires new approaches that integrate multi-modal data with physics-based modeling and artificial intelligence to accelerate scientific discovery and materials innovation. This seminar introduces Scientific Artificial Intelligence (Sci-AI) approaches for discovering materials constitutive laws, modeling complex materials behavior, and designing new materials with targeted properties. The central theme is the tight coupling of physical principles, data-driven learning, and interpretability, enabling AI models to move beyond black-box prediction toward reliable scientific inference. The talk is organized around three interconnected themes. First, a hierarchical symbolic AI framework is presented for the automated discovery of physically interpretable constitutive laws directly from data, allowing the model to identify governing mechanisms, enforce dimensional consistency and physical constraints, and balance accuracy with model complexity. Second, a physics-informed, image-based encoder–decoder architecture is introduced to accelerate materials behavior modeling by learning compact latent representations of complex microstructural and field data, while seamlessly fusing high-fidelity simulations, in-situ imaging, and external sensing information across multiple length and time scales. Third, a generative AI framework is developed for inverse materials design, enabling the synthesis of physically plausible microstructures conditioned on nonlinear material properties, processing constraints, and performance targets, thereby closing the loop from data and modeling to design and manufacturing. Overall, this seminar highlights recent advances in Sci-AI for computational science and engineering and demonstrates how physics-guided machine learning can bridge data and models to enable advanced materials modeling and design.

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Abstract: I will discuss results and ideas at the interface of stochastic processes, statistical modeling, and the movement ecology of wild animals. Recently published results demonstrate broad differences in the tendency for carnivores to utilize heavily traveled routeways within their home ranges and ways in which range-residency can be incorporated in dynamic models of population growth. New ideas focus on opportunities for investigating learning from animal movement data and the potential influences of vision on correlated movement.

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Abstract: The titular question occurred to me while learning Rosendal’s groundbreaking framework for understanding coarse geometries of topological groups, especially as they apply to questions surrounding what geometric group theorists know as the Milnor–Schwarz Lemma. The key case to understand is for connected groups, where I remain mostly mystified. In fact, the goal of the talk is mostly to situation the question, explain why I think it is exciting, and some neat, if confounding, tools and examples arising from Brazas’s work on free (Graev) topological groups.

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Abstract: This workshop will bring together several leading world experts in the field of operator algebras, which has seen sustained interest over the last century. The primary focus will be on recent important developments surrounding the theory of C*-algebras and finitary approximations. There will also be a spotlight on key standing problems in the area, particularly on structure and classification of operator algebras associated to groups and dynamical systems, and qualitative and quantitative approximation properties.

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Abstract: TBA

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Abstract: A translation surface is a closed surface that is obtained by gluing edges of a polygon in parallel. The group $GL_2(R)$ acts on the collection translation surfaces of a fixed genus $g$. For a fixed translation surface $S$ and $t>0$, we obtain a probability measure on the collection of translation surfaces by rotating $S$ with a uniform angle and then multiplying by diag$(e^t, e^-t)$. Alternatively, we can talk on expanding a piece of horospherical orbit. We prove equidistribution of this sequence of measures as $t \to \infty$. This resolves a conjecture of Forni, and extends a result of Eskin and Mirzakhani that (in particular) showed our result with a Cesàro average. We will also discuss an application of this result to billiards with rational angles.

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Abstract: Nonparametric tests for functional data are a challenging class of tests to work with because of the potentially high dimensional nature of the data. One of the main challenges for considering rank-based tests, like the Mann-Whitney or Wilcoxon Rank Sum tests (MWW), is that the unit of observation is typically a curve. Thus any rank-based test must consider ways of ranking curves. While several procedures, including depth-based methods, have recently been used to create scores for rank-based tests, these scores are not constructed under the null and often introduce additional, uncontrolled for variability. We therefore reconsider the problem of rank-based tests for functional data and develop an alternative approach that incorporates the null hypothesis throughout. Our approach first ranks realizations from the curves at each measurement occurrence, then calculates a summary statistic for the ranks of each subject, and finally re-ranks the summary statistic in a procedure we refer to as a doubly ranked test. We propose two summaries for the middle step: a sufficient statistic and the average rank. As we demonstrate, doubly rank tests are more powerful while maintaining ideal type I error in the two sample, MWW setting. We also extend our framework to more than two samples, developing a Kruskal-Wallis test for functional data which exhibits good test characteristics as well. Finally, we illustrate the use of doubly ranked tests in functional data contexts from material science, climatology, and public health policy.

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Abstract: It was understood long ago that the first nonzero eigenvalue of the
Laplacian on a closed hyperbolic surface cannot exceed that of the
hyperbolic plane, asymptotically as the diameter goes to infinity. But
until recently, it was not known whether there exist hyperbolic surfaces
that attain this bound. In joint work with Magee and Puder, we show
something much stronger: random covering spaces of an arbitrary
hyperbolic surface have an optimal spectral gap with probability 1-o(1).
I will aim to explain how this question reduces to an unusual problem of
random matrix theory, whose resolution was made possible by recent
advances in the theory of strong convergence.