Mehdi Lejmi CUNY, Bronx Community College
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Conformally Kahler Hermitian metrics of constant Riemannian scalar curvature and J-invariant Ricci are
called conformally Kahler Einstein-Maxwell metrics. In this talk, we discuss deformations and possible construction
of such metrics on blow-ups. This is a joint work in progress with Abdellah Lahdili.

Georgios Moschidis, University of California Berkeley
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The AdS instability conjecture provides an example of weak turbulence appearing in the dynamics of the Einstein equations in the presence of a negative cosmological constant. The conjecture claims the existence of arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time.
In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein-scalar field system. The construction of the unstable initial data will require carefully designing a family of initial configurations of localized matter beams and estimating the exchange of energy taking place between interacting beams over long periods of time, as well as estimating the decoherence rate of those beams. I will also discuss possible paths for extending these ideas to the vacuum case.

Alex Margolis, Vanderbilt University
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A well-known problem in group theory, dating back to work of Furstenberg and Mostow, is to determine which locally compact groups contain a fixed finitely generated group as a uniform lattice. We will consider a geometric generalisation of this problem: classifying cobounded quasi-actions on a fixed finitely generated group.
We introduce the concept of the topological completion of a quasi-action. This is a locally compact group, well-defined up to a compact normal subgroup, reflecting the geometry of the quasi-action. We exhibit a dichotomy in the class of hyperbolic groups, where either topological completions are always connected (when the group is a lattice in a Lie group) or are totally disconnected. We give applications to understanding the large scale geometry of finitely generated groups.

Soon-Yi Kang, Kangwon National University
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The canonical basis of the space of modular functions on the modular group of genus zero form a Hecke system. From this fact, many important properties of modular functions were derived.
In this talk, we show that the Niebur-Poincare basis of the space of Harmonic Maass functions also forms a Hecke system. As its applications, we show several arithmetic properties of modular functions on the higher genus modular curves such as divisibility of Fourier coefficients of modular functions of arbitrary level and arithmetic of divisor polar harmonic Maass forms.
This is a joint work with Daeyeol Jeon and Chang Heon Kim.

Giang Tran, University of Waterloo
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Random feature methods have been successful in various machine learning tasks, are easy to compute, and come with theoretical accuracy bounds. They serve as an alternative approach to standard neural networks since they can represent similar function spaces without a costly training phase. However, for accuracy, random feature methods require more measurements than trainable parameters, limiting their use for data-scarce applications or problems in scientific machine learning. In this work, we propose the sparse random feature expansion method, which enhances the compressive sensing approach to allow for more flexible functional relationships between inputs and a more complex feature space. We provide generalization bounds on the approximation error for functions in a reproducing kernel Hilbert space depending on the number of samples and the distribution of features. The error bounds improve with additional structural conditions, such as coordinate sparsity, compact clusters of the spectrum, or rapid spectral decay. We show that the sparse random feature expansion method outperforms shallow networks for well-structured functions and applications to scientific machine learning tasks.

Michael Landry, Washington University in St. Louis
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I will describe work concerning the Thurston norm on the second homology of a 3-manifold and its interaction with foliations and flows. This norm is a 3-manifold invariant with connections to many areas: geometric group theory, foliation theory, Floer theory, and more. There are some beautiful clues due to Thurston, Fried, Mosher, Gabai, McMullen, and others that indicate there should be a dictionary between the combinatorics of the norm’s polyhedral unit ball and the geometric/topological structures existing in the underlying manifold. The picture is incomplete, and mostly limited to the case when the manifold fibers as a surface bundle over the circle. I will explain some new results which hold not just in the fibered case but also the more general setting of manifolds admitting veering triangulations (introduced by Agol). The main focus will be on joint work with Yair Minsky and Samuel Taylor, in which we use a veering triangulation to define a polynomial invariant which generalizes McMullen’s Teichmuller polynomial.

Cain Edie-Michell, Vanderbilt University
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Mahya Ghandehari, University of Delaware
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Yash Lodha, Korea institute for advanced study
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David Penneys, Ohio State University
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Caroline Turnage-Butterbaugh, Carleton College
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