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April 21, 2021 (Wednesday), 11:00 am

Number Theory Seminar

Arithmetic properties of the Fourier coefficients of weakly holomorphic modular functions of arbitrary level

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.

April 21, 2021 (Wednesday), 4:10 pm

Computational Analysis Seminar

Generalization Bounds for Sparse Random Feature Expansions

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.

April 21, 2021 (Wednesday), 4:10 pm

Topology & Group Theory Seminar

3-manifolds, surfaces, and the veering polynomial

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.

April 23, 2021 (Friday), 4:10 pm

Subfactor Seminar

Talk Title TBA

Cain Edie-Michell, Vanderbilt University
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April 28, 2021 (Wednesday), 4:10 pm

Computational Analysis Seminar

Talk Title TBA

Mahya Ghandehari, University of Delaware
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April 28, 2021 (Wednesday), 4:10 pm

Topology & Group Theory Seminar

Talk Title TBA

Yash Lodha, Korea institute for advanced study
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April 30, 2021 (Friday), 4:10 pm

Subfactor Seminar

Talk Title TBA

David Penneys, Ohio State University
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May 5, 2021 (Wednesday), 11:00 am

Number Theory Seminar

Talk Title TBA

Caroline Turnage-Butterbaugh, Carleton College
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