Organizers: Han Li, Vincent Martinez, Azita Mayeli
Friday, 2:00-3:15pm EST, (currently hosted remotely; please email vrmartinez-at-hunter-dot-cuny-dot-edu or amayeli-at-gc-dot-cuny-dot-edu for passcode)
These seminars are devoted to various topics in harmonic analysis and the analysis of partial differential equations. Topics of focus include, but are not limited to, sampling and frames theory, Riesz bases and signal processing, Fuglede Conjecture, singular integral theory, oscillatory integral operators, restriction and Kakeya-type estimates, decoupling, analysis in data science, dispersive equations, hydrodynamic equations and the mathematics of turbulence, regularity theory, stochastic PDEs, ergodic theory, long-time behavior of dynamical systems. If you would like to present a talk or added to our mailing list, please contact one of the corresponding co-organizers.
Spring 2022 Schedule
March 4 Zoom (Recording)
Mark Magsino (Ohio State University, Department of Mathematics)
Singular Values of Random Subensembles of Frame Vectors
Frame theory studies redundant representations in a Hilbert space. In finite dimensions, this is simply a spanning set but there are many interesting and useful frames in these settings. One application involves compressed sensing, which is a method for efficient acquisition and reconstruction of signals using underrepresented systems. However, verifying the key property of compressed sensing frames is NP-hard which makes constructing them difficult. One way around this is to examine random subensembles of these frames and try to control their singular values. We will show that the singular values of random subensembles of so-called equiangular tight frames are closely linked to the Kesten-McKay distribution. This is joint work with Dustin Mixon and Hans Parshall.
March 11 Zoom (Recording)
Gareth Speight (University of Cincinnati, Department of Mathematical Sciences)
Whitney Extension and Lusin Approximation in Carnot Groups
Lusin's theorem states that any measurable function can be approximated by a continuous function, except on a set of small measure. Analogous results for higher smoothness give conditions under which a function may admit a Lusin type approximation by Cm functions. Such results can often be obtained as a consequence of a suitable Whitney extension theorem. We review what is known in the Euclidean setting then describe some recent extensions to Carnot groups, a family of non-Euclidean spaces that nevertheless have a rich geometric structure. Based on joint work with Marco Capolli, Andrea Pinamonti, and Scott Zimmerman.
Hiatus
March 25 Zoom (Recording)
Weilin Li (NYU Courant Institute of Mathematics)
Function approximation with one-bit Bernstein and one-bit neural networks
The celebrated universal approximation theorems for neural networks (NNs) typically state that every sufficiently nice function can be arbitrarily well approximated by a neural network with carefully chosen real parameters. Motivated by recent questions regarding NN compression, we ask whether it is possible to represent any reasonable function with a quantized NN -- a NN whose parameters are only allowed to be selected from a small set of allowable parameters. We answer this question in the affirmative. Our main theorem shows that any continuously differentiable multivariate function can be approximated by a one-bit quadratic NN (a NN with quadratic activation whose nonzero weights and biases are only allowed to contain +1 or -1 entries) and the rate of approximation of our scheme is able to exploit any additional smoothness of the target function. A key component of our work is a novel approximation result by linear combinations of multivariate Bernstein polynomials, with only +1 and -1 coefficients. Joint work with Sinan Gunturk.
April 1 Zoom (Recording)
Itay Londner (Weizmann Institute of Science, Department of Mathematics)
Tiling the integers with translates of one tile: the Coven-Meyerowitz tiling conditions
It is well known that if a finite set of integers A tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A+B=ℤp of a finite cyclic group. Coven and Meyerowitz (1998) proved that when the tiling period M has at most two distinct prime factors, each of the sets A and B can be replaced by a highly ordered ''standard'' tiling complement. It is not known whether this behaviour persists for all tilings with no restrictions on the number of prime factors of M. In joint work with Izabella Laba (UBC), we proved that this is true for all sets tiling the integers with period M=(pqr)2. In my talk I will discuss this problem and introduce some ideas from the proof.
April 8 Zoom (Recording)
Shahaf Nitzan (Georgia Tech, Department of Mathematics)
The uncertainty principle in finite dimensions
I will give a survey of some results related to the talks title, and discuss a couple of new observations in the area. The talk is based on joint work with Jan-Fredrik Olsen and Michael Northington.
Spring Break
April 29 Zoom (Recording)
Weinan Wang (The University of Arizona, Department of Mathematics)
Local well-posedness for the Boltzmann equation with very soft potential and polynomially decaying initial data
We consider the local well-posedness of the spatially inhomogeneous non-cutoff Boltzmann equation when the initial data decays polynomially in the velocity variable. We consider the case of very soft potentials γ + 2s < 0. Our main result completes the picture for local well-posedness in this decay class by removing the restriction γ + 2s > -3/2 of previous works. It is based on the Carleman decomposition of the collision operator into a lower order term and an integro-differential operator similar to the fractional Laplacian.
Hiatus
May 13 Zoom (Recording)
Arie Israel (UT Austin, Department of Mathematics)
The norm of linear extension operators for Cm(ℝn)
We will describe a new proof of the finiteness principle for Whitney's extension problem. As a byproduct, we obtain the existence of linear extension operators with an improved bound on the norm of the operator. We discuss connections to the algorithmic problem of interpolation of data. This is joint work with Jacob Carruth and Abraham Frei-Pearson.
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