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Organizers: Han Li, Vincent Martinez, Azita Mayeli, Max Yarmolinsky
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**Friday, 1:00-2:15pm EST, (currently hosted remotely; please email vrmartinez-at-hunter-dot-cuny-dot-edu or amayeli-at-gc-dot-cuny-dot-edu for passcode)**

**Fall 2021 Schedule**

**October 1 Zoom**
(Recording)

Anuj Kumar (IU Bloomington, Department of Mathematics)

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On well-posedness and smoothing of solutions to the generalized SQG equations in critical Sobolev spaces, Part I
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This talk is based on recent works in which we study the dissipative generalized surface quasi-geostrophic equations in a supercritical regime where the order of the dissipation is small relative to order of the velocity, and the velocities are less regular than the advected scalar by up to one order of derivative. The existence and uniqueness theory of these equations in the borderline Sobolev spaces is addressed, as well as the instantaneous Gevrey class smoothing of their corresponding solutions. These results appear to be the first of its kind for a quasilinear parabolic equation whose coefficients are of higher order than its linear term. The main tool is the use of an approximation scheme suitably adapted to preserve the underlying commutator structure. We also study a family of inviscid generalized SQG equations where the velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these equations in borderline Sobolev spaces is addressed.

**October 8 Zoom**
(Recording)

Vincent Martinez (CUNY Hunter, Department of Mathematics and Statistics)

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On well-posedness and smoothing of solutions to the generalized SQG equations in critical Sobolev spaces, Part II
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We will continue the discussion about the issue of well-posedness at critical regularity for a family for active scalar equations with increasingly singular constitutive law.

**October 15 Zoom**
(Recording)

Bing-Ying Lu (Universitat Bremen, Applied Analysis Group)

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Universality near the gradient catastrophe point in the semiclassical sine-Gordon equation
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We study the semiclassical limit of the sine-Gordon (sG) equation with below threshold pure impulse initial data of Klaus-Shaw type. The Whitham averaged approximation of this system exhibits a gradient catastrophe in finite time. In accordance with a conjecture of Dubrovin, Grava and Klein, we found that in an O(ε^{4/5}) neighborhood near the gradient catastrophe point, the asymptotics of the sG solution are universally described by the Painlevé I tritronquée solution. A linear map can be explicitly made from the tritronquée solution to this neighborhood. Under this map: away from the tritronquée poles, the first correction of sG is universally given by the real part of the Hamiltonian of the tritronquée solution; localized defects appear at locations mapped from the poles of tritronquée solution; the defects are proved universally to be a two parameter family of special localized solutions on a periodic background for the sG equation. We are able to characterize the solution in detail. Our approach is the rigorous steepest descent method for matrix Riemann--Hilbert problems, substantially generalizing Bertola and Tovbis's results on the nonlinear Schrödinger equation to establish universality beyond the context of solutions of a single equation. This is joint work with Peter D. Miller.

**October 22 Zoom**
(Recording)

Suen Chun Kit Anthony (The Education University of Hong Kong, Department of Mathematics and Information Technology)

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Vanishing parameter limit for a class of active scalar equations
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In this talk, we study an abtract class of active scalar equations which depend on some viscosity parameters κ and ν. We examine the wellposedness of the equations in different scenarios and address the convergence of solutions as κ or ν vanishes. We further discuss some physical applications of the general results obtained from such abtract class of active scalar equations.

**October 29 Zoom**
(Recording)

Hau-Tieng Wu (Duke University, Department of Mathematics)

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Some recent progress in diffusion based manifold learning
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Diffusion based manifold learning has been actively developed and applied in past decades. However, there are still many interesting practical and theoretical challenges. I will share some recent progress in this direction, particularly from the angle of robustness and scalability and the associated theoretical support under the manifold setup. If time permits, its clinical application will be discussed.

**November 5 Zoom**
(Recording)

Deniz Bilman (University of Cincinnati, Department of Mathematical Sciences)

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High-Order Rogue Waves and Solitons, and Solutions Interpolating Between Them
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It is known from our recent work that both fundamental rogue wave solutions (with Peter Miller and Liming Ling) and multi-pole soliton solutions (with R. Buckingham and D.S. Wang) of the nonlinear Schr&odinger (NLS) equation exhibit the same asymptotic behavior in the limit of large order in a shrinking region near the peak amplitude point, despite the quite different boundary conditions these solutions satisfy at infinity. We show how rogue waves and solitons of arbitrary orders can be placed within a common analytical framework in which the ''order'' becomes a continuous parameter, allowing one to tune continuously between types of solutions satisfying different boundary conditions. In this scheme, solitons and rogue waves of increasing integer orders alternate as the continuous order parameter increases. We show that in a bounded region of the space-time of size proportional to the order, these solutions all appear to be the same when the order is large. However, in the unbounded complementary region one sees qualitatively different asymptotic behavior along different sequences. In this talk we focus on the behavior in this exterior region. The asymptotic behavior is most interesting for solutions that are neither rogue waves nor solitons. This is joint work with Peter Miller.

**November 12 Zoom**
(Recording)

(Break)

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**November 19 Zoom**
(Recording)

Goetz Pfander (Catholic University of Eichstatt-Ingolstadt, Department of Mathematics)

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Exponential bases for partitions of intervals
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Fourier series form a cornerstone of analysis; it allows the expansion of a complex valued 1-periodic function in the orthogonal basis of integer frequency exponentials (for the space of square integrable functions on the unit interval). A simple rescaling argument shows that by splitting the integers into evens and odds, we obtain orthogonal bases for functions defined on the first, respectively the second half of the unit interval. We shall generalize this curiosity and show that, given any finite partition of the unit interval into subintervals, we can split the integers into subsets, each of which forms a basis (not necessarily orthogonal) for functions on the respective subinterval. In addition, novel fundamental results in the theory of Fourier series will be discussed.

**December 3 Zoom**
(Recording)

Tao Zhang (Guangzhou University, School of Mathematics and Information Science)

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Fuglede's conjecture holds in ℤ_{p}×ℤ_{pn}
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Fuglede's conjecture states that for a subset Ω of a locally compact abelian group *G* with positive and finite Haar measure, there exists a subset of the dual group of *G* which is an orthogonal basis of *L*^{2}(Ω) if and only if it tiles the group by translation. In this talk, we consider the Fuglede's conjecture in the group *ℤ _{p}×ℤ_{pn}*. I will talk about the main idea of our proof.

**December 10 Zoom**
(Recording)

Tuan Pham (Brigham Young University, Department of Mathematics)

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Conservation of frequencies and applications to the well-posedness problem of the Navier-Stokes equations
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A well-known conserved quantity of the Navier-Stokes equations is the total energy. This conservation law has been used extensively in the local regularity theory, especially since the groundbreaking work of Caffarelli-Kohn-Nirenberg in 1982. In the Fourier space, the Navier-Stokes equations are naturally associated with a stochastic cascade as noted by Le Jan and Sznitman in 1997. In the dynamic of this stochastic cascade, the initial frequency is conserved. In this talk, I will explain how the conservation of frequencies can be used to study the well-posedness problem of the Navier-Stokes equations. A notable application is that any initial data in *L*^{2} whose Fourier transform is supported in the half-space produces a unique global mild solution.

**December 17 Zoom**
(Recording)

Peter Balazs (Director of the Acoustics Research Institute, Austrian Academy of Sciences)

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Continuous Frames and Reproducing Kernels
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Frame theory has become a tremendously active research field, with connection to many mathematical disciplines but also applications. In short, frames are a sequence of elements that allow stable representation of elements in a Hilbert space. One generalization of the original definition considers not a sequence indexed by a discrete set, but a function indexed by a continuous set. This will be the topic of this talk, in particular, how closely this concept is intertwined with reproducing kernel Hilbert spaces (RKHS). We start with a short motivation why frame theory is important, also for applications. We introduce the basic definitions. We show recent developments which focus on the various facets of the interplay of continuous frames and RKHS. In particular, we analyze the structure of the reproducing kernel of a RKHS using frames and reproducing pairs. We show that finite redundancy of a continuous frame implies atomic structure of the underlying measure space. This implies that all the attempts to extend the notion of Riesz basis to general measure spaces are fruitless since every such family can be identified with a discrete Riesz basis, by using the RKHS structure of the range of the analysis operator. This can also be used to formulate a result that classifies all dual functions to a given continuous frame. Finally, we will give general kernel theorems for operators acting between coorbit spaces, which are Banach spaces associated to continous frame representations and contain most of the usual function spaces (Besov spaces, modulation spaces, etc.). This collects work with K. Gröchenig, M. Speckbacher and N. Teofanov.

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