Date Speaker Topic
January 16 Organizational Meeting Organizational Meeting

 

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January 23

TBA

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January 30

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February 6

TBA

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February 13

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February 20

Martin Schmoll, Clemson University

 Probabilistic Frames and Wasserstein Distances

Abstract: We take a look at probabilistic frames from the perspective of Wasserstein distances. The frame condition can be captured using Wasserstein distances, hence Wasserstein distances appear naturally in the theory of probabilistic frames. Adapting classic results of Olkin and Pukelsheim and an estimate of Gelbrich we analyze probabilistic 2-frames via their frame operator. We deduce Wasserstein optimality of particular linear maps between sets of probabilistic frames with fixed frame operator and apply this insight to extend and simplify some known results. Since Parseval probabilistic frames are path connected, the previous implies that the set of probabilistic frames with any given frame operator is connected. We finally talk about the set of transport duals and some of its generalizations. We show, there are transport duals that cannot be obtained by push-forward.

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February 27

Cody Stockdale, Clemson University

On the theory of compact Calderón-Zygmund operators

Abstract: While the boundedness properties of Calderón-Zygmund singular integral operators are classical in harmonic analysis, a theory for compact CZ operators has more recently been established. We present new developments in the theory of compact CZ operators. In particular, we give a new formulation of the T1 theorem for compactness of CZ operators, which, compared to existing compactness criteria, more closely resembles David and Journé’s original T1 theorem for boundedness and follows from a simpler argument. Additionally, we discuss the extension of compact CZ theory to weighted Lebesgue spaces via sparse domination methods. This talk is based on joint works with Mishko Mitkovski, Paco Villarroya, and Brett Wick.

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March 6 (3:00 – 4:00 pm)

Elodie Pozzi, Saint Louis University

The Role of Hardy Spaces to Solve some Divergence Equations in Two-Dimensions

Abstract: It is well-known that the real part u of an analytic function f on a simply connected domain satisfies \Delta(u)=0, where \Delta denotes the Laplacian operator. In the unit disk, f has an L^p-extension on the unit circle if it meets a Hardy condition, placing f in the Hardy space H^p(D). Solving \Delta(u)=0 in D with boundary value h in L^p(T) is equivalent to finding f in H^p(D) such that Re(f)=h on T. In this talk, we will consider the divergence equation \tex{div}(\sigma \ast \text{grad}(u))=0 on a simply connected domain \Omega for \sigma in a Sobolev space and will discuss the link with a generalized version of the Hardy space. The generalized Hardy spaces have been introduced in a work by L. Bers, L. Niremberg, and of I. Vekua. We will give results on the solvability of this equation depending on the type of Sobolev space and the simply connected domain Omega. This is based on a joint work with Laurent Baratchart (INRIA Université Côte d’Azur) and Emmanuel Russ (Aix-Marseille Université).

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March 6 (5:00 – 6:00 pm)

Brett Wick, Washington University Saint Louis

The Corona Problem

Abstract: The Corona problem has served as a major motivation for many results in complex function theory, operator theory, and harmonic analysis. In its simplest form, the question asks whether, given N bounded analytic functions f_1,\ldots,f_N on the unit disc such that \inf (\left\vert f_1\right\vert+\cdots+\left\vert f_N\right\vert)\geq\delta>0, it is possible to find N bounded analytic functions g_1,\ldots,g_N such that f_1g_1+\cdots+f_Ng_N =1. While the Corona problem is well-understood in the context of one complex variable, it remains highly challenging in the case of several complex variables. In this talk, we will discuss some generalizations of the Corona problem to matrix-valued functions and their connections to geometry, to function spaces on the unit ball in several complex variables, and to setting a related question in the quaternions. The resolution of these questions relies on tools and methods from complex analysis, harmonic analysis, operator theory, and partial differential equations.

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March 13 Danny Riley, Tufts University Optimal Configurations for the Discrete and Continuous p-Frame Potentials

Abstract: The discrete p-frame potentials are a generalization of the “frame potential” that Benedetto and Fickus developed to completely characterize the finite unit-norm tight frames in any dimension. For a fixed finite number of points N constrained to the unit sphere in d-dimensional Euclidean space, the p-frame potential is the sum of the pairwise inner products each raised to the power p. The problem of determining the optimal configurations — that is, the configurations which minimize the potential — is unsolved for the vast majority of combinations of p, N, and d, including cases as “easy” as N=5, p=5, on the circle in d=2! In this talk, I will cover the methodologies used to determine the known optimal configurations, from strategies as classic as Lagrange multipliers to linear programming bounds and expansions in terms of the ultraspherical polynomials. I will conclude with an extension of this problem to probability measures, demonstrate its connection to the discrete setting, and describe conjectures for both problems. .

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March 20

Spring Break

 No Talk

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March 27

TBA

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April 3

TBA

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April 10

Richard Huber, Technical University of Denmark

 The L2-Optimal Discretization of Tomographic Projection Operators

Abstract: Tomographic inverse problems are a cornerstone of medical investigations, allowing the visualization of patients’ interior features. While the infinite-dimensional operators modeling the measurement process (e.g., the Radon transform) are well understood, in practice one can only observe finitely many measurements and employ finitely many computations in reconstruction. Thus, proper discretization of these operators is crucial; principle criteria are the approximation quality (discretization error) and the computational complexity. For iterative reconstruction approaches, the computation of the forward operator and the adjoint (called the backprojection) is often a major driver of the methods’ computational complexity. Different discretization approaches show distinct strengths regarding the approximation quality of the forward- or backward projections, respectively, which commonly leads to the use of the ray-driven forward and the pixel-driven backprojection (creating a non-adjoint pair of operators). Using such unmatched projection pairs in iterative methods can be problematic, as theoretical convergence guarantees of many iterative methods are based on matched operators. We present a novel theoretical framework for designing an L2-optimal Galerkin discretization of the forward projection. Curiously, this optimized scheme is also the optimal discretization for the backprojection. In particular, this approach grants a matched discretization framework for which both the forward and backward discretization (being the optimal choices) converge in the strong operator topology, thus eliminating the need for unmatched operator pairs. In the parallel beam case, this optimal discretization is the well-known strip model for discretization, while in the fanbeam case, a novel weighted strip model is optimal.

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April 17

TBA

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April 24

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