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# Archive: Inverse Problems and Analysis

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 Dr. William Ross, University of Richmond Dr. William Ross, University of Richmond Ewing 336 Title: The range and valence of a real Smirnov function

Abstract: As an excellent example of operator theory informing complex analysis, we cover how one can use symmetric unbounded Toeplitz operators on the Hardy space to describe the possible ranges (and valences) of a class of analytic functions (the real Smirnov class) on the open unit disk. This is joint work with Tim Ferguson. 5/9/2018 2:00:00 PM 5/9/2018 3:00:00 PM False Dr. Junshan Lin, Auburn University Dr. Junshan Lin, Auburn University Ewing 336 Title: Electromagnetic Field Enhancement in Subwavelength Metallic Structures: Asymptotic Analysis and Numerical Approach

Abstract: Since the discovery of the extraordinary optical transmission through nanohole arrays in metallic films by Ebbesen, a wealth of research has been sparked in the experimental and theoretical investigation of localized electromagnetic field enhancement in subwavelength nanostructures. This remarkable phenomenon can lead to potentially significant applications in near-field imaging, bio-sensing, etc. However, there has been a long debate on the interpretation of the enhancement effect since Ebbesen’s work. In addition, a quantitative analysis of the field enhancement in subwavelength structures is still widely open. In this talk, using two-dimensional slits as a prototype, I will present mathematical studies of the field enhancement in the subwavlength structures. Based upon the layer potential technique, asymptotic analysis and homogenization theory, the enhancement mechanisms for both the single slit and an array of slits are studied quantitatively. Numerical approach for computing the resonances of such metallic structures will also be discussed briefly. 4/18/2018 2:00:00 PM 4/18/2018 3:00:00 PM False Dr. Robert Martin, University of Cape Town Dr. Robert Martin, University of Cape Town Ewing 336 Title: Function Spaces obeying a time-varying bandlimit

Abstract:
In communication engineering, Shannon sampling theory provides a crucial bridge between continuous and discrete representations of information: Any signal containing no frequencies greater than A>0 in magnitude can be stably (and perfectly) reconstructed from the values it takes on any real sequence with equidistant spacing $\pi/A$ using the Shannon sampling formula. This key property of A-bandlimited signals is applied ubiquitously in signal processing to approximate, discretize and later reconstruct audio or video signals.
We will apply operator theory to construct spaces of functions which obey a "time-varying bandlimit", and which can be reconstructed perfectly using generalized Shannon-type formulas. Applications to signal processing will be discussed. 4/11/2018 2:00:00 PM 4/11/2018 3:00:00 PM False David Penneys, Ohio State University David Penneys, Ohio State University EWG 336 Title: Classifying small quantum symmetries

Abstract: Unitary fusion categories generalize the representation categories of quantum groups, and thus we say that fusion categories encode quantum symmetries. In order to represent unitary fusion categories as categories of Hilbert spaces, we are naturally led to von Neumann algebras and subfactors. Every unitary fusion category can be realized as a category of bimodules associated to a finite index subfactor, so we say that subfactors are universal hosts for quantum symmetries. In one sense, subfactors of small index are the simplest subfactors. I will discuss the small index subfactor classification program and the search for exotic examples of quantum symmetries. 3/23/2018 2:00:00 PM 3/23/2018 3:00:00 PM False