Analysis and Applications Seminar

This seminar welcomes faculty and students (undergraduate and graduate) from the Pikes Peak region who are interested in applications of contemporary research in analysis (of all sorts). It is intended to have a very informal format, with several seminars in the format of a workshop rather than lecture format, introducing the audience to topics that are of current interest and are normally not covered in the standard math curriculum.

While all areas of applications are welcomed, each semester will have a special focus. In Spring 2022 seminar the focus is on the mathematical developments related to deep learning and data science.  

Wednesdays 4:30-5:45pm
UC 309, UCCS campus

Please contact Dr. Radu Cascaval (radu@uccs.edu) if you are interested to join this seminar or need more info. Limited number of parking passes will be made available to non-UCCS individuals attending this seminar.

TITLES & ABSTRACTS:  

Fall 2018 seminars:

Dec 5, 2018 Seminar
Speaker: Dr. Barbara Prinari, UCCS
Title: Discrete solitons for the focusing Ablowitz-Ladik equation with non-zero boundary conditions via inverse scattering transform  

Abstract: Soliton solutions of the focusing Ablowitz-Ladik (AL) equation with nonzero boundary conditions at infinity are derived within the framework of the inverse scattering transform (IST). After reviewing the relevant aspects of the direct and inverse problems, explicit soliton solutions will be discussed which are the discrete analog of the Tajiri-Watanabe and Kuznetsov-Ma solutions to the focusing NLS equation on a finite background. Then, by performing suitable limits of the above solutions, discrete analog of the celebrated Akhmediev and Peregrine solutions will also be presented. These solutions, which had been recently derived by direct methods, are obtained for the first time within the framework of the IST, thus providing a spectral characterization of the solutions and a description of the singular limit process.

Nov 14, 2018 Seminar
Speaker: Dr. Robert Jenkins, Colorado State University - Fort Collins
Title: Semiclassical soliton ensembles 

Abstract:  Equations like the Korteweg-de Vries (KdV) and the nonlinear Schroedinger equation exhibit interesting and complicated dynamics when the dispersive length scales in the problem are small compared to those of the initial wave profile; this is the relevant scaling regime for many problem is optical fibers. In this talk I'll discuss one way to analyze such problems for integrable PDEs using the inverse scattering transform (IST) that approximates initial data by an increasingly large sum of solitons. I'll talk both about NLS and some more recent work of mine on the resonant three wave interaction equations. There will be lots of pictures to help clear up the technical details!
 

Oct 17, 2018 Seminar
Speaker: Dr. Geraldo de Souza, Auburn University
Title: Fourier series, Wavelets, Inequalities, Geometry and Optimization 

Abstract: This talk will have two parts. In the first part, I will start with motivation and comments to some important problems in Analysis. Each problem has led to important discovery, such as Wavelets, technique of convergence of Fourier, among others. The second part I will talk about Inequalities. In general, I view the second part of this presentation as simple or perhaps an elementary approach to the subject (even though it is a new idea). On the other hand, this talk will show some interesting observations that are part of the folklore of mathematics. I will go over some very common and important inequalities in analysis that we see in the course of Analysis and even in Calculus. I will give some different views of different proofs, using Geometry, Graphing and some of them “a new analytic proof” by using optimization of functions of two variables (this is very interesting).

Oct 3, 2018 Seminar 
Speaker: Dr. Robert Carlson, UCCS
Title: An elementary trip from the Gauss hypergeometric function to the Poschl-Teller potential in quantum mechanics 

Abstract: A simple transformation takes the (G) equation for the Gauss hypergeometric function to the (J) equation for Jacobi polynomials. J has an (unusual) adjoint equation (H) (of Heun type) with an extra singular point. H has eigenfunctions that can be expressed in terms of the Gauss hypergeometric function.  Another change of variables lets us rediscover a ‘solvable’ (Poschl-Teller) Schrodinger equation. The methods use the kinds of techniques we often teach in Math 3400.

Sep 12, 2018 Seminar 
Speaker: Dr. Sarbarish Chakravarty, UCCS
Title: Beach Waves and KP Solitons

Abstract: In this talk, I will give a brief overview of the soliton solutions of the KP equation, and discuss how these solutions can describe shallow water wave patterns on long flat beaches.

Apr 11, 2018 Seminar

Speaker: Dr. Greg Fasshauer, Colorado School of Mines
Title: An Introduction to Kernel-Based Approximation Methods 

Abstract:  I will start with a few historical remarks, and then motivate the use of kernel-based approximation as a numerical approach that generalizes standard polynomial-based methods. Examples of kernels and their use in data fitting problems will be provided along with an overview of some of the concerns and issues associated with the use of kernel methods.
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Mar 14, 2018 Seminar

Speaker: Dr. Ethan Berkove, Laffayette College (Joint with Rings and Wings Seminar)
Title: Short Paths and Long Titles: Travels through the Sierpinski carpet, Menger sponge, and beyond.  

Abstract:  Sierpinski carpet and Menger sponge are fractals which can be thought of as two and three dimensional versions of the Cantor set.  Like the Cantor set, each is formed by starting with a shape (a square for the carpet, a cube for the sponge) and then recursively removing certain subsets of it.  Unlike the Cantor set, what remains is connected in the following sense: given any two points s and f in the carpet or sponge, there is a path from s to f that stays in the carpet or sponge.  In this talk, we’ll discuss what we know about the shortest path from s to f in the carpet, sponge, and even higher dimensional versions of these fractals.  The proofs required a surprising (at least to us) breadth of techniques, from combinatorics, geometry, and even linear programming.  (Joint work with Derek Smith)
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Feb 28, 2018 Seminar

Speaker: Dr. Radu Cascaval, UCCS
Title: Traffic Flow Models. A Tutorial

Abstract: We present several traffic flow models, both at the micro- and macro-scale, including for multi-lane traffic. Problems of controlling the traffic will be described and numerical simulations will illustrate possible solutions.

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Dec 8, 2017  Seminar

Speaker: Dr. Barbara Prinari, UCCS
Title: Solitons and rogue waves for a square matrix nonlinear Schrodinger equation with nonzero boundary conditions

Abstract:  In this talk we discuss the Inverse Scattering Transform (IST) under nonzero boundary conditions for a square matrix nonlinear Schrodinger equation which has been proposed as a model to describe hyperfine spin F = 1 spinor Bose-Einstein condensates with either repulsive interatomic interactions and anti-ferromagnetic spin-exchange interactions, or attractive interatomic interactions and ferromagnetic spin-exchange interactions. Emphasis will be given to a discussion of the soliton and rogue wave solutions one can obtain as a byproduct of the IST.

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Nov 17, 2017  Seminar

Speaker: Dr. Oksana Bihun, UCCS
Title: New properties of the zeros of Krall polynomials

Abstract:  We identify a class of remarkable algebraic relations satisfied by the zeros of the Krall orthogonal polynomials that are eigenfunctions of linear differential operators of order higher than two. Given an orthogonal polynomial family p_n(x), we relate the zeros of the polynomial p_N with the zeros of p_m for each m <=N (the case m = N corresponding to the relations that involve the zeros of pN only). These identities are obtained by exacting the similarity transformation that relates the spectral and the (interpolatory) pseudospectral matrix representations of linear differential operators, while using the zeros of the polynomial p_N as the interpolation nodes. The proposed framework generalizes known properties of classical orthogonal polynomials to the case of non-classical polynomial families of Krall type. We illustrate the general result by proving new remarkable identities satisfied by the Krall-Legendre, the Krall-Laguerre and the Krall-Jacobi orthogonal polynomials.

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Oct 27, 2017  Seminar

Speaker: Dr. Radu C. Cascaval, UCCS
Title: What do Analysis and Scientific Computation have in common ...

Abstract:  Analysis, the world of the infinitesimally small, is thought to be one of the last standing outposts where humans can fight the computational invasion. In spite of this fact, computational sciences continue to benefit greatly from advances in analysis. This talk will illustrate this relationship, in particular functional analysis connections to numerical spectral methods, meshless methods, and their applications to numerical solutions to PDEs.
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Sept 29, 2017  Seminar

Speaker: Dr. Fritz Gesztesy, Baylor University
Title: The eigenvalue counting function for Krein-von Neumann extensions of elliptic operators

Abstract:  We start by providing a historical introduction into the subject of Weyl-asymptotics for Laplacians on bounded domains in n-dimensional Euclidean space, and a brief introduction into the basic principles of self-adjoint extensions.  Subsequently, we turn to bounds on eigenvalue counting functions and derive such a bound for Krein-von Neumann extensions corresponding to a class of uniformly elliptic second order PDE operators (and their positive integer powers) on arbitrary open, bounded, n-dimensional subsets \Omega in R^n. (No assumptions on the boundary of \Omega are made; the coefficients are supposed to satisfy certain regularity conditions.)  Our technique relies on variational considerations exploiting the fundamental link between the Krein-von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of the corresponding differential operator suitably extended to all of R^n. We also consider the analogous bound for the eigenvalue counting function for the corresponding Friedrichs extension.  This is based on joint work with M. Ashbaugh, A. Laptev, M. Mitrea, and S. Sukhtaiev.
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