Mathematical Data Science Seminar

Dec 6, 2024 
Speaker: Seth Minor, Applied Math Dep, Cu Boulder
Title: Discovering Closed-Form Weather Models from Data

Abstract: The multiscale and turbulent nature of Earth's atmosphere has historically rendered accurate weather modeling a hard problem. Recently, there has been an explosion of interest surrounding data-driven approaches to weather modeling (e.g., GraphCast), which in many cases boasts both improved forecast accuracy and computational efficiency when compared to traditional methods. However, many of the new data-driven approaches employ highly parameterized neural networks, which often result in uninterpretable models and, in turn, a limited gain in scientific understanding. In this talk, we address a current research direction that addresses the interpretability problem in data-driven weather modeling. In particular, we cover a data-driven approach for explicitly discovering the governing PDEs symbolically, thus identifying mathematical models with direct physical interpretations. In particular, we use a weak-form sparse regression method dubbed the Weak Sparse Identification of Nonlinear Dynamics (WSINDy) algorithm to learn models from simulated and assimilated weather data.
 

Nov 15, 2024  
Speaker: Jenny Russell, Director, UCCS Institutional Research
Title: How DATA Can Improve the Decision Making Process

Abstract: There is no shortage of data in today’s world – but data alone is not enough. The true value of data emerges when it is communicated in ways that resonate with diverse audiences, especially those who may find data overwhelming or uninteresting. This presentation explores ways to transform data into compelling stories that engage stakeholders, facilitate understanding, and ultimately influence decisions. By utilizing various visualization techniques, narrative structures, and impactful examples, we can break down complex data and provide actionable insights. We will discuss how to make data accessible, relatable, and impactful, regardless of the industry or audience.
 

Nov 1, 2024 
Speaker: Meng Li, Data Scientist, Booz Allen Hamilton
Title: Mathematics in Data Science: Key Concepts and Applications

Abstract:  Mathematics forms the backbone of data science, providing the tools and frameworks necessary for creating predictive models, optimizing decision-making, and deriving actionable insights from data. This talk offers a high-level overview of key mathematical methods integral to data science, such as linear algebra for dimensionality reduction, probability and statistics for model assessment, optimization techniques for algorithm training, and calculus for understanding model dynamics. We will explore how these methods are applied across industries, including finance, healthcare, and marketing, illustrating real-world applications from portfolio optimization to customer behavior prediction. This session is designed for both new and experienced data enthusiasts looking to understand the mathematical principles that drive data science innovation.
 

Oct 4, 2024 
Speaker: Mihno Kim, Colorado College
Title: An Introduction to Handling Missing Values

Abstract:  Almost all real-life data contains missing values, yet most modeling techniques were developed based on having complete data. While accounting for the missing values is a popular research area, they are a greatly overlooked problem in practice because it is a preliminary step before the primary analysis. This talk will address the importance of properly handling the missing values. Different types of missing values will be introduced, along with several techniques that are easy to use.
 

Sept  20, 2024 
Speaker: Troy Butler, CU Denver and NSF
Title: Transforming Displacements to Distributions with a Machine-Learning Framework

Abstract:  In general, any uncertainty quantification (UQ) analysis for a computational model of an engineered or physical system governed by principles of mechanics seeks to use principles of probability theory to identify, classify, and quantify sources of uncertainty between simulated predictions and observational data. A significant UQ challenge is that both aleatoric (i.e., irreducible) and epistemic (i.e., reducible) sources of uncertainty can plague the modeling, manufacturing, and observation of a system. Aleatoric uncertainty may arise from the intrinsic variability of material properties represented as model parameters such as Young's modulus while epistemic uncertainty may arise from an inability to perfectly measure the true state of system responses such as displacements when subjected to known forces. In this talk, we combine two novel frameworks to quantify both types of uncertainty in the modeling of engineered systems. The data-consistent (DC) framework is utilized to quantify aleatoric uncertainties in system properties appearing as model parameters for a given Quantities of Interest (QoI) map. The Learning Uncertain Quantities (LUQ) framework is a three-step machine-learning enabled process for transforming noisy spatio-temporal data into samples of a learned QoI map to enable DC-based inversion. We focus discussion primarily on the LUQ framework to highlight key aspects of the mathematical foundations, the implications for learning QoI maps from a combination of data and simulations, and also to develop quantitative sufficiency tests for the data. Illustrative examples are used throughout the presentation to provide intuition for each step while the final two examples demonstrate the full capabilities of the methods for problems involving the manufacturing of shells of revolution motivated by real-world applications.

Sept  13, 2024 
Speaker: Radu Cascaval, UCCS
Title: An Introduction to PINNs (Physics-Informed Neural Networks)

Abstract:  PINNs have gained popularity in the field of Scientific Machine Learning due to the wide range of applications where PINNs have been found to be effective. In particular solving PDEs, inverse problems and optimization problems. In this seminar we will introduce a few such application, together with a description of the DeepXDE package. 

Spring 2024 Abstracts:

April 26, 2024 
Speaker: Jonathan Thompson, UCCS Math 
Title: Supervised and Unsupervised Learning via the Kernel Trick

The field of machine learning has garnered extraordinary interest over the last decade as a result of powerful hardware, improved algorithms, and new theoretical insights. In this talk, we offer an elementary introduction to the study of kernel methods by way of extending supervised and unsupervised learning algorithms (such as support vector machines and principal component analysis) to support non-linear predictive models embedded in an infinite-dimensional Hilbert space. In doing so, we utilize the so-called "kernel trick", which allows us to learn reduced-order decision boundaries for high-dimensional non-linear data.

April 19, 2024 
Speaker: Dustin Burchett, MITRE Corp.
Title: Network Resiliency and Optimization: The Graph Conductance Problem

This talk aims to provide an entry-level understanding of networks and topologies, with a specific focus on analyzing resiliency as a function of the nodes present in the network. A significant portion of the talk will be dedicated to the “graph conductance problem”, one possible measurement of network resiliency. The conductance of a cut in a given topology can be easily computed. However, the conductance of a graph is the minimum conductance of all possible cuts, which is an Np-complete nonlinear combinatorial optimization problem. An overview of optimization problems are provided, as well as candidate approaches to solving the conductance problem. Solution approximations are showcased, which highlight key nodes providing resiliency in the network.

This talk will provide attendees with a basic understanding of the relationships between network topologies and optimization problems, equipping them with the knowledge to measure and enhance network resiliency.
 

April 12, 2024 
Speaker: Caroline Kellackey, US Navy
Title: Advanced Framework for Simulation, Integration and Modeling (AFSIM) Survivability Study

In this talk, we will explore the AFSIM Survivability Study, which focuses on utilizing the Advanced Framework for Simulation, Integration and Modeling (AFSIM) software to analyze the survivability of military systems. We will delve into the concept of survivability and how it is measured using the probability of a missile reaching its target. The Monte Carlo method will be discussed as a means to obtain numerical results, and the calculation of the number of scenarios in a study will be explored. Additionally, we will walk through the process of designing a missile using AFSIM, considering factors such as altitude, speed, flight path, and end-game maneuvers. Practical instructions on conducting a study using AFSIM will be provided, including scenario selection, input file generation, running the experiment, and data analysis. Finally, we will discuss how to interpret and present the findings, examining the relationship between speed, altitude, and the probability of survival through graphical representations.
 

April 5, 2024 
Speaker: Justin Garrish, Colorado School of Mines
Title: Quantitative Assessment of Metabolic Health: Bayesian Hierarchical Models Uniting Dynamical Systems and Gaussian Processes

Diseases such as diabetes and cystic fibrosis disrupt the body's ability to regulate plasma glucose, resulting in chronic health issues across multiple body systems, often requiring long-term management. Through data collected under controlled settings, clinicians and researchers can utilize differential equations (DE)-based models to analyze the physiological response to glucose and generate indices of metabolic health. While such models have proven invaluable in clinical metabolism research, their efficacy is often limited outside of specific conditions. To address this challenge, we propose integrating Gaussian processes into established DE-based models within a Bayesian framework to construct robust hybrid models capable of providing reliable indices of metabolic health with associated uncertainty quantification. In this presentation, we first explore the necessary background in Gaussian processes and Bayesian statistics, emphasizing their connection to positive definite kernel-based approximation methods.  Then, through illustrative examples, we discuss mathematical and statistical modeling, numerical implementation, and clinical interpretation of results in human metabolism studies.

March 22, 2024 
Speaker: Dr. Radu Cascaval, UCCS
Title: Mathematics in Data Science and the UCCS Math Clinic

We will provide an overview of various mathematical tools (from the field of linear algebra and optimization) in machine learning and data science, including regularizations and kernel methods. We then describe a few applications that have being investigated during the UCCS Math Clinic in recent years.  

Spring 2022 abstracts:

Feb 9, 2022 
Speaker: Dr. Radu Cascaval, UCCS
Title: Mathematical Analysis of Deep Learning and Kernel Methods

Kernel methods have become an important tool in the realm of machine learning and found a wide applicability in classification tasks such as support vector machines and deep learning. This seminar will provide an overview of such methods and how mathematical analysis can aid in understanding their success.  

Feb 23, 2022
Speaker: Dr. Denis Silantyev, UCCS
Title: Obtaining Stokes wave with high-precision using conformal maps and spectral methods on non-uniform grids

Two-dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth has a class of solutions called Stokes waves which is fully nonlinear periodic gravity waves propagating with the constant velocity. We developed a new highly efficient method for computation of Stokes waves. The convergence rate of the numerical approximation by a Fourier series is determined by the complex singularity of the travelling wave in the complex plane above the free surface. We study this singularity and use an auxiliary conformal mapping which moves it away from the free surface thus dramatically speeding up Fourier series convergence of the solution. Three options for the auxiliary conformal map are described with their advantages and disadvantages for numerics. Their efficiency is demonstrated for computing Stokes waves near the limiting Stokes wave (the wave of the greatest height) with 100-digit precision. Drastically improved convergence rate significantly expands the family of numerically accessible solutions and allows us to study the oscillatory approach of these solutions to the limiting wave in great detail.
 

March 30, 2022
Speaker: Dr. John Villavert, Univ Texas Rio Grande Valley
Title: Methods for superlinear elliptic problems

We give an elementary overview of several nonlinear elliptic (and parabolic) PDEs that arise from well-known problems in analysis and geometry. We discuss existence, non-existence (including Liouville theorems) and qualitative results for the equations and introduce some powerful geometric and topological techniques used to establishing these results. 
We shall attempt to highlight the underlying ideas in the techniques and illustrate how we can refine them to handle more general problems involving differential and integral equations. 
 
April 27, 2022
Speaker: Dr. Cory B. Scott, Colorado College
Title: Machine Learning for Graphs

The recent rise in Deep Learning owes much of its success to a small handful of techniques which made machine learning models drastically more efficient on image and video input. These techniques are directly responsible for the explosion of image filters, face recognition apps, deepfakes, etc. However, they all rely on the fact that image and video data lives on a grid of pixels (2D for images, 3D for video). If we want to analyze data that doesn't have a rigid grid-like structure - like molecules, social networks, biological food webs or traffic patterns - we need some more tricks. One of these techniques is called a Graph Neural Network (GNN). In this talk, we will talk about GNNs in general, and demonstrate a couple of cool applications of these models.

Feb 20, 2019
Speaker: Dr. Daniel Appelo, CU Boulder
Title: What’s new with the wave equation? 

The defining feature of waves is their ability to propagate over vast distances in space and time without changing shape. This unique property enables the transfer of information and constitutes the foundation of today’s communication based society. To see that accurate propagation of waves requires high order accurate numerical methods, consider the problem of propagating a wave in three dimensions for 100 wavelengths with 1% error. Using a second order method this requires 0.2 trillion space-time samples while a high order method requires many orders of magnitude fewer samples. 

In the first part of this talk we present new arbitrary order dissipative and conservative Hermite methods for the scalar wave equation. The degrees-of-freedom of Hermite methods are tensor-product Taylor polynomials of degree m in each coordinate centered at the nodes of Cartesian grids, staggered in time. The methods achieve space-time accuracy of order O(2m). Besides their high order of accuracy in both space and time combined, they have the special feature that they are stable for CFL = 1, for all orders of accuracy. This is significantly better than standard high-order element methods. Moreover, the large time steps are purely local to each cell, minimizing communication and storage requirements.

In the second part of the talk we present a spatial discontinuous Galerkin discretization of wave equations in second order form that relies on a new energy based strategy featuring a direct, mesh-independent approach to defining interelement fluxes. Both energy-conserving and upwind discretizations can be devised. The method comes with optimal a priori error estimates in the energy norm for certain fluxes and we present numerical experiments showing that optimal convergence for certain fluxes.  

Mar 6, 2019
Speaker: Dr. Richard Wellman, Colorado College
Title: Scalable semi-supervised learning with operators in Hilbert space

There is preponderance of semi-supervised learning problems in science and industry, but there is a dearth of applicable semi-supervised algorithms. The LaplaceSVM Semi-Supervised Support Vector Machine is a learning algorithm that demonstrates state of the art performance on benchmark semi-supervised   data sets. However, this algorithm does not scale. In this talk we’ll discuss the mathematical foundations of the LaplaceSVM and show the kernel is a solution of a non-homogenous self-adjoint operator equation. It can be shown certain Galerkin spectral approximations are themselves valid reproducing kernels that encode the underlying Riemannian geometry. The spectral kernels have excellent scalability metrics and interesting mathematical properties. We discuss both the mathematics and experimental results of the resultant semi-supervised algorithm.

Apr 6, 2019
Speaker: Dr. Mahmoud Hussein,  CU Boulder
Title: Exact dispersion relation for strongly nonlinear elastic wave propagation

Wave motion lies at the heart of many disciplines in the physical sciences and engineering. For example, problems and applications involving light, sound, heat or fluid flow are all likely to involve wave dynamics at some level. In this seminar, I will present our recent work on a class of problems involving intriguing nonlinear wave phenomena‒large-deformation elastic waves in solids; that is, the “large-on-small” problem.

Specifically, we examine the propagation of a large-amplitude wave in an elastic one-dimensional medium that is undeformed at its nominal state. In this context, our focus is on the effects of inherent nonlinearities on the dispersion relation. Considering a thin rod, where the thickness is small compared to the wavelength, I will present an exact formulation for the treatment of a nonlinearity in the strain-displacement gradient relation. As examples, we consider Green Lagrange strain and Hencky strain. The ideas presented, however, apply generally to other types of nonlinearities. The derivation starts with an implementation of Hamilton’s principle and terminates with an expression for the finite-strain dispersion relation in closed form. The derived relation is then verified by direct time-domain simulations, examining both instantaneous dispersion (by direct observation) and short-term, pre-breaking dispersion (by Fourier transformations), as well as by perturbation theory. The results establish a perfect match between theory and simulation and reveal that an otherwise linearly nondispersive elastic solid may exhibit dispersion solely due to the presence of a nonlinearity. The same approach is also applied to flexural waves in an Euler Bernoulli beam, demonstrating qualitatively different nonlinear dispersive effects compared to longitudinal waves. Finally, I will present a method for extending this analysis to a continuous thin rod with a periodic arrangement of material properties. The method, which is based on a standard transfer matrix augmented with a nonlinear enrichment at the constitutive material level, yields an approximate band structure that accounts for the finite wave amplitude. Using this method, I will present an analysis on the condition required for the existence of spatial invariance in the wave profile.

This work provides insights into the fundamentals of nonlinear wave propagation in solids, both natural and engineered-a problem relevant to a range of disciplines including dislocation and crack dynamics, geophysical and seismic waves, material nondestructive evaluation, biomedical imaging, elastic metamaterial engineering, among others.

May 8, 2019 
Speaker: Dr. Michel Calvisi, Mechanical and Aerospace Eng, UCCS
Title: The Curious Dynamics of Translating Bubbles: An Application of Perturbation Methods and Potential Flow Theory

When subject to an acoustic field, bubbles will translate and oscillate in interesting ways. This motion is highly nonlinear and its understanding is essential to the application of bubbles in diagnostic ultrasound imaging, microbubble drug delivery, and acoustic cell sorting, among others. This talk will review some of the interesting physics that occur when bubbles translate in an acoustic field, including Bjerknes forces, the added mass effect, and nonspherical shape oscillation. Such nonspherical shape modes strongly affect the stability and acoustic signature of encapsulated microbubbles (EMBs) used for biomedical applications, and thus are an important factor to consider in the design and utilization of EMBs. The shape stability of an EMB subject to translation is investigated through development of an axisymmetric model for the case of small deformations using perturbation analysis. The potential flow in the bulk volume of the external flow is modeled using an asymptotic analysis. Viscous effects within the thin boundary layer at the interface are included, owing to the no-slip boundary condition. The results of numerical simulations of the evolutions equations for the shape and translation of the EMB demonstrate the counterintuitive result that, compared to a free gas bubble, the encapsulation actually promotes instability when a microbubble translates due to an acoustic wave. 

Fall 2018 abstracts:

Sep 12, 2018
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.

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Oct 3, 2018 
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.

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Oct 17, 2018 
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).

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Nov 14, 2018 
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!

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Dec 5, 2018 
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.

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Apr 11, 2018

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

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
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 11, 2017
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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