Extended Abstracts 2021/2022: Methusalem Lectures (Trends in Mathematics, 3)

Preface
Contents
Part I Geometric Analysis
	Analysis on Noncompact Manifolds and Index Theory: Fredholm Conditions and Pseudodifferential Operators
		1 Introduction
		2 Motivation and Some Classical Results
			2.1 A Classical Case: Smooth, Compact Manifolds
			2.2 Motivating Example: Cylindrical Ends
				2.2.1 Polar Coordinates
				2.2.2 The Black-Scholes Operator
				2.2.3 Generalized Spherical Coordinates
				2.2.4 Manifolds with Cylindrical Ends
		3 Fredholm Operators on Manifolds with Nice Ends (Lie Manifolds)
			3.1 Lie Manifolds
				3.1.1 Definition of Lie Manifolds
				3.1.2 Lie Manifolds and Geometry
				3.1.3 Differential Operators
			3.2 Formulation of the Main Result: Fredholm Conditions on ``Nice\'\' Manifolds
			3.3 Examples and Applications
		4 Pseudodifferential Operators and Problems
			4.1 Problems
		References
	Singular Value Decomposition for the X-Ray Transforms on the Reduced Heisenberg Group, and a Two-Radius Theorem
		1 Introduction
		2 Definitions
			2.1 Heisenberg Group
			2.2 X-Ray Transform
		3 Spectral Decomposition of Ir, Ir* and Ir* Ir
		4 Intertwining Differential Operators
		5 Ongoing Work
		Appendix
			Entry Functions on the Heisenberg Group
			Special Functions
		References
	Nonlocal Functionals with Non-standard Growth
		1 Introduction
		2 Nonlocal Functionals with Orlicz Growth
		3 Nonlocal Functionals with Variable Exponent
		4 Nonlocal Double Phase Functionals
		5 Nonlocal Functionals with Orthotropic Structure
		References
	A Variational Approach to the Hot Spots Conjecture
		1 Introduction
		2 A Non-standard Variational Principle
		3 Application to the Hot Spots Conjecture
		References
	Endpoint Sobolev Inequalities for Vector Fields and Cancelling Operators
		1 Sobolev Inequalities for Vector Fields
		2 Sobolev Estimates for Cancelling Operators
		3 Duality Estimates for Cocancelling Operators
		4 Further Results
		References
	Scattering of Maxwell Potentials on Curved Spacetimes
		1 Introduction
		2 Background Spacetimes and Field Equations
			2.1 CSCD Spacetimes
			2.2 Field Equations
		3 Main Results
		4 Remarks
			4.1 Conformal Scale
			4.2 Spaces of Data
			4.3 Goursat Problem
			4.4 Role of Timelike Conformal Symmetry
		References
Part II Analysis and PDEs
	Remark on the Ill-Posedness for KdV-Burgers Equation in Fourier Amalgam Spaces
		1 Introduction
		2 Proof of Theorem 1
		References
	-Convergence for the Bi-Laplace-Beltrami Equationon Hypersurfaces
		1 Introduction
		2 Auxiliary Materials
		3 Variational Formulation of Model Problem and  Convergence
		References
	Bounded Weak Solutions with Orlicz Space Data: An Overview
		1 Introduction: Uniformly Elliptic Operators
		2 Degenerate Elliptic Operators
		3 Further Remarks
		References
	Recent Progress on the Mathematical Theory of Wave Turbulence
		1 Introduction
			1.1 Description of the Theory
				1.1.1 Scope of the Theory
				1.1.2 Applications
			1.2 Mathematical Literature
		2 Main Results
			2.1 Discussions
			2.2 Related Works
			2.3 Main Ideas in the Proof
		References
	Laplace-Beltrami Equation on Lipschitz Hypersurfaces in the Generic Bessel Potential Spaces
		1 Introduction and Formulation of the Main Results
		Introduction and formulation of the problems
		2 Generic Bessel Potential Spaces and Mellin Convolutions
		3 Localization and the Model Problems, Proof of the Main Result
		References
	On the Convergence Fourier Series and Greedy Algorithm by Multiplicative System
		1 Introduction
		2 New Results: (Theorems)
		References
	Asymptotics of Harmonic Functions in the Absence of Monotonicity Formulas
		1 Introduction
		2 Alternative for Motonicity Formula: Convergence of Doubling Index
		3 Uniqueness of Blowup and Expansion Formula
		References
	Semiregular Non-commutative Harmonic Oscillators: Some Spectral Asymptotic Properties
		1 Introduction
		2  Non-commutative Harmonic Oscillators (NCHOs)
		3 Jaynes-Cumming Model Hamiltonian Analytical Study
			3.1 The JC-Model by Semiregular NCHOs
			3.2 The JC-Model for One Atom with N-Level and One Cavity-Mode in the -Configuration
			3.3 The JC-Model for an N-Level Atom and n=N-1 Cavity-Modes in the -Configuration
			3.4 The JC-Model for an N-Level Atom and n=N-1 Cavity-Modes in the -Configuration
			3.5 The JC-Model for an N-Level Atom and n=N-1 Cavity-Modes in the So-Called -Configuration
		4 The Asymptotics of the Eigenvalues Counting Function
			4.1 The Semiregular Metric Globally Elliptic System Class (SMGES Class)
			4.2 The Decoupling Theorem
			4.3 The Weyl Law
		References
	Global Compactness, Subcritical Approximation of the Sobolev Quotient, and a Related Concentration Result in the HeisenbergGroup
		1 Critical Sobolev Embeddings in the Heisenberg Group
		2 Subcritical Approximation of the Sobolev Quotient
		3 Struwe\'s Global Compactness in the Heisenberg Group
		4 Asymptotics of the Optimal Functions
		5 Localization of the Energy Concentration
		References
	A Note on Carleson-Hunt Type Theorems for Vilenkin-Fourier Series
		1 Introduction
		2 Preliminaries
		3 On Martingale Inequalities
		4 a.e. Convergence of Vilenkin-Fourier Series
		5 Almost Everywhere Divergence of Vilenkin-Fourier Series
		References
	Self-Similar Gravitational Collapse for Polytropic Stars
		1 Introduction
		2 Strategy of Proof
		References
	Control of Parabolic Equations with Inverse Square Infinite Potential Wells
		1 Critically Singular Heat Equations
		2 Existing Results
		3 The Main Result
		4 The Hilbert Uniqueness Method
		5 The Global Carleman Estimate
		6 Additional Remarks
		References
	On Geometric Estimates for Some Problems Arising from Modeling Pull-in Voltage in MEMS
		1 Introduction
		2 Preliminaries
		3 The Pull-in Voltage for the Newtonian Potential
		References
	A Note on Fractional Powers of the Hermite Operator
		1 Introduction
		2 Fractional Powers  H-s  of the Hermite Operator
		3 More on Fractional Powers of the Hermite Operator
		References
	Non-Standard Version of Egorov Algebra of Generalized Functions
		1 Introduction
		2 Notations and Set-Theoretical Framework
		3 Non-Standard Version of Egorov Algebra *E(Ω)
		4 Non-Standard Delta-Function
		5 Embedding of Distributions Into *E(Ω)
		6 Regular Algebra
		References
	Density Conditions for Coherent State Subsystems of Nilpotent Lie Groups
		1 Introduction
		2 Coherent State Systems for Nilpotent Lie Groups
		3 Density Conditions for Lattice Subgroups
		4 Necessary Density Conditions for Irregular Point Sets
			4.1 Approximate Lattices
			4.2 General Point Sets
		5 Open Problems and Questions
		References
	Space-Time Mixed Norm Estimates in Riemannian Symmetric Spaces of Non-Compact Type
		1 Introduction
		2 Strichartz Inequality
		3 Smoothing Property
		4 Conclusion
		References
	Analysis on Compact Symmetric Spaces: Eigenfunctions and Nonlinear Schrödinger Equations
		1 Compact Globally Symmetric Spaces
		2 Eigenfunction Bounds and Spherical Functions
		3 NLS on CGSSs
		4 Conclusion
		References
Part III Applied Mathematics
	On Empirical Bayes Approach to Inverse Problems
		1 Introduction
		2 Compact G* G, Inverse Signal-in-White-Noise Model
			2.1 Constructing an Empirical Bayes Posterior
			2.2 Main Result
		3 Further Discussion
			3.1 Uncertainty Quantification (UQ) Problem
			3.2 The Case of Non-Linear G
			3.3 Examples of Inverse Problems Originated from PDEs
				3.3.1 Heat Equation: Recovery of Initial Condition
				3.3.2 Elliptic PDE: Recovery of a Functional Coefficient
				3.3.3 Parabolic PDE: Recovery of a Functional Coefficient
		References
	The Interferon Influence on the Infection Wave Propagation
		1 Introduction
		2 IFN Action in Infected Cells
		3 IFN Action in Uninfected Cells
		4 IFN Redistribution
		5 Conclusions and Perspectives
		References
	Machine Learning-Based Analysis of Human Motions for Parkinson\'s Disease Diagnostics
		1 Introduction
		2 Problem Statement
		3 Proposed Solution
			3.1 Motion Mass
			3.2 Machine Learning Workflow
		4 Discussion
		5 Conclusions
		References