This book consists of five chapters presenting problems of current research in mathematics, with its history and development, current state, and possible future direction. Four of the chapters are expository in nature while one is based more directly on research. All deal with important areas of mathematics, however, such as algebraic geometry, topology, partial differential equations, Riemannian geometry, and harmonic analysis. This book is addressed to researchers who are interested in those subject areas.
Young-Hoon Kiem discusses classical enumerative geometry before string theory and improvements after string theory as well as some recent advances in quantum singularity theory, Donaldson–Thomas theory for Calabi–Yau 4-folds, and Vafa–Witten invariants.
Dongho Chae discusses the finite-time singularity problem for three-dimensional incompressible Euler equations. He presents Kato's classical local well-posedness results, Beale–Kato–Majda's blow-up criterion, and recent studies on the singularity problem for the 2D Boussinesq equations.
Simon Brendle discusses recent developments that have led to a complete classification of all the singularity models in a three-dimensional Riemannian manifold. He gives an alternative proof of the classification of noncollapsed steady gradient Ricci solitons in dimension 3.
Hyeonbae Kang reviews some of the developments in the Neumann–Poincare operator (NPO). His topics include visibility and invisibility via polarization tensors, the decay rate of eigenvalues and surface localization of plasmon, singular geometry and the essential spectrum, analysis of stress, and the structure of the elastic NPO.
Danny Calegari provides an explicit description of the shift locus as a complex of spaces over a contractible building. He describes the pieces in terms of dynamically extended laminations and of certain explicit “discriminant-like” affine algebraic varieties.
Author(s): Nam-Gyu Kang, Jaigyoung Choe, Kyeongsu Choi, Sang-hyun Kim
Series: KIAS Springer Series in Mathematics, 1
Publisher: Springer
Year: 2022
Language: English
Pages: 205
City: Singapore
Preface
Contents
Enumerative Geometry, Before and After String Theory
1 Introduction
2 Classical Enumerative Geometry
2.1 Schubert Calculus
2.2 Enumerating Rational Curves in Plane
2.3 Enumerating Rational Curves in Hypersurfaces
2.4 Moduli Space and Intersection Theory
3 Enter String Theory
4 Modern Enumerative Geometry
4.1 Step 1: Moduli Space
4.2 Step 2: Virtual Fundamental Class
4.3 Step 3: Virtual Invariants
4.4 More Virtual Invariants
5 Recent Developments
5.1 Cohomological, K-Theoretic and Motivic Refinements and Wall Crossing
5.2 Quantum Singularity Theory
5.3 Donaldon-Thomas Theory for Calabi-Yau 4-Folds
5.4 Vafa-Witten Invariant
References
On the Singularity Problem for the Euler Equations
1 Introduction
2 The Local in Time Well-Posedness
3 The BKM Type Blow-Up Criterion
4 On the Type I Blow-Up
5 Type I Blow-Up and the Energy Concentrations
6 The Boussinesq Equations
References
Singularity Models in the Three-Dimensional Ricci Flow
1 Background on the Ricci Flow
2 Ancient Solutions and Noncollapsing
3 Classification of Ancient Solutions in Dimension 2
4 Structure of Ancient κ-Solutions in Dimension 3
5 Classification of Ancient κ-Solutions in Dimension 3
6 Preservation of Symmetry Under the Ricci Flow
7 Improvement of Symmetry on a Neck
8 Asymptotic Behavior of Noncollapsed Steady Gradient Ricci Solitons in Dimension 3
9 Rotational Symmetry of Noncollapsed Steady Gradient Ricci Solitons in Dimension 3 – The Proof of Theorem 5.1
References
Spectral Geometry and Analysis of the Neumann-Poincaré Operator, a Review
1 Introduction
2 NP Operator and Plasmon Resonance
3 Analysis of Cloaking by Anomalous Localized Resonance
3.1 CALR in Two Dimensions
3.2 CALR in Three Dimensions
4 Concavity and Negative Eigenvalues
4.1 A Concavity Condition for Negative Eigenvalues
4.2 NP Spectrum on Tori
4.3 Further Results
5 NP Spectrum on Polygonal Domains
6 Spectral Structure of Thin Domains
7 Analysis of Field Concentration
8 Conclusion
References
Sausages and Butcher Paper
1 Introduction
1.1 Apology
1.2 Other Work
2 The Shift Locus
3 Elaminations
3.1 Pinching
3.2 Push over and Amalgamation
4 Butcher Paper
4.1 Böttcher Coordinates
4.2 Holomorphic 1-Form
4.3 Horizontal/Vertical Foliations
4.4 Construction of the Dynamical Elamination
5 Formal Shift Space
5.1 Dynamical Elaminations
5.2 Realization
5.3 Squeezing
5.4 Rotation
6 Degree 2
7 Degree 3
7.1 The Tautological Elamination
7.2 Topology of mathcalS3
7.3 Geometry and Topology of X3
8 Degree 4 and Above
8.1 Weyl Chamber
8.2 Two Partitions
8.3 Monkey Prisms, Monkey Turnovers
8.4 K(π,1)
8.5 Degree d
8.6 Tautological Elaminations
8.7 Completed Tautological Elamination
9 Sausages
9.1 Sausages: The Basic Idea
9.2 Definitions
9.3 Moduli Spaces
9.4 The Sausage Map
9.5 Sausages and Combinatorics of the Tautological Elamination
10 Fundamental Groups
10.1 Braid Groups
10.2 Shift Automorphisms
References
