2024 MATRIX Annals, Part I.

Format:

Book

Author/Creator:

Wood, David R.

Contributor:

Etheridge, Alison.

de Gier, Jan.

Joshi, Nalini.

Series:

MATRIX Book Series

MATRIX Book Series ; v.7

Language:

English

Physical Description:

1 online resource (737 pages)

Edition:

1st ed.

Place of Publication:

Cham : Springer, 2026.

Summary:

MATRIX is Australia's residential mathematical research institute.It facilitates new collaborations and mathematical advances through intensive residential research programs, each 1-4 weeks in duration.

Contents:

Intro

Preface

MATRIX Program: Elliptic Partial Differential Equations, Geometry, and the Calculus of Variations

MATRIX Program: New Deformations of Quantum Field and Gravity Theories

MATRIX Program: Dynamics and Computation

MATRIX Program: Nijenhuis Geometry and Integrable Systems II

MATRIX-RIMS TandemWorkshop: Evolutionary Partial Differential Equations and Applications

MATRIX Program: Instabilities of Flows in Porous Media

MATRIX Program: Harmonic Analytic Connections

MATRIX Program: Low Dimensional Topology: Invariants of Links, Homology Theories, and Complexity

MATRIX Program: Addressing New Modelling and Data Challenges revealed by the COVID-19 Pandemic

Contents

Part I Refereed Articles

Chapter 1 MATRIX Program: New Deformations of Quantum Field and Gravity Theories

Stress-energy tensor deformations, Ricci flows and black holes

1 Introduction

2 Two-dimensional TT deformations

2.1 The TT dressing mechanisms and the gravity picture

2.2 Global coordinate transformations and the Burgers' equation

2.3 Combining TT and √TT flows

3 An eigenvalue-based approach to higher-dimensional stress tensor flows

3.1 Defining TT-like deformations in higher dimensions

3.2 Dynamical gravity theories and the Palatini frame

3.3 Higher-dimensional √TT-like flows

4 Ricci flows in the space of metrics

4.1 Ricci flows from TT-like flows

4.2 Einstein-Ricci solitons and the singular conformal limit

4.3 Classical (d-1)-form field theories as vacuum energy theories

5 TT and √TT-deformed black holes in d = 4

5.1 TT-deformed electrovacuum solutions

5.2 Ricci flows of the extremal Bekenstein-Hawking entropy

5.3 √TT-deformed electrovacuum solutions

5.4 Charged black holes in asymptotically curved space-time

6 Conclusions

References.

Field-dependent metrics and higher-form symmetries in duality-invariant theories of non-linear electrodynamics

2 Two Notions of Coupling to a Field-Dependent Metric

3 Duality Invariance and Field-Dependent Metrics

4 Application to Higher-Form Symmetries

4.1 One-Form Symmetries in Non-Linear Electrodynamics

4.2 Construction of Non-Trivial Currents in Maxwell Theory

4.3 Generalization to Interacting Theories

4.4 Remarks on Generalized Currents

5 Conclusion

References

Supergravity component reduction with computer algebra

2 Component Reduction with Cadabra

3 Optimization

4 Other Reduction Procedures

4.1 Testing

4.2 Representation Conversion

4.3 Degauging

5 Repository

5.1 Class Documentation

Equivariant localization and gluing rules in 4d N = 2 higher derivative supergravity

2 Equivariant localization in supergravity

3 Off-shell supergravity, CKS backgrounds and higher derivative invariants

4 Omega background and proof of [36]

5 Gluing rules and black holes

6 M2-brane partition functions

7 List of open questions

AdS3 integrability, tensionless limits, and deformations: A review

2 AdS3 strings in uniform lightcone gauge

2.1 Superstring action and integrability

2.2 Gauge fixing and lightcone Hamiltonian

3 Symmetries, S matrix, and spectrum

3.1 Symmetries

3.2 S matrix

3.3 Spectrum

4 Small-tension spectra and dual CFT

4.1 The case of k = 0 (pure-RR background)

4.2 The case of k = 1 (symmetric-product orbifold CFT)

5 Deformations

5.1 Overview of integrable deformations

5.2 Homogeneous and inhomogeneous Yang-Baxter deformations

5.3 Deformed backgrounds

6 The bi-Yang-Baxter+WZ deformation

6.1 Deformed background.

6.2 Deformed symmetries and S-matrix

7 Elliptic deformation

8 Outlook

The incomprehensible simplicity of PST

2 Six-dimensional 2-form gauge theory

2.1 Non-chiral 2-form gauge theory

2.2 Hamiltonian formulation

2.3 First-order form of the Lagrangian density for the two-form gauge field and its splitting into chiral and anti-chiral part

3 Properties of the first-order chiral 2-form field Lagrangian

4 PST construction of Lorentz-covariant Lagrangians for chiral forms

5 Coupling to gravity and interacting chiral-form theories

5.1 M5-brane

5.2 Non-linear conformal chiral two-form electrodynamics

6 Conclusion

Chapter 2 MATRIX Program: Nijenhuis Geometry and Integrable Systems II

Thoughts about potentials with finite-band spectrum and finite-dimensional reductions of integrable systems

2 Preliminaries

2.1 Benenti systems

2.2 Neumann system, geodesic flow on ellipsoid and stationary solutions of the KdV equation as Benenti systems

3 When solutions of one Benenti system are solutions of another

4 Stationary solutions corresponding to the Neumann system and finite-bandness of the corresponding Sturm-Liouville-Hill problem

4.1 The property of the polynomial C(t) for solutions of KdV coming from the solution of Neumann system.

4.2 Proof that the Sturm-Liouville-Hill problem corresponding to the Neumann system has finite-band spectrum.

5 Conclusion and outlook

Research problems on relations between Nijenhuis geometry and integrable systems

1 Basics of Nijenhuis Geometry

1.1 What does Nijenhuis geometry study?

1.2 Singular points in Nijenhuis geometry

1.3 Useful properties of a Nijenhuis operator

1.4 What is a Benenti system?

2 Study of BKM systems

2.1 What are BKM systems?.

2.2 Research direction 1: Looking for other integrable systems which are BKM systems

2.3 Research direction 2: Applying established methods of the theory of integrable systems to BKM systems

2.4 Finite-gap type solutions of BKM systems and separation of variables

3 Frobenius and F-manifolds

3.1 F-structures

3.2 Frobenius manifolds

4 Nijenhuis operators: conservation laws and geodesically compatible metrics

4.1 Main definitions and basic results

4.2 Research problems

5 Superintegrable systems and Nijenhuis geometry

5.1 Questions on Killing tensors and related conservation laws.

6 Projective invariance of geometric Poisson brackets

6.1 Basic definitions

6.2 Weigthed tensor fields, projectely invariant equations, and projective invariance of Poisson brackets for k = 2, 3

6.3 Questions related to interplay of projective differential geometry and geometric Poisson structures

On Haantjes tensors for second-order superintegrable systems

2 Operator fields with n+1 conservation laws

3 Killing tensor fields

3.1 The Smorodinski-Winternitz I system

3.2 Second-order superintegrable systems with Haantjes-zero Killing tensor fields

4 Killing tensor fields in non-degenerate maximally superintegrable systems of second order

4.1 Three-dimensional non-degenerate systems

4.2 Abundant systems in any dimension

Classification of differentially non-degenerate left-symmetric algebras in dimension 3

1.1 Nijenhuis Geometry

1.2 Left-symmetric algebras

1.3 Relation between LSA and Nijenhuis operators

1.4 Main Result

2 Method

2.1 Algorithm

2.2 The 2-dimensional case

2.3 The 3-dimensional case Proof of the main result

3 Discussion

4 Appendix: code.

Chapter 3 MATRIX-RIMS Tandem Workshop: Evolutionary Partial Differential Equations and Applications

Diffusion laws and reaction term determine boundary conditions

1.1 Renormalized problem

Global boundedness in 1D tumor invasion system with chemotaxis effect

2 A priori estimate

Existence, uniqueness, and uniform boundedness of solutions to a reaction-diffusion equation involving a convolution term and mu

2 Reaction-diffusion approximation

Elastic curves and self-intersections

2 Geometric perspective on elastica theory

2.1 Bending energy

2.2 Elastica: The Euler-Lagrange equation

2.3 Uniqueness and dimensional rigidity

2.4 Curvature equation in general dimensions

2.5 Curvature equation for planar elasticae

2.6 Curvature equation for spatial elasticae

2.7 Explicit formula for curvature

2.8 Explicit formula for planar elasticae

3 Analytic perspective on elastica theory

3.1 Existence of minimizers: Direct method

3.2 Regularity

3.3 Method of Lagrange multipliers

4 Li-Yau type inequality and related problems

4.1 Li-Yau type inequality

4.2 Elastic knot

4.3 p-Elastica

4.4 Elastic flow

A.1 Jacobi elliptic integrals

A.2 Jacobi elliptic functions

Semi-discrete linear geometric flows of closed polygons

2.1 Setup

2.2 Semi-discrete polyharmonic flows of polygons

2.3 Matrix properties

3 Self-similar solutions

3.1 Self-similar solutions by scaling

3.2 Self-similar solutions by translation

3.3 Self-similar solutions by rotation

4 General solutions and long-time behaviour

5 Semi-discrete Yau curvature difference flow

Chapter 4 MATRIX Program: Low Dimensional Topology: Invariants of Links, Homology Theories, and Complexity.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

3-032-16202-5

9783032162021

OCLC:

1584002781