4-manifolds: Introduction Surfaces in 4-manifolds Complex surfaces Kirby calculus: Handelbodies and Kirby diagrams Kirby calculus More examples Applications: Branched covers and resolutions Elliptic and Lefschetz fibrations Cobordisms, $h$-cobordisms and exotic ${\mathbb{R}}^{4,}$s Symplectic 4-manifolds Stein surfaces Appendices: Solutions Notation, important figures Bibliography Index.

4-manifolds and Kirby calculus

A Strong coupling test of S duality

Geodesics in subriemannian manifolds: Dido meets Heisenberg Chow's theorem: Getting from A to B A remarkable horizontal curve Curvature and nilpotentization Singular curves and geodesics A zoo of distributions Cartan's approach The tangent cone and Carnot groups Discrete groups tending to Carnot geometries Open problems Mechanics and geometry of bundles: Metrics on bundles Classical particles in Yang-Mills fields Quantum phases Falling, swimming, and orbiting Appendices: Geometric mechanics Bundles and the Hopf fibration The Sussmann and Ambrose-Singer theorems Calculus of the endpoint map and existence of geodesics Bibliography Index.

A Tour of Subriemannian Geometries, Their Geodesics and Applications

Effective field theories in type I and II superstring theories for D-branes located at points in the orbifold C~2/Z_n are supersymmetric gauge theories whose field content is conveniently summarized by a `quiver diagram,' and whose Lagrangian includes non-metric couplings to the orbifold moduli: in particular, twisted sector moduli couple as Fayet-Iliopoulos terms in the gauge theory. These theories describe D-branes on resolved ALE spaces. Their spaces of vacua are moduli spaces of smooth ALE metrics and Yang-Mills instantons, whose metrics are explicitly computable. For U(N) instantons, the construction exactly reproduces results of Kronheimer and Nakajima.

D-branes, quivers, and ALE instantons

The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N=4 super Yang-Mills theory in four dimensions. The key ingredients are electric-magnetic duality of gauge theory, mirror symmetry of sigma-models, branes, Wilson and 't Hooft operators, and topological field theory. Seemingly esoteric notions of the geometric Langlands program, such as Hecke eigensheaves and D-modules, arise naturally from the physics.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cntp/2007/0001/0001/CNTP-2007-0001-0001-a001.pdf

Electric-Magnetic Duality And The Geometric Langlands Program

1. Four-manifolds 2. Connections 3. The Fourier transform and ADHM construction 4. Yang-Mills moduli spaces 5. Topology and connections 6. Stable holomorphic bundles over Kahler surfaces 7. Excision and glueing 8. Non-existence results 9. Invariants of smooth four-manifolds 10. The differential topology of algebraic surfaces Appendix References Index

The Geometry of Four-Manifolds

On decrit la construction d'une famille particuliere de 4-varietes hyper-Kahler: les espaces asymptotiquement localement euclidiens ALE On decrit une 4-variete de Riemann avec juste une extremite qui a l'infini ressemble au quotient R 4 /Γ de l'espace euclidien R 4 par un groupe fini Γ d'identifications

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The construction of ALE spaces as hyper-Kähler quotients

1. Statement of the result The genus of a smooth algebraic curve of degree d in CP is given by the formula g = (d − 1)(d − 2)/2. A conjecture sometimes attributed to Thom states that the genus of the algebraic curve is a lower bound for the genus of any smooth 2-manifold representing the same homology class. The conjecture has previously been proved for d ≤ 4 and for d = 6, and less sharp lower bounds for the genus are known for all degrees [5,6,7,10,14]. In this note we confirm the conjecture. Theorem 1. Let Σ be an oriented 2-manifold smoothly embedded in CP so as to represent the same homology class as an algebraic curve of degree d. Then the genus g of Σ satisfies g ≥ (d − 1)(d − 2)/2.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1994/0001/0006/MRL-1994-0001-0006-a014.pdf

The Genus of Embedded Surfaces in the Projective Plane

Preface 1. Outlines 2. The Seiberg-Witten equations and compactness 3. Hilbert manifolds and perturbations 4. Moduli spaces and transversality 5. Compactness and gluing 6. Floer homology 7. Cobordisms and invariance 8. Non-exact perturbations 9. Calculations 10. Further developments References Glossary of notation Index.

/pdf/monopoles-and-three-manifolds-50dh4atlw9.pdf

Peter Kronheimer

Papers

The Geometry of Four-Manifolds

The construction of ALE spaces as hyper-Kähler quotients

The Genus of Embedded Surfaces in the Projective Plane

Monopoles and Three-Manifolds

Gauge theory for embedded surfaces, II