Frontiers in geometry and topology / Paul M., Lenhard L. Ng, and Peter S. Ozsváth, editors.

Format:

Book

Contributor:

M., Paul, editor.

Ng, Lenhard L., editor.

Ozsváth, Peter S., editor.

Series:

Proceedings of symposia in pure mathematics ; Volume 109.

Proceedings of Symposia in Pure Mathematics Series ; Volume 109

Language:

English

Subjects (All):

Topology--Congresses.

Topology.

Physical Description:

1 online resource (320 pages)

Edition:

First edition.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2024]

Summary:

This volume contains the proceedings of the summer school and research conference "Frontiers in Geometry and Topology", celebrating the sixtieth birthday of Tomasz Mrowka, which was held from August 1-12, 2022, at the Abdus Salam International Centre for Theoretical Physics (ICTP). The summer school featured ten lecturers and the research conference featured twenty-three speakers covering a range of topics. A common thread, reflecting Mrowka's own work, was the rich interplay among the fields of analysis, geometry, and topology. Articles in this volume cover topics including knot theory; the topology of three and four-dimensional manifolds; instanton, monopole, and Heegaard Floer homologies; Khovanov homology; and pseudoholomorphic curve theory.

Contents:

Cover

Title page

Contents

Preface

Summer School Program

Research Conference Program

Reminiscences on the Occasion of Tom Mrowka's 60th Birthday

Lectures on families of Dirac operators and applications

Introduction

1. Dirac operators, geometry and topology

2. K-theory and the index theorem for families

3. Dirac operators and the Weinstein conjecture in dimension three

Acknowledgments

References

Lectures on generalised Seiberg-Witten equations

1. "New" Four-dimensional gauge theories

2. The framework of generalised Seiberg-Witten equations

3. The analysis of generalised Seiberg-Witten equations

Acknowledgements

A splitting formula in instanton Floer homology

1. Introduction

2. Reduced instanton Floer (co)homology

3. Proof of Theorem A

Finite group actions on 4-manifolds and equivariant bundles

2. Some motivating questions

3. The -signature formula

4. Congruence relations for the -signature formula

5. Equivariant line bundles

6. The proof of Theorem A

7. Equivariant (2) bundles

8. Equivariant index computation

Exotic families of embeddings

2. Embeddings into ⁴

3. Stabilization and sums

4. Families and invariants

5. Specific choices

6. Proof of Theorem 1.3

7. Families of submanifolds

An instanton take on some knot detection results

2. The figure eight and the cinquefoils

3. Other determinant-5 knots

4. HOMFLY homology of nearly fibered knots

Examples of homology 3-spheres whose Chern-Simons function is not Morse-Bott

2. The character varieties of and and their image in the character variety of the separating torus.

3. Proof of Part (1)

4. Proof of Part (2)

5. Further discussion and other examples

Blowup formulas for Segre and Verlinde numbers of surfaces and higher rank Donaldson invariants

2. Background material

3. Virtual blowup formulas

4. Verlinde formulas for =± and Donaldson invariants in arbitrary rank

5. -theoretic Donaldson invariants and Segre invariants for =

6. Generating functions for Verlinde and Segre formulas in low rank.

A note on PL-disks and rationally slice knots

Khovanov homology of strongly invertible knots and their quotients

2. Conventions for Khovanov homology

3. A map on annular Khovanov homology

4. The localization theorem for strongly invertible knots

5. An application to slice disks

6. An analogue in Heegaard Floer homology

On almost complex embeddings of rational homology balls

2. Embeddings into homotopy complex projective planes

3. An auxiliary 4-manifold and its intersection lattice

4. Embeddings into the complex projective plane

Acknowledgment

Explicitly describing fibered 3-manifolds through families of singularly fibered surfaces

2. Braid conventions

3. Background: constructing fibrations

4. Small movies

5. Proof of Theorem 1.2

6. Example of a fibration

On the minimal genus problem in four-manifolds

2. Homology ²× ²'s

3. Elliptic surfaces

4. The genus function for \CP²#2\overline{\CP}² and \CP²#3\overline{\CP}²

5. The Akhmedov-Park exotic \CP²#2\overline{\CP}² manifold

6. The Baldridge-Kirk exotic \CP²#3\overline{\CP}² manifold

References.

The Gysin sequence and the ( ) homology of (2, )

2. \SU( ) representation spaces

3. ( ) homology

4. Proof of Observation 1.1

Heegaard Floer multicurves of double tangles

2. Review and conventions

3. Computations in bordered sutured Heegaard Floer theory

4. Proof of the Main Theorem

5. Satellites, thinness, and A-links

6. Growth of knot Floer and Khovanov homology under cabling

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Description based on print version record.

Includes bibliographical references.

Other Format:

Print version: M., Paul Frontiers in Geometry and Topology

ISBN:

9781470477585