Singularities and Low Dimensional Topology

The special semester 'Singularities and low dimensional topology' in the Spring of 2023 at the Erds Center (Budapest) brought together algebraic geometers and topologists to discuss and explore the strong connection between surface singularities and topological properties of three- and four-dimensional manifolds. The semester featured a Winter School (with four lecture series) and several focused weeks. This volume contains the notes of the lecture series of the Winter School and some of the lecture notes from the focused weeks. Topics covered in this collection range from algebraic geometry of complex curves, lattice homology of curve and surface singularities to novel results in smooth four-dimensional topology and grid homology, and to Seiberg-Witten homotopy theory and 'spacification' of knot invariants. Some of these topics are already well-documented in the literature, and the lectures aim to provide a new perspective and fresh connections. Other topics are rather new and have been covered only in research papers. We hope that this volume will be useful not only for advanced graduate students and early-stage researchers, but also for the more experienced geometers and topologists who want to be informed about the latest developments in the field.

Provides a glimpse into the interaction of complex singularity theory and low dimensional topology Gives a detailed introduction to two instances of 'spacification': Seiberg-Witten homotopy and Khovanov homotopy Puts several recent results and constructions into perspective and shows applications of those

Auteur

Javier Fernandez de Bobadilla earned his Ph.D. in 2001, from Nijmegen University. He is currently an Ikerbasque Research Professor at the Basque Center for Applied Mathematics. His interests are at the study of singularities and degenerations from topological, geometric and algebraic methods.

Marco Marengon received his PhD in 2017, and he is currently a senior research fellow at the Rényi Institute in Budapest. His expertise is in low-dimensional topology, including the topology of smooth 4-manifolds and knot invariants such as Heegaard Floer homology and Khovanov homology.

András Némethi received his PhD in 1991, and currently he is a professor at the Rényi Institute and the Eötvös Loránd University in Budapest. His expertise is in singularity theory of complex algebraic (or analytic) varieties and its connections with low dimensional topology. Since 2020 he is member of the Hungarian Academy of Sciences.

András Stipsicz received his PhD in 1994, and currently he is the director (and a professor) of the Rényi Institute in Budapest. His expertise lies in low dimensional topology, especially in the smooth topology of four-manifolds and invariants of three- and four-manifolds, including Seiberg-Witten and Heegaard Floer invariants. Since 2016 he is member of the Hungarian Academy of Sciences.

Contenu

Preface.- Chapter 1. Heegaard Floer Homology by C.-M. Michael Wong, Sarah Zampa.- Chapter 2. A Knot-theoretic Tour of Dimension Four by Kyle Hayden, Márton Beke.- Chapter 3. Severi Strata of Plane Curve Singularities by Duco van Straten, László Koltai.- Chapter 4. Lattice Cohomology by András Némethi, Tamás Ágoston.- Chapter 5. A Tour of Monopole Floer Spectra by Matthew Stoffregen.- Chapter 6. Spectra in Khovanov and knot Floer theories by Sucharit Sarkar, Marco Marengon, András Stipsicz.

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