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This volume is based on lecture series of two Summer Schools in 2024: the Simons Collaboration Summer School "New structures in low-dimensional topology" (Budapest, Hungary) and the Georgia Topology Summer School "Knotted surfaces in four-manifolds" (Athens, Georgia, USA). These notes provide a glimpse to several novel methods and results in low dimensional topology. Indeed, the lectures on "Instanton Floer homology and applications" (by Mrowka and Baldwin) give a detailed account on instanton invariants, apply it in the sutured setting, and provide results regarding the minimal genus problem. Novel invariants are discussed in the lectures of Gukov and Park and provide a close connection to theoretical physics. The lectures of Lobb and Greene on the square peg problem give an up-to-date account regarding the solution of this simple-looking, more than 100 years old problem on the plane. The lectures of Maggie Miller describe knotted surfaces in the 4-dimensional sphere, while the lectures of Mark Hughes provide a diagrammatic approach to the same problem. Arunima Ray's lectures also deal with surfaces, but in this case, the embedding is not necessarily smooth, only 'locally flat'. Kyle Hayden s lectures connect link homologies to the study of surfaces in four-dimensional spaces. Finally, the lectures of Stipsicz recall the construction of invariants for four-dimensional manifolds and examine the genus function of a four-manifold using these tools.
List of contents
Chapter 1. Instanton Floer homologies and applications.- Chapter 2. New quantum invariants from braiding Verma modules.- Chapter 3. An introduction to symplectic geometry and instricption problems.- Chapter 4. Concordance of surfaces in 4-manifolds.- Chapter 5. Constructing locally flat surfaces in 4-manifolds.- Chapter 6. Knotted surfaces in 4-manifolds and their diagrams.- Chapter 7. Link homologies and knotted surfaces.- Chapter 8. Four lectures on smooth four-manifolds.
About the author
Aaron Lauda
is a professor of mathematics at USC with a joint appointment in physics & astronomy and has a membership in the USC Center for Quantum Information Science and Technology. His research spans representation theory and low-dimensional topology, emphasizing categorification, diagrammatic methods, and higher category theory. He explores links to knot homology, quantum groups, and applications to quantum computation.
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.
Gordana Matić
is a professor of mathematics at the University of Georgia. She has been elected a fellow of the American Mathematical Society in 2015 for her contributions to low-dimensional and contact topology. Her interests are in general in the topology of 3- and 4-manifolds and include contact and symplectic topology, gauge theory and Heegaard Floer homology, smooth surfaces in 4-manifolds and other related topics
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 has been a member of the Hungarian Academy of Sciences.
Summary
This volume is based on lecture series of two Summer Schools in 2024: the Simons Collaboration Summer School "New structures in low-dimensional topology" (Budapest, Hungary) and the Georgia Topology Summer School "Knotted surfaces in four-manifolds" (Athens, Georgia, USA). These notes provide a glimpse to several novel methods and results in low dimensional topology. Indeed, the lectures on "Instanton Floer homology and applications" (by Mrowka and Baldwin) give a detailed account on instanton invariants, apply it in the sutured setting, and provide results regarding the minimal genus problem. Novel invariants are discussed in the lectures of Gukov and Park and provide a close connection to theoretical physics. The lectures of Lobb and Greene on the square peg problem give an up-to-date account regarding the solution of this simple-looking, more than 100 years old problem on the plane. The lectures of Maggie Miller describe knotted surfaces in the 4-dimensional sphere, while the lectures of Mark Hughes provide a diagrammatic approach to the same problem. Arunima Ray's lectures also deal with surfaces, but in this case, the embedding is not necessarily smooth, only 'locally flat'. Kyle Hayden’s lectures connect link homologies to the study of surfaces in four-dimensional spaces. Finally, the lectures of Stipsicz recall the construction of invariants for four-dimensional manifolds and examine the genus function of a four-manifold using these tools.
