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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.
