Below is the schedule of talks, as an embedded google calendar. All summer school talks will take place in room 328, Boyd Graduate Research Center, and all conference talks will be in room 202, Physics Building. Course 1: Knotted surfaces and their diagrams Instructor: Mark Hughes, TA: Geunyoung Kim Abstract: This course will serve as an introduction to knotted surfaces both in S^4 and in arbitrary 4-manifolds. After describing classical families of 2-knots, we will describe a number of practical techniques for representing and manipulating these surfaces, including broken surface diagrams, banded unlink diagrams, and bridge trisections. We'll also discuss several theorems that establish when two knotted surfaces are equivalent, and outline their proofs. Our primary goal throughout the course will be to develop intuition for working with surfaces in 4-manifolds, with numerous examples given. Problem sets and supplemental slides Course 2: Direct and indirect constructions of locally flat surfaces in 4-manifolds Instructor: Arunima Ray, TA: Daniel Hartman Abstract: There are two main approaches to building locally flat surfaces in 4-manifolds: direct methods applying Freedman-Quinn's disc embedding theorem, and indirect methods using surgery theory. Notably the second method also requires the disc embedding theorem, but only indirectly. In this minicourse we will give an introduction to both methods, by sketching the proofs of the following results: every primitive second homology class in a closed, simply connected 4-manifold is represented by a locally flat torus [Lee-Wilczynski]; Alexander polynomial one 1-knots are topologically slice [Freedman-Quinn]; and, if time permits, 2-knots in the 4-sphere with infinite cyclic fundamental group of the complement are topologically unknotted [Freedman-Quinn]. Lecture Notes Videos: Lecture 1, Lecture 2, Lecture 3, Lecture 4 Course 3: Link homologies and knotted surfaces Instructor: Kyle Hayden, TA: Yikai Teng Abstract: Link homology theories from Heegaard Floer homology and Khovanov homology have become indispensable tools for studying knots and links, including powerful 4-dimensional obstructions (e.g., slice genus bounds). This course will discuss what these toolkits say about surfaces in 4-space themselves, via the homomorphisms assigned to link cobordisms. After introducing these two main theories (with a focus on their shared formal properties), we will discuss hands-on computational techniques in the Khovanov setting and applications of knot Floer homology and Khovanov homology to questions about knotted surfaces and 4-manifolds. Lecture 1 Supplemental Slides, Problem Sets Course 4: TBA Instructor: András Stipsicz, TA: Jin Miyazawa Abstract: TBA
