2026 Graduate Summer School Course Descriptions
Week 1 (June 28 - July 4):
Week 2 (July 5 - 11):
Week 3 (July 12 - 18):
Graduate Program Lecturers: Dave Auckly, Kansas State University; Anthony Conway, University of Texas at Austin; Hisaaki Endo, Institute of Science Tokyo; David Gay, University of Georgia; Marco Golla, Université de Nantes; Joshua Greene, Boston College; Tye Lidman, North Carolina State University; Robert Lipshitz, University of Oregon; Maggie Miller, University of Texas at Austin; Lisa Piccirillo, University of Texas at Austin; and Laura Starkston, University of California, Davis.
Dave Auckly, Kansas State University
Title: Notions of equivalence and complexity for knotted surfaces.
Abstract: There are many distinct notions of equivalence between embedded surfaces. For example, one could consider surfaces to be equivalent when they are smoothly isotopic, stably isotopic after taking the connected sum with some well-understood embedding, or just regularly homotopic. Each of the weaker notions of equivalence suggests a corresponding notion of complexity. It could be the minimal number of summands in a connected sum, or the minimal number of components in the double point set of a regular homotopy. This last is analogous to the crossing number of a classical knot. Indeed, any two tame embeddings of the circle into three-space are regularly homotopic, and the minimal number of components in the double point set is the crossing number. In this course, we will give careful definitions of these notions of equivalence and prove theorems relating to the various types of equivalence. Many examples will be explored. In general, it is difficult to compute the complexity of a pair of surfaces. Nevertheless, it is possible to get good lower bounds for some of these complexities via modern invariants of embedded surfaces. This course will address gauge-theoretic approaches to these bounds.
Anthony Conway, University of Texas at Austin
Title: Knotted surfaces in topological 4-manifolds.
Abstract: These lectures will be concerned with classifications of locally flat surfaces in topological 4-manifolds. In addition to surveying the literature on the topic, the plan is to cover recent developments in the area as well as connections with the search for exotic surfaces. Along the way, we will also overview some helpful tools for working in the topological category such as the s-cobordism theorem and elements of surgery theory.
Prerequisites: Some algebraic topology (mostly homology and cohomology, occasionally vector bundles) and manifold topology (e.g. immersions and embeddings).
Hisaaki Endo, Institute of Science Tokyo
Title: Lefschetz fibrations
Abstract: In this lecture course, we will first review the definition and basic properties of Lefschetz fibrations, and describe the theorem that states the correspondence between the isomorphism classes of Lefschetz fibrations and the equivalence classes of monodromy representations. We will then introduce chart description, a method for illustrating monodromy representations, and outline their applications to stabilization theorems and an invariant of hyperelliptic Lefschetz fibrations.
Prerequisites: Familiarity with basic differential topology (smooth manifolds, vector bundles, isotopies, tubular neighborhoods etc.).
David Gay, University of Georgia
Title: Diffeomorphisms of 4-manifolds coming from surfaces in 4-manifolds
Abstract: We know very little about the smooth mapping class groups of 4-manifolds, i.e. self-diffeomorphisms modulo smooth isotopy, although we do have some very interesting examples of exotic behavior and some interesting diffeomorphisms where we are not sure whether or not they are isotopic to the identity. I will show how to construct diffeomorphisms from embedded surfaces (usually with some auxiliary data) and show how to think about both relations and obstructions to relations in these smooth mapping class groups from this "surfaces-in-4-manifolds" perspective.
Joshua Greene, Boston College
Title: Lagrangian surfaces in 4-manifolds
Abstract: This course will survey properties of Lagrangian surfaces in some simple symplectic 4-manifolds. Topics will include different notions of isotopy, basic invariants, constructions, obstructions, surgery, and pseudoholomorphic curve techniques. One application is to inscription problems in the plane.
Prerequisites: an acquaintance with the introductory symplectic geometry books by Cannas da Silva and by McDuff and Salamon is useful but not necessary.
Tye Lidman, North Carolina State University & Lisa Piccirillo, University of Texas at Austin
Title: Techniques in exotica
Abstract: For a smooth 4-manifold topologist, constructing exotic manifolds or surfaces often distills to two steps: constructing candidates and obstructing diffeomorphisms. In this course, we present several techniques for construction and obstruction, and show how these techniques can be applied to produce both exotic manifolds and surfaces.
Prerequisites: Chapter 1.1-3 and Chapters 4 and 5 of ``4-Manifolds and Kirby Calculus" by Gompf-Stipsicz.
Robert Lipshitz, University of Oregon
Title: Khovanov homology and surfaces
Abstract: Khovanov homology of surfaces in 4-space has recently become an important tool, both because of its direct applications to distinguishing smooth surfaces and the indirect role it plays in the construction of the skein lasagna invariants of 4-manifolds. In this series, we will recall the construction of Khovanov homology for knots in 3-space and surfaces in 4-space, use the latter to distinguish some exotic surfaces, and survey the definition and known examples and applications of the skein lasagna module of a 4-manifold.
Maggie Miller, University of Texas at Austin
Title: Constructions of knotted surfaces
Abstract: In this introductory course, we will go over many common constructions of knotted surfaces, as well as some diagrammatic theories for how to describe and manipulate these examples (in S^4 and also in other 4-manifolds).
Prerequisites: fundamental group, homology. Recommended pre-reading: ``Knots and Links” by Rolfsen, chapters 4,5, and 9.
Laura Starkston, University of California, Davis
Title: Symplectic surfaces in 4-manifolds
Abstract: Symplectic structures add some geometric information while maintaining a fair amount of topological flexibility. We will look at how questions about exotic isotopy classes change when imposing symplectic geometric conditions. We will see what new tools can be used given the symplectic structure, and how smooth topological tools must be constrained. We will also look at connections to complex algebraic curves and some singularity theory.