Topology

This talk focuses on applications of real Seiberg-Witten theory to knotted surfaces in S^4. It is divided into two parts. In the first, we discuss how to construct exotic unknotted RP^2s. In the second, we discuss current work in progress relating the real Bauer-Furuta invariant to cobordism…

One of the distinguishing features of higher algebra is the difficulty of constructing quotients. In this talk I will explain a new technique for constructing algebra structures on quotients. This allows us to prove for example that 𝕊/8 is an E1-algebra

Unknotting number is a fundamental measure of how complicated a knot is, measuring how `far' it is from the unknot via crossing changes. It is a challenging invariant to compute; a vast array of tools have been applied to its calculation, and many conjectureshave grown up…

We use Floer homology to study some more exotic constructions in four-dimensions. This is joint work with Lisa Piccirillo

Given a simply-connected 4-manifold X with boundary S^3, when is a knot K sliced by a simple disc in X, i.e. when does K bound a disc in X whose complement has abelian fundamental group? When X is the 4-ball, Freedman proved that this occurs if and only if the knot has Alexander polynomial one.…

For any fixed rational number p/q, Dehn surgery gives a map from the set of knots in the 3-sphere to the set of closed orientable 3-manifolds. In 1978, Gordon conjectured that these maps are never injective. I will discuss some results which demonstrate non-injectivity for some special cases of…

Recently, 3&4 manifolds with finite group actions has become a popular topic. Real manifolds form the simplest class among them. Following the strategy of Manolescu, Kronheimer-Mrowka, respectively, Konno- Miyazawa-Taniguchi and Li introduced two versions of real Seiberg-Witten-Floer…

Link Floer homology is a powerful invariant of links due to Ozsváth and Szabó. One of its most striking properties is that it detects each link’s Thurston norm, a result due to Ozsváth and Szabó. In this talk I will discuss generalizations of this result to the context of 4-ended tangles, as…

Heegaard Floer homology was originally defined over the integers by Ozsvath and Szabo using choices of coherent orientations on the moduli spaces. In this talk I will explain how to construct orientations in a more canonical way, by using a coupled Spin structure on the Lagrangian tori. This…

We give a combinatorial description of link Floer homology. Then, we outline the Manolescu-Sarkar construction for the link Floer stable homotopy type, and give the two ways in which it extends over the full grid. We find both resulting Steenrod squares, and give an example where one of them is…