Topology

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...

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...

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Skein lasagna modules have become a popular tool in the study of 4-dimensional topology. The original paper of Morrison-Walker-Wedrich rather focuses on the theoretical aspects of Khovanov homology. In particular, they improve the functoriality of Khovanov homology from R^3 to S^3, and from...

It has been known since the '60's that all 2-knots are slice, meaning that all 2-spheres embedded in the 4-sphere bound embedded 3-balls into the 5-ball, and thus are concordant to the unknotted sphere. However, when we restrict to the class of "periodic" 2-knots, meaning 2-knots which are...

Since the 1980s, mathematicians have discovered uncountably many "exotic" embeddings of R^2 in R^4, i.e., embeddings that are topologically but not smoothly isotopic to the standard xy-plane. However, until today, there have been no direct, computable invariants that could detect such exotic...

We study the topological components of the surface group representations into SL(2,R) and PSL(2,R). Some components correspond to geometric structures on surface, especially discrete and faithful ones to hyperbolic structures. Utilizing the signature formula established by Kim-Pansu-...

In the past two decades, the field of topological data analysis (TDA) has significantly advanced scientific research and raised new mathematical inquiries. Nonetheless, key challenges in statistical methodology and domain-specific interpretations persist. In this talk, I will discuss my...

We study a variant of the unoriented link Floer homology defined by Ozsváth, Stipsicz, and Szabó, and construct a spectral sequence from various versions of Khovanov homology to the corresponding versions of unoriented link Floer homology over the field with two elements. In particular, we...

The $\hat{Z}$-invariants of three-manifolds introduced by Gukov, Pei, Putrov, and Vafa have influenced many areas of mathematics and physics. However, their TQFT structure is not yet fully understood. In this talk, I will present a framework for decorated Spin-TQFTs and construct one based on...

Abstract: Inspired by a categorification of a numerical invariant of 3-manifolds, series invariants for closed manifolds and for knot complements were introduced. This in turn motivated an extension of the series invariant of the former case to Lie superalgebras. It was recently generalized to...

Abstract: Algebraic K-theory of smooth compact manifolds provides a homotopical lift of the classical h-cobordism theorem and serves as a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. I will...

A result by Ozsvath and Szabo states that the knot Floer complex of an L-space knot is a staircase. In this talk, we will discuss a similar result for two-component L-space links: the link Floer complex of such links can be thought of as an array of staircases. We will describe an algorithm to...