Georgia Institute of Technology College of Sciences

Engel structures are maximally non-integrable rank-two plane fields on four-dimensional manifolds. They are closely related to contact geometry, but their global behavior is still much less understood.

In contact topology, complex tangencies of real hypersurfaces in complex manifolds give a fundamental source of contact structures, often with strong rigidity properties. This motivates the Engel analogue: can a compact four-dimensional submanifold of $\mathbb C^3$ have complex tangencies forming an Engel structure?

In this talk, I will explain how to construct such examples in the case of embeddings $M \times S^1 \subset \mathbb C^3$. The main idea is to start from a standard construction of Engel structures on circle bundles over $3$-manifolds, and then realize these Engel distributions as complex tangencies of a suitable embedding into $\mathbb C^3$.  This gives the first compact examples of submanifolds of $\mathbb C^3$ whose complex tangencies are Engel, answering a question of Yakov Eliashberg. This is joint work with E. Fernández and Á. del Pino.

Moduli spaces are central objects in modern topology and geometry, serving as powerful tools for extracting invariants from underlying manifolds. Gauge theory provides a prolific source of such spaces, utilizing techniques from geometry, analysis, and algebra to probe their structure. In this talk, we survey key gauge-theoretic moduli spaces with an emphasis on how $S^1$-actions can be used to study their topological properties. In particular, we apply these methods to hyperkähler ALE spaces, discussing their applications to the McKay correspondence and symplectic and contact homologies.

In this talk I will outline some computations and applications involving Khovanov and Lee skein lasagna modules, including the detection of some exotic pairs of 4-manifolds.  This work is joint with Qiuyu Ren.  If time allows I will also discuss a new version of the lasagna module which should ease the computational complexity for manifolds with 1-handles.  This work is joint with Qiuyu Ren, Ian Sullivan, Paul Wedrich, and Melissa Zhang.

In work in progress with Ryan Stees, we show that every closed, oriented 3-manifold can be obtained by 0-surgery on a link. Since the 0-surgery of a link can capture the data of many of the typical isotopy and concordance invariants of a link, particularly in the pairwise linking number 0 case, this result gives us a nice lens through which to study both 3-manifolds and links. However, 0-surgery on a link is certainly not a complete link invariant, and we also give multiple constructions for non-isotopic (and even non-concordant) links with homeomorphic 0-surgeries.