Georgia Institute of Technology College of Sciences

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.

Abstract: The Teichmuller space T(S) of a closed surface S is a moduli space where each point represents a hyperbolic metric on the surface S. Interpreted appropriately, each of these hyperbolic metrics is encoded by a representation of the fundamental group of S to PSL(2,R), the group of isometries of the hyperbolic plane. This talk concerns a similar story with the Lie group PSL(2,R) replaced by the exceptional split real Lie group G2’ of type G2. That is, we shall “geometrize” surface group representations to G2’ as holonomies of some (explicitly constructed) locally homogenous (G,X)-manifolds. Along the way, we encounter pseudoholomorphic curves in a non-compact pseudosphere that carry a (T,N,B)-framing analogous to that of space curves in Euclidean 3-space. These curves play a key role in the construction. Time permitting, we discuss how this specific G_2’ recipe relates to a broader construction that unifies other approaches to geometrize representations in rank two. This talk concerns joint work with Colin Davalo.