Georgia Institute of Technology College of Sciences

Dehn surgery is a fundamental construction in topology where one removes a neighborhood of a knot from the three-sphere and reglues to obtain a new three-manifold. The Cosmetic Surgery Conjecture predicts two different surgeries on the same non-trivial knot always gives different three-manifolds. We discuss how gauge theory, in particular, the Chern-Simons functional, can help approach this problem. This technique allows us to solve the conjecture in essentially all but one case. This is joint work with Ali Daemi and Mike Miller Eismeier.

This thesis adopts the immersed-curve perspective to analyze the knot Floer complexes of (1,1)-satellite knots. The main idea is to encode the chain model construction through what we call a planar (1,1)-pairing. This combinatorial and geometric object captures the interaction between the companion and the pattern via the geometry of immersed and embedded curves on a torus (or its planar lift).

We discuss the proof of the following Theorem

Assume $E$ is a $C^{p}$ real Banach manifold, and $f:E\circlearrowleft$, $f\circ f=f$ is a $C^{p}$ retraction, where $1\leq p\leq +\infty$. Then the retract $f(E)$ is a $C^{p}$ sub Banach manifold of $E$.

The Jones polynomial was first defined by Vaughan Jones as a "trace function" on an algebra discovered via operator algebras. It was discovered that the polynomial satisfies certain skein relations. The HOMFLY polynomial was discovered through both skein relations and a "lift" of the trace function on the Jones algebra to the "Hecke algebra". Another 2-variable polynomial called the Kauffman polynomial was discovered purely via skein relations. In this talk, we discuss how the process started by Jones was reversed for this polynomial.

I'll talk about some specialized kirby calculus constructions: immersed surface complements and round handles. I'll prove using kirby calculus that S2xS2 minus an appropriate smooth embedded S2vS2 is diffeomorphic to R4. Maybe that is obvious, but the point is we can find nice diagrams where you see everything explicitly.

TBD

Instanton Floer homology, introduced by Floer in the 1980s, has become a power tool in the study of 3-dimensional topology. Its application has led to significant achievements, such as the proof of the Property P conjecture. While instanton Floer homology with complex coefficients is widely studied and conjectured to be isomorphic to the hat version of Heegaard Floer homology, its counterpart with integral coefficients is less understood.

We say that a Riemannian manifold has good higher expansion if every rationally null-homologous i-cycle bounds an i+1 chain of comparatively small volume. The interactions between expansion, spectral geometry, and topology have long been studied in the settings of graphs and surfaces. In this talk, I will explain how to construct rational homology 3-spheres which are good higher expanders.