Georgia Institute of Technology College of Sciences

Every closed 3-manifold admits foliations, where the leaves are surfaces. For a given 3-manifold, which surfaces can appear as leaves? Kerékjártó and Richards gave a classification up to homeomorphism of noncompact surfaces, which includes surfaces with infinite genus and infinitely many punctures. In their 1985 paper "Every surface is a leaf", Cantwell--Conlon prove that for every orientable noncompact surface L and every closed 3-manifold M, M has a foliation where L appears as a leaf. We will discuss their paper and construction and the surrounding context.

The 2011 PhD thesis of Farris demonstrated that the ECH of a prequantization bundle over a Riemann surface is isomorphic as a Z/2Z-graded group to the exterior algebra of the homology of its base, the only known computation of ECH to date which does not rely on toric methods. We extend this result by computing the Z-grading on the chain complex, permitting a finer understanding of this isomorphism. We fill in some technical details, including the Morse-Bott direct limit argument and some writhe bounds.

In this talk we will survey some of the developments of Cheeger and Colding’s conjecture on a sequence of n dimensional manifolds with uniform two sides Ricci Curvature bound, investigated by Anderson, Tian, Cheeger, Colding and Naber among others. The conjecture states that every Gromov-Hausdorff limit of the above-mentioned sequence, which is a metric space with singularities,  has the singular set with Hausdorff codimension at least 4.

A surface of genus $g$ has many symmetries. These form the surface’s mapping class group $Mod(S_g)$, which is finitely generated. The most commonly used generating sets for $Mod(S_g)$ are comprised of infinite order elements called Dehn twists; however, a number of authors have shown that torsion generating sets are also possible. For example, Brendle and Farb showed that $Mod(S_g)$ is generated by six involutions for $g \geq 3$.

Cosmetic surgeries (purely cosmetic surgeries) are two distinct surgeries on a knot that produce homeomorphic 3-manifolds (as oriented manifolds). It is one of the ways Dehn surgeries on knots could fail to be unique. Gordon conjectured that there are no nontrivial purely cosmetic surgeries on nontrivial knots in S^3. We will recap the progress of the problem over time, and mainly discuss Ni and Wu's results in their paper "Cosmetic surgeries on knots in S^3".

We study braided embeddings, which is a natural generalization of closed braids in three dimensions. Braided embeddings give us an explicit way to construct lots of higher dimensional embeddings; and may turn out to be as instrumental in understanding higher dimensional embeddings as closed braids have been in understanding three and four dimensional topology. We will discuss two natural questions related to braided embeddings, the isotopy and lifting problem.

In 1941, Hopf gave a proof of the fact that the rational cohomology of a compact connected Lie group is isomorphic to the cohomology of a product of odd dimensional spheres. The proof is natural in the sense that instead of using the classification of Lie groups, it utilizes the extra algebraic structures, now known as Hopf algebras. In this talk, we will discuss the algebraic background and the proof of the theorem.

Continuing the theme of Hopf algebras, we will discuss a recipe by Reshetikhin and Turaev for link invariants using representations of quantum groups, which are non-commutative, non-cocommutative Hopf algebras. In the simplest case with the spin 1/2 representation of quantum sl2, we recover the Kauffman bracket and the Jones polynomial when combined with writhe. Time permitting, we will also talk about colored Jones polynomials and connections to 3-manifold invariants.

In the setting of manifolds with connected torus boundary, we can reinterpret bordered invariants as immersed curves in the once punctured torus. This machinery, due to Hanselman, Rasmussen, and Watson, is particularly useful in the context of knot complements. We will show how a type D structure can be viewed as a multicurve in the boundary of a manifold, and we will consider how the operation of cabling acts on this new invariant.