The seminar meets on Wednesdays and/or Fridays, 3-4:30pm in MS 5148. Please contact Kristen Hendricks or Ciprian Manolescu for more information.

Winter 2015:

Spring 2015:

Summer 2015:

Abstracts:

Lidman: The Cabling conjecture predicts that the only knots in the three-sphere with non-prime surgeries are cable knots. We describe an approach to this problem using standard techniques from 3- and 4-dimensional contact/symplectic topology. This is joint work with Steven Sivek.

Sarkar: We will start our story with the Jones polynomial, a revolutionary knot invariant introduced by V. Jones in 1984. We will then talk about Khovanov homology of knots, which is a "categorification" of the Jones polynomial constructed by M. Khovanov. Finally, we will discuss a recent stable homotopy level refinement of Khovanov homology, which is joint work with R. Lipshitz, and a more algebraic topological reformulation of this invariant using the Burnside category, which is joint work with T. Lawson and R. Lipshitz. Along the way, we will mention topological applications of these three knot invariants.

Hanselman: An L-space is defined to be a rational homology 3-sphere with minimal Heegaard Floer homology. However, the condition of being an L-space may have a topological interpretation independent of Heegaard Floer theory. To see if this is true, we need to better understand which 3-manifolds are L-spaces. I will discuss one approach using cut and paste arguments and bordered Floer homology. Specifically, we can determine exactly when splicing two integer framed knot complements produces an L-space. This extends a result of Hedden and Levine, who showed that splicing 0-framed knot complements never produces an L-space. I will also discuss work towards a more general gluing statement.

Liu: In this talk, we will have an overview of the structure of Heegaard Floer homology and its topological applications during the last decade.

Hendricks: In joint work with C. Manolescu, we use the conjugation symmetry on the Heegaard Floer complexes to define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to Z_4-equivariant Seiberg-Witten Floer homology. From this we obtain two new invariants of homology cobordism, explicitly computable for surgeries on L-space knots and quasi-alternating knots, and two new concordance invariants of knots, one of which (unlike other invariants arising from Heegaard Floer homology) detects non-sliceness of the figure-eight knot.

McCoy: The unknotting number is a classical knot invariant which is easy to define but generally hard to compute. It is now known that an alternating knot has unknotting number one if and only if it has an unknotting crossing in every alternating diagram. This talk will explain the main ideas behind the proof of this result.

Scaduto: The Vafa-Witten equations are a set of gauge-theoretic PDE on a 4-manifold. The S-duality conjecture in physics predicts that the moduli space of Vafa-Witten solutions encodes the euler characteristic of the moduli space of anti-self-dual instantons. This talk will introduce the equations and discuss their basic properties.

Lin: In this talk, we will discuss the Kobayashi-Hitchin correspondence for the ASD equations/Hitchin equations/Vafa-Witten equations. Very roughly speaking, these results give 1-1 correspondence between the solutions of the gauge theoretic PDE and the holomorphic structures on a vector bundle. After recalling the basic facts and definitions in complex geometry, we will state the results and sketch the proof of the Kobayashi-Hitchin correspondence for the ASD equations by Donaldson. In the end, we will discuss how to generalize this idea to prove the similar results for the Vafa-Witten equations (following the paper by Yuuji Tanaka).

He: In this talk we will discuss some details about Witten's gauge theory approach to Jones polynomial. Roughly speaking, Witten conjectures that the coefficient of Jones polynomial is counting numbers of solutions of Kapustin-Witten Equation with specific singular boundary condition. At First, we will review some properties that taught by C. Manolescu last quarter. We will also discuss some difficulties of well-defining this problem. In addition, we will briefly describe the R. Mazzeo and E. Witten proof of the conjecture for empty knot.

Carlson: Since work of Kervaire and Milnor in 1963, the exotic sphere problem had been reduced to stable homotopy theory, with the only remaining question on the relationship whether there are half as many or exactly as many exotic spheres as stable maps of spheres in certain dimensions. This question was always known to be equivalent to the existence of framed manifolds in singly even dimension with Kervaire invariant one. All dimensions not of the form 2^k-2 were long settled, and homotopy theorists built many conditional results on the assumption that manifolds of Kervaire invariant one did exist in all such dimensions, but this conjecture was refuted in all but the case k=7 by Hill, Hopkins, and Ravenel in 2009. I'll talk about the invariant itself, some of the related work in the '60s, a dash of the homotopy theoretic consequences of existence deduced in the '70s and '80s, and try to devote close to half my time to discussing HHR's proof. This uses techniques of equivariant stable homotopy theory, which has some significantly novel aspects I'll describe relative to the classical situation. Further questions coming out of the Kervaire solution itself belong primarily to homotopy theory, but I'll at least vaguely indicate the existence of other product of equivariant homotopy theory of further interest in differential topology, e.g. the theory of topological modular forms.

Fan: In this talk, I will give an introduction to the Ozsvath-Szabo tau-invariant of a knot using grid diagrams. Then, I will show Sarkar's reproof of the theorem that the absolute value of the tau-invariant is a lower bound for the slice-genus. As an application, I will talk about Sarkar's combinatorial proof the Milnor Conjecture, which claims that the unknotting number of the (p,q)-torus knot is (p-1)(q-1)/2.

Miller: Osvath and Szabo conjectured that an irreducible rational homology 3-sphere is an L-space if and only if it admits no co-orientable taut foliation; Boyer, Gordon, and Watson recently extended this conjecture to say that those hold if and only if M's fundamental group is not left-orderable. I will talk about what these notions mean and some basic results that might lead one to make these conjectures, including the fact that L-spaces support no co-orientable taut foliation. If time provides, I will talk about the current status of these conjectures.

Zemke: In this talk we will discuss naturality issues with respect to cobordism maps in Heegaard Floer homology. We will discuss cobordism maps associated to cobordisms with embedded graphs. We will focus on the hat flavor, but we will talk about the necessary modifications for the other flavors. As an application, we compute the pi_1 action on the various flavors of Heegaard Floer homology.

Lidman: Knot contact homology is an invariant of knots arising from constructions in contact and symplectic topology by counting holomorphic curves. We prove that the degree zero piece of knot contact homology detects every torus knot as well as some other topological aspects of knots. This is joint work with Cameron Gordon.

Manion: We will discuss some of the motivation behind a new method for computing HFK, due to Ozsvath and Szabo and inspired by bordered Floer homology, as well as the definition and some basic properties of this method. If time permits, we will show how this theory can be used for computations in the case of 3-strand pretzel knots.

Jianfeng Lin: In 2003, Manolescu defined the Seiberg-Witten-Floer stable homotopy type for rational homology three-spheres. In this talk, I will explain how to construct similar invariants for a general three-manifold and discuss some applications of these new invariants. This is a joint work with Tirasan Khandhawit and Hirofumi Sasahira.

Juncheng Nan: The stabilizer of a fixed point class of a map is the fixed subgroup of the induced fundamental group homomorphism based at a point in the class. A theorem of Jiang, Wang and Zhang is used to prove that if a map of a graph satisfies a strong remnant condition, then the stabilizers of all its fixed point classes are trivial. Consequently, if $\phi_{p, f}$ is the $n$-valued lift to a covering space $p$ of a map $f$ with strong remnant of a graph, then the Nielsen numbers are related by the equation $N(\phi_{p, f}) = n \cdot N(f)$.