Abstract: In topology, smooth cobordism forms a natural equivalence relation on knots. In recent years, there has been a great deal of interest in understanding Lagrangian cobordisms between Legendrian knots, which are smooth cobordisms that satisfy additional geometric conditions imposed by symplectic geometry. Lagrangian cobordisms are not well understood, but are known to be significantly different than smooth cobordisms. For example, Lagrangian corbordisms form a non-symmetric relation on Legendrian knots. There has been some progress on answering qualitative questions about Lagrangian corbordisms such as: For a fixed pair of Legendrians, does there exist a Lagrangian cobordism between them? One can also ask quantitative questions: What is the “length” or “width” of a Lagrangian cobordism? I will give examples of pairs of Legendrians where Lagrangian cobordisms are flexible in that they mimic what one would expect of a smooth cobordism. I will also give examples of other pairs of Legendrians where Lagrangian cobordisms are more rigid than what one would experiences in the smooth world. This is joint work with Joshua M. Sabloff.
Abstract: Let L and L' be links in thickened (closed orientable) surfaces. A concordance between L and L' is a pair (S,M), consisting of a compact orientable 3-manifold M (with appropriate boundary), and S a disjoint union of annuli properly embedded in M x I, cobounding L and L'. Given such a concordance, how complex need the 3-manifold M be? We shall show that there exist representatives of the same concordance class that are not concordant if one restricts to 3-manifolds that are Morse-theoretically simple. We use an augmented version of Khovanov homology to detect such links. These links provide counterexamples to an analogue of the Slice-Ribbon Conjecture.
Abstract: Intersection cohomology is an invariant of singular spaces that was introduced by Goresky and MacPherson in the late 70’s. It has since become a central tool in geometric representation theory. I will give a brief introduction to intersection cohomology and the related theory of parity sheaves, introduced in joint work with Juteau and Williamson.
February 13, 2020: Andy Manion "Heegaard Floer algebras, hypertoric varieties, and the amplituhedron"
Abstract: Recently, Ozsvath-Szabo showed that their 2016 theory of "bordered knot Floer homology" does indeed compute knot Floer homology. This theory has a rich algebraic structure with many relationships to other areas of mathematics. I will sketch the physical framework into which Heegaard Floer homology is supposed to fit, along with the expected role of Ozsvath-Szabo's bordered knot Floer homology in the overall framework. Then I will discuss a new observation that apparently goes beyond the existing physical framework, namely that Ozsvath-Szabo's algebras from bordered knot Floer homology can be viewed as convolution algebras for certain hypertoric varieties whose associated polytopes are relatives of the "amplituhedron" as introduced by Arkani-Hamed and Trnka to compute scattering amplitudes in maximally supersymmetric gauge theory. This is joint work with A. Lauda and A. Licata.
Abstract: Annular Rasmussen invariants are invariants of annular links which generalize the Rasmussen s invariant and come from an integer bifiltration on Khovanov-Lee homology. In this talk we will explain some connections between the annular Rasmussen invariants and other topological information. Additionally we will state theorems about restrictions on the possible values of annular Rasmussen invariants and a computation of the invariants for all 3-braid closures, or conjugacy classes of 3-braids. Time permitting, we will sketch some proofs.
Abstract: The L-Space Conjecture is taking the low-dimensional topology community by storm. It aims to relate seemingly distinct Floer homological, algebraic, and geometric properties of a closed 3-manifold Y. In particular, it predicts a 3-manifold Y isn't "simple" from the perspective of Heegaard-Floer homology if and only if Y admits a taut foliation. The reverse implication was proved by Ozsváth and Szabó. In this talk, we'll present a new theorem supporting the forward implication. Namely, we'll build taut foliations for manifolds obtained by surgery on positive 3-braid closures. Our theorem provides the first construction of taut foliations for every non-L-space obtained by surgery along an infinite family of hyperbolic L-space knots. As an example, we'll construct taut foliations in every non-L-space obtained by surgery along the P(-2,3,7) pretzel knot. No background in Floer homology or foliation theories will be assumed.
Abstract: If $\Sigma_n(K)$ is the n-fold cyclic branched cover of $K$, what happens to the rank of HF or HFK as n goes to infinity? This talk will start with a brief overview of what happens to the size of $H_1(\Sigma_n(K))$ as n goes to infinity. We will then discuss what is known about the Heegaard Floer case as well as a few results to at least bound what happens in this case.
Abstract: A smooth knot (or link) K in the three-space $\mathbb R^3$ is called
Legendrian if the restriction of the 1-form $\alpha=x\,dy+dz$
on K vanishes, where x,y,z are the standard coordinates in $\mathbb
R^3$. If $\alpha|_K$ is everywhere non-vanishing on K, then K is
Classification of Legendrian and transverse knots up to respectively Legendrian and transverse isotopy is an important unsolved problem of contact topology. A number of useful invariants have been constructed in the literature, but there are still small complexity examples in which the existing methods do not suffice to decide whether or not the given Legendrian (or transverse) knots are equivalent.
We propose a totally new approach to the equivalence problem for Legendrian and transverse knots, which allows one to practically distinguish between non-equivalent Legendrain (or transverse) knots in small complexity cases, and gives rise to a complete algorithmic solution in the general case.
The talk is based on joint works (recent and in progress) with Maxim Prasolov and Vladimir Shastin.
Abstract: We discuss a recent proof of a conjecture of Viterbo regarding uniform bounds on the spectral norm of exact Lagrangian submanifolds inside a fixed cotangent disk bundle of the n-dimensional torus. It uses methods of symplectic cohomology, and works more generally for a new class of smooth manifolds.
Abstract: In this talk, I would like to discuss the question of "which integral homology spheres can be embedded into complex 2-plane as the boundaries of Stein submanifolds". This question was first considered and explored in detail by Gompf. At that time, he made a fascinating conjecture that: a Brieskorn homology sphere, with either orientation, cannot be embedded into complex 2-plane as the boundary of a Stein submanifold. In this talk, I will provide partial progress (some completed work as well as report on some recent work in progress) towards resolving this conjecture.
Abstract: In mathematics, various objects admit canonical decompositions into prime components: we have the fundamental theorem of arithmetic for integers, the fundamental theorem of algebra for polynomials, the Jordan normal form, the ergodic decomposition, etc., etc. For 3-manifolds we have a two-level decomposition: the prime decomposition (the Kneser-Milnor theorem) and the JSJ decomposition. A similar two-level decomposition is known for knots: Schubert's theorem on decomposition into primes and the JSJ decomposition for knot complements. It turns out that the knots (but not the links) have also a third-level decomposition: each prime knot has a canonical decomposition into two-strand Conway irreducible tangles. We will discuss this decomposition into tangles. An interesting application of this decomposition is a complete classification of mutant knots.