May 24, 2018: Yuli Rudyak "Arnold conjecture on symplectic fixed points"
Abstract:
The Arnold conjecture claims that for every Hamiltonian symplectomorphism $\phi: M \to M$ of a closed symplectic manifold $(M,\omega)$, the number $Fix \phi$ of fixed points of $\phi$ is at least the minimal number of critical points of a smooth function on $M$. For every $(M,\omega)$ with $\pi_2(M)=0$, Floer and Hofer performed a certain analytical reduction of the problem and used this reduction in order to estimate $Fix \phi$ by the cup-length of $M$.
Using the Floer--Hofer reduction, Rudyak and Oprea have completed the proof of the Arnold conjecture in case $\pi_2(M)=0$ by proving a certain result of Lusternik--Schnirelmann type.
May 3, 2018: Semen Podkorytov "r-similar maps"
Abstract: For each natural r, we define an equivalence relation, called r-similarity, on [X, Y], the set of homotopy classes of maps X -> Y. We prove the following. Let f, g: X -> Y be two r-similar maps and h: Y -> Z be a map 1-similar to a constant map. Then the compositions hf and hg are (2r+1)-similar.
April 5, 2018: Ina Petkova "Knot Floer homology and the gl(1|1) link invariant"
Abstract: The Reshetikhin-Turaev construction for the standard representation of the quantum group gl(1|1) sends tangles to C(q)-linear maps in such a way that a knot is sent to its Alexander polynomial. After a brief review of this construction, I will give an introduction to tangle Floer homology — a combinatorial generalization of knot Floer homology which sends tangles to (homotopy equivalence classes of) bigraded dg bimodules. Finally, I will discuss how to see tangle Floer homology as a categorification of the Reshetikhin-Turaev invariant. This is joint work with Alexander Ellis and Vera Vertesi.
March 29, 2018: Slava Krushkal "Engel relations in 4-manifold topology"
Abstract: I will discuss geometric classification techniques in the theory of topological 4-manifolds, surgery and the s-cobordism theorem, which are known to hold for a certain class of fundamental groups and are open in general. Starting with an introduction to the 4-dimensional topological surgery conjecture, this talk will focus on recent results on the construction of new universal surgery models. The construction relies on geometric applications of the group-theoretic 2-Engel relation. (Joint work with Michael Freedman)
February 22 and March 1, 2018: Vladimir Chernov "Causality and Linking in globally hyperbolic and causally simple spacetimes. (parts 1 and 2)"
Abstract: In the first part of the talk we recall the notions of a globally hyperbolic spacetime X and of the associated contact manifold of light rays $N_X$. Conjectures on relations of causality in such spacetimes and of linking in $N_X$ of spheres of light rays through the two points were first formulated by Low (for topological linking) and later by Natario and Tod (for Legendrian linking). These conjectures were solved by Nemirovski and the author. In the second talk we formulate the generalization of the Legendrian Low conjecture of Natario and Tod (proved by Nemirovski and myself before) to the case of causally simple spacetimes. We prove a weakened version of the corresponding statement. In all known examples, a causally simple spacetime X can be conformally embedded into some globally hyperbolic $\tilde X$ and the space of light rays $N_X$ is an open submanifold of the space of light rays in $N_{\tilde X}$. If this is always the case, this provides an approach to solving the conjectures relating causality and linking in causally simples spacetimes.
February 1 and February 8, 2018: David Freund "Multistring Based Matrices"
Abstract: Flat virtual links can be interpreted as combinatorial models for curves on surfaces. Using a canonical surface representation of a chord diagram, Turaev associated a matrix to a flat virtual knot that captures invariants of the knot. Our first talk will focus on the development of this based matrix and its computation, justifying that the construction does not naturally generalize to flat virtual links. In the second talk, we construct a generalization of Turaev's based matrix to flat virtual links that successfully generates analogous invariants.
January 25, 2018: Yu Pan "Wrapped Floer Homology for exact Lagrangian fillings and cobordisms"
Abstract: I will give a brief introduction of the Legendrian contact homology, which is an invariant of Legendrian knots Λ defined in the spirit of Symplectic Field Theory. With the similar idea, the wrapped Floer homology for exact Lagrangian fillings gives an isomorphism between the linearized contact homology of Λ and the singular homology of the Lagrangian filling. The wrapped Floer homology for exact Lagrangian cobordisms also gives relations between linearized contact homology of boundary Legendrian knots and the singular homology of the Lagrangian cobordism. At the end, I would like to mention an on-going project with Dan Rutherford about the wrapped Floer theory for immersed exact Lagrangian fillings.
January 18, 2018: Inanc Baykur "Symplectic and exotic 4-manifolds via positive factorizations"
Abstract: We will discuss new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes which yield novel constructions of interesting symplectic and smooth 4-manifolds, such as small symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz fibrations and pencils.
January 11, 2018: John Baldwin "Khovanov homology detects the trefoil"
Abstract: In 2010, Kronheimer and Mrowka proved that Khovanov homology detects the unknot, answering a "categorified" version of the famous open question: Does the Jones polynomial detect the unknot? An even more difficult question is: Does the Jones polynomial detects the trefoils? The goal of this talk is to outline our proof that Khovanov homology detects the trefoils, answering a "categorified" version of this second question. Our proof, like Kronheimer and Mrowka's, relies on a relationship between Khovanov homology and instanton Floer homology. More surprising, however, is that it also hinges fundamentally on several ideas from contact and symplectic geometry. This is joint work with Steven Sivek.
November 16, 2017: David Freund "Complexity of Virtual Multistrings"
Abstract: A virtual $n$-string $\alpha$ is a collection of $n$ closed curves on an oriented surface $M$. Associated to $\alpha$, there are two natural measures of complexity: the genus of $M$ and the number of intersection points. By considering virtual $n$-strings up to equivalence by virtual homotopy, i.e., homotopies of the component curves and stabilizations/destabilizations of the surface, a natural question is whether these quantities can be minimized simultaneously. We show that this is possible for non-parallel virtual $n$-strings and that, moreover, such a representative can be obtained by monotonically decreasing genus and the number of intersections from any initial representative.
November 9, 2017: Nathan Dowlin "Khovanov Homology, Unknotting Number, and the Knight Move Conjecture"
Abstract: I will discuss a version of Khovanov homology which has interesting torsion under the basepoint action. It turns out that this torsion gives a lower bound for the unknotting number, and is closely related to the page at which the Lee spectral sequence collapses. In particular, for knots with $u(K)<3$, I will show that the Lee spectral sequence must collapse at the $E_2$ page. An immediate corollary is that the Knight Move Conjecture is true when $u(K)<3$.
November 2, 2017: Biji Wong "A Heegaard Floer theory for 3-orbifolds"
Abstract: Using Bordered Floer, we construct an invariant of 3-orbifolds Y^orb with singular set a link that generalizes HF-hat for 3-manifolds. We show that when the singular set is a nullhomologous knot, the invariant behaves likes HF-hat in that it categorifies the order of H_1^orb(Y^orb). This is work in progress.
October 26, 2017: Adam Levine "Concordance of knots in homology spheres"
Abstract: Every knot in the 3-sphere bounds a non-locally flat piecewise-linear (PL) disk in the 4-ball, but Akbulut showed in 1990 that the same is not true for knots in the boundary of an arbitrary contractible 4-manifold. We strengthen this result by showing that there exists a knot K in a homology sphere Y (which is the boundary of a contractible 4-manifold) such that K does not bound a PL disk in any homology 4-ball bounded by Y. In more recent work (joint with Jen Hom and Tye Lidman), we show that the group of knots in homology spheres modulo non-locally-flat PL concordance is infinitely generated and contains an infinite cyclic subgroup.
October 19, 2017: Melissa Zhang "Annular Khovanov homology and 2-periodic links"
Abstract:
Topologists often encounter spaces with interesting symmetries. By analyzing the symmetries of an object through the regularities of its algebraic invariants, we are able to learn more about the object and its relationship with smaller, less complex objects. For example, by using the right tools, we can easily see that for a topological space X equipped with a cyclic action, the rank of the singular homology of X is at least that of the fixed point set $X^{fix}$.
In low-dimensional topology, knots and links are ubiquitous and far-reaching in their associations. One particular interesting algebraic invariant of links is Khovanov homology, a combinatorial homology theory whose graded Euler characteristic is the Jones polynomial. In this talk, we consider links exhibiting 2-fold symmetry and prove a rank inequality for a variant of Khovanov homology.
September 28, 2017: Ina Petkova "An introduction to knot Floer homology"
Abstract: Knot Floer homology is a powerful invariant of knots and links, developed by Ozsvath and Szabo in the early 2000s. Among other properties, it detects the genus, detects fiberedness, and gives a lower bound to the 4-ball genus. The original definition involves counting homomorphic curves in a high-dimensional manifold, and as a result the invariant can be hard to compute. In 2007, Manolecu-Ozsvath-Sarkar came up with a purely combinatorial description of knot Floer homology. We'll discuss this combinatorial definition and work out a few small examples. (This is an expository talk.)