Thursday 3:30-4:30 PM
201 Kemeny Hall
Note: Special meeting times are marked in red.
Schedule
 Date Speaker Title May 24 Yuli Rudyak(University of Florida) Arnold conjecture on symplectic fixed points May 3 Semen Podkorytov(Steklov Mathematical Institute -- St Petersburg) TBA Apr. 5 Sergey Melikhov(Steklov Mathematical Institute -- Moscow) TBA Mar. 29 Slava Krushkal(UVA) TBA Jan. 25 Yu Pan(MIT) Wrapped Floer Homology for exact Lagrangian fillings and cobordisms Jan. 18 Inanc Baykur(UMass Amherst) Symplectic and exotic 4-manifolds via positive factorizations Jan. 11 John Baldwin(Boston College) Khovanov homology detects the trefoil Nov. 16 David Freund(Dartmouth) Complexity of Virtual Multistrings Nov. 9 Nathan Dowlin(Columbia) Khovanov Homology, Unknotting Number, and the Knight Move Conjecture Nov. 2 Biji Wong(UQAM) A Heegaard Floer theory for 3-orbifolds Oct. 26 Adam Levine(Duke) Concordance of knots in homology spheres Oct. 19 Melissa Zhang(Boston College) Annular Khovanov homology and 2-periodic links Sept. 28 Ina Petkova(Dartmouth) An Introduction to knot Floer homology
Abstracts

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#### 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.)