The seminar meets Fridays 3:30-4:30 pm in Science Center 507. The organizers are Peter Kronheimer, Fan Ye, and Gage Martin. If you'd like to get future announcements, you could send email to gagemartin@math.harvard.edu.
For those without Harvard IDs, the 5th floor of the Science Center can be accessed by elevator during 3:00-6:00 pm.
Title: Symplectic annular Khovanov homology and knot symmetry
Abstract: Khovanov homology is a combinatorially-defined invariant which has proved to contain a wealth of geometric information. In 2006 Seidel and Smith introduced a candidate Lagrangian Floer analog of the theory, which has been shown by Abouzaid and Smith to be isomorphic to the original theory over fields of characteristic zero. The relationship between the theories is still unknown over other fields. In 2010 Seidel and Smith showed there is a spectral sequence relating the symplectic Khovanov homology of a two-periodic knot to the symplectic Khovanov homology of its quotient; in contrast, in 2018 Stoffregen and Zhang used the Khovanov homotopy type to show that there is a spectral sequence from the combinatorial Khovanov homology of a two-periodic knot to the annular Khovanov homology of its quotient. (An alternate proof of this result was subsequently given by Borodzik, Politarczyk, and Silvero.) These results necessarily use coefficients in the field of two elements. This inspired investigations of Mak and Seidel into an annular version of symplectic Khovanov homology, which they defined over characteristic zero. In this talk we introduce a new, conceptually straightforward, formulation of symplectic annular Khovanov homology, defined over any field. Using this theory, we show how to recover the Stoffregen-Zhang spectral sequence on the symplectic side. Our construction also gives an analog of recent results of Lipshitz and Sarkar for the Khovanov homology of strongly invertible knots. This is joint work with Cheuk Yu Mak and Sriram Raghunath.
Title: Instanton Floer homology from Heegaard diagrams
Abstract: Heegaard Floer homology and monopole Floer homology are known to be isomorphic thanks to the monumental work of Taubes et al. But is there a simpler, more axiomatic explanation? And how is instanton Floer homology related to these other theories? I'll talk about work in progress with Zhenkun Li, Steven Sivek, and Fan Ye motivated by these questions. In particular, I'll sketch the construction of a chain complex that computes sutured instanton homology, which is isomorphic as a vector space to the Heegaard Floer chain complex of the sutured manifold. We are currently trying to prove that the differentials on the two sides agree.
Title: Framed real monopole Floer homology
Abstract: Seiberg-Witten theory has an analogue for 3- and 4-manifolds with involutions called real Seiberg-Witten theory. This theory can be used to construct invariants of links and embedded surfaces by passing to double branched covers. This talk will focus on a framed version of real Seiberg-Witten-Floer homology. It turns out this invariant of links has rather surprising properties not seen in ordinary Seiberg-Witten theory. I will explain why it is special and how it is related to some recent developments.
Title: Integer surgeries rational homology cobordant to lens spaces
Abstract: This talk is about a project that investigates when positive integer surgery on a knot in S^3 is smoothly rational homology cobordant to a lens space. This merges two classic areas of low-dimensional topology together: The study of lens space surgeries, and the study of cobordism. We give a complete answer to what lens spaces can be obtained this way, and some results on surgery slopes. Various tools are involved, including Greene’s changemaker lattice coming from Heegaard Floer d-invariant.
Title: A way to prove the Four Color Theorem using gauge theory
Abstract: In this talk, I show how ideas coming out of gauge theory can be used to prove that certain configurations in the list of "633 unavoidable's" are reducible. In particular, I show how to prove the most important initial example, the Birkhoff diamond (four “adjacent" pentagons), is reducible using our filtered $3$- and $4$-color homology. In this context reducible means that the Birkhoff diamond cannot show up as a “tangle" in a minimal counterexample to the 4CT. This is a new proof of a 111-year-old result that is a direct consequence of a special (2+1)-dimensional TQFT. I will then indicate how the ideas used in the proof might be used to reduce the unavoidable set of 633 configurations to a much smaller set.
This is joint work with Ben McCarty.
Title: Khovanov skein lasagna modules and exotica
Abstract: The Khovanov skein lasagna module S(X;L) is a smooth invariant of a 4-manifold X with link L in its boundary. In this talk I will outline the construction of Khovanov skein lasagna modules, as well as new computations and applications including the detection of some exotic 4-manifolds. This work is joint with Qiuyu Ren.
Title: Deformation inequivalent symplectic structures and Donaldson's four-six question
Abstract: Studying symplectic structures up to deformation equivalences is a fundamental question in symplectic geometry. Donaldson asked: given two homeomorphic closed symplectic four-manifolds, are they diffeomorphic if and only if their stabilized symplectic six-manifolds, obtained by taking products with CP^1 with the standard symplectic form, are deformation equivalent? I will discuss joint work with Amanda Hirschi on showing how deformation inequivalent symplectic forms remain deformation inequivalent when stabilized, under certain algebraic conditions. This gives the first counterexamples to one direction of Donaldson’s “four-six” question and the related Stabilizing Conjecture by Ruan. In the other direction, I will also discuss more supporting evidence via Gromov-Witten invariants.
Title: Connectivity in the space of framed hyperbolic 3-manifolds
Abstract: I will show that the space of framed infinite volume hyperbolic 3-manifolds is connected but not path connected. This space is equipped with the geometric topology, in which two framed manifolds are close if they are almost isometric on large neighborhoods of their framed basepoints. The proof of connectivity will be an application of the density theorem. I will then use a combination of results on representations of Kleinian groups and Chabauty spaces of subgroups to construct an infinite family of path components of this space.
Title: Sectorial Decompositions of Symmetric Products and Homological Mirror Symmetry
Abstract: Symmetric products of Riemann surfaces play a crucial role in symplectic geometry and low-dimensional topology. They are essential ingredients for defining Heegaard Floer homology and serve as important examples of Liouville manifolds when the surfaces are open. In this talk, I will discuss ongoing work on the symplectic topology of these spaces through Liouville sectorial methods, along with examples as applications of this decomposition construction to homological mirror symmetry.
Title: Iterated Mazur pattern and linear independence
Abstract: This talk will discuss the use of involutive knot Floer homology to ascertain the linear independence in the smooth concordance group of various classes of iterated Mazur pattern satellite knots. Part of this work is joint with Sungkyung Kang.
Title: Braid positivity, taut foliations, and unknot detection
Abstract: The L-space conjecture predicts that three seemingly different ways to measure the "size" of a 3-manifold are equivalent. In particular, it predicts that a manifold with the "extra" geometric structure of a taut foliation also has "extra" Heegaard Floer homology (meaning that the manifold is not an L-space). In this talk, I'll discuss the motivation for this conjecture, and describe some results which produce taut foliations by leveraging special properties of positive braid knots. Along the way, we will produce some novel obstructions to braid positivity. I will not assume any background knowledge in Floer or foliation theories; all are welcome!
Title: Torus links and colored Heegaard Floer homology
Abstract: For any knot K in the three-sphere, we define new knot invariants labeled by integers which are invariant under full twists. These invariants, called colored Heegaard Floer homology of K, is the colimit of the cobordism maps from the (n, mn)-cable of K to (n, (m+1)n)-cable by letting m goes to infinity. We will focus on the explicit computation of colored Heegaard Floer homology for the family of L-space knots, particularly for unknots from their Alexander polynomials.
Title: Signs in Heegaard Floer
Abstract: I will discuss joint work in progress with Manolescu to prove
naturality of Heegaard Floer invariants away from characteristic 2. We
start by using Perutz's construction that identifies Heegaard Floer
groups as Lagrangian Floer cohomology groups, bypassing Ozsvath and
Szabo's specialised construction. The main issue from this perspective
is that the definition of Lagrangian Floer cohomology groups away from
characteristic 2 requires making additional choices for each pair of
Lagrangians, most importantly that of Pin structures following de Silva,
Fukaya-Oh-Ohta-Ono, and Seidel. Ensuring that the Floer groups are
independent of choices up to an unknown isomorphism then amounts to
ensuring the connectedness of the space of choices required in the
construction. We identify a specific space of choices, which is homotopy
equivalent to real projective space, for which the strategy implemented
by Juhász-Thurston-Zemke in characteristic 2 readily gives independence
of the groups up to overall sign. The most laborious part of our work
then amounts to removing this sign ambiguity; there is one particularly tricky commutative diagram to check.