The seminar meets Fridays 3:30-4:30 pm in Science Center 507. The organizers are Peter Kronheimer, Fan Ye, and Joshua Wang. If you'd like to get future announcements, you could send email to fanye@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.
Spring 2023 Schedule:
Jan. 27: Francesco Lin (Columbia University)
Feb. 3: Shaoyun Bai (Simons Center)
Feb. 10: Kristen Hendricks (Rutgers University)
Feb. 17: Artem Kotelskiy (Stony Brook University)
Feb. 24: Christopher Scaduto (University of Miami)
Mar. 3: Matthew Hogancamp (Northeastern University)
Mar. 10: Thomas Mark (University of Virginia)
Mar. 17: Spring recess, no seminar
Mar. 24: Allison Miller (Swarthmore College)
Mar. 31: May be cancelled
Apr. 7: Langte Ma (Stony Brook University)
Apr. 14: Matthew Stoffregen (Michigan State University)
Apr. 21: Laura Wakelin (Imperial College London)
Apr. 28: no seminar
May. 5: Joshua Greene (Boston College)
May. 12: cancelled
Abstracts:
Jan. 27:
Title: Homology cobordism and the geometry of hyperbolic three-manifolds (Francesco Lin)
Abstract: A major challenge in the study of the structure of the three-dimensional homology cobordism group is to understand the interaction between hyperbolic geometry and homology cobordism. In this talk, I will discuss how monopole Floer homology can be used to study some basic properties of certain subgroups of the homology cobordism group generated by hyperbolic homology spheres satisfying some natural geometric constraints.
Feb. 3:
Title: Arnold conjecture and refinements of Floer homology (Shaoyun Bai)
Abstract: We show that for any closed symplectic manifold, the number of 1-periodic orbits of any non-degenerate Hamiltonian is bounded from below by an integral version of total Betti number which takes account of torsions of all characteristics. The proof is based on a perturbation scheme (FOP perturbations) which produces pseudo-cycles from normally complex derived orbifolds, and a regularization procedure of moduli spaces of J-holomorphic curves (extending recent work of Abouzaid-McLean-Smith) which produces coherent smooth structures on the moduli spaces of Floer trajectories. I will outline the proof and indicate how these results may fit into other topics including Floer homotopy theory. This is based on joint work with Guangbo Xu.
Feb. 10:
Title: Naturality issues in involutive Heegaard Floer homology (Kristen Hendricks)
Abstract: Heegaard Floer homology is an invariant of 3-manifolds, and knots and links within them, introduced by P. Oszváth and Z. Szabó in the early 2000s. Because of its relative computability by the standards of gauge and Floer theoretic invariants, it has enjoyed considerably popularity. However, it is not immediately obvious from the construction that Heegaard Floer homology is natural, that is, that it assigns to a basepointed 3-manifold a well-defined module over an appropriate base ring rather than an isomorphism class of modules, and well-defined cobordism maps to 4-manifolds with boundary. This situation was improved in the 2010s when A. Juhász, D. Thurston, and I. Zemke showed naturality of the various versions of Heegaard Floer homology. In this talk we consider involutive Heegaard Floer homology, a refinement of the theory introduced by C. Manolescu and I in 2015, whose definition relies on Juhász-Thurston-Zemke naturality but which is itself not obviously natural even given their results. We prove that involutive Heegaard Floer homology is a natural invariant of basepointed 3-manifolds together with a framing of the basepoint, and has well-defined maps associated to cobordisms, and discuss some consequences and implications. This is joint work with J. Hom, M. Stoffregen, and I. Zemke.
Feb. 17:
Title: On SO(3) representation spaces of spatial trivalent graphs (Artem Kotelskiy)
Abstract: Given a spatial trivalent graph G, we will first review some of the objects and results from Kronheimer and Mrowka's theory about the instanton invariant J#(G). Motivated by the Tutte relation and the 4-color theorem, we will then proceed to studying decompositions of graphs into two 4-ended tangle-graphs along a 4-punctured sphere. The resulting Lagrangian Floer theory happens to be on the pillowcase. Viewing J# invariant from this angle, we will propose several modifications to the construction of representation varieties: two different reductions at two basepoints, and a passage to the equivariant theory (which corresponds to the wrapped Floer theory on the pillowcase). The advantage is that the resulting equivariant invariants are as simple as possible, and can be used to recover the initial unreduced invariants via mapping cones. Based on this we will speculate on the existence of the corresponding instanton-theoretic curve invariants in the pillowcase, and indicate an open-ended strategy for how to study the Tutte relation for J#(G). This is joint work in progress with Fan Ye.
Feb. 24:
Title: Skein exact triangles in equivariant singular instanton theory (Christopher Scaduto)
Abstract: Given a knot or link in the 3-sphere, its Murasugi signature is an integer-valued invariant which can easily be computed from a diagram. Work of Herald and Lin gives an alternative description of knot signatures, as signed counts of SU(2)-representations of the knot group which are traceless around meridians. There is a version of singular instanton homology for links which categorifies the Murasugi signature. We construct unoriented skein exact triangles for these Floer groups, categorifying the behavior of the Murasugi signature under unoriented skein relations. More generally, we construct skein exact triangles in the setting of equivariant singular instanton theory. This is joint work with Ali Daemi.
Mar. 3:
Title: The annular Bar-Natan category and handle-slides (Matthew Hogancamp)
Abstract: Khovanov homology can be upgraded to an invariant of pairs (K,V) where K is a framed knot and V is an object of the annular Bar-Natan category (ABN). In this context, the pair (K,V) is called a colored knot, and its Khovanov invariant is called colored Khovanov homology. In my talk I will discuss recent joint work with David Rose and Paul Wedrich, in which we construct an object in ABN (more accurately, an ind-object therein), called a Kirby color, whose associated colored Khovanov invariant satisfies the important handle-slide relation. Our work gives an annular perspective on the Manolescu-Neithalath 2-handle formula for sl(2) skein lasagna modules.
Mar. 10:
Title: Fillable contact structures from positive surgery (Thomas Mark)
Abstract: For a Legendrian knot K in a closed contact 3-manifold, we describe a necessary and sufficient condition for contact n-surgery along K to yield a weakly symplectically fillable contact manifold, for some integer n>0. When specialized to knots in the standard 3-sphere this gives an effective criterion for the existence of a fillable positive surgery, along with various obstructions. These are sufficient to determine, for example, whether such a surgery exists for all knots of up to 10 crossings. The result also has certain purely topological consequences, such as the fact that a knot admitting a lens space surgery must have slice genus equal to its 4-dimensional clasp number. We will mainly explore these topologically-flavored aspects, but will give some hints of the general proof if time allows.
Mar. 17: Spring break.
Mar. 24:
Title: Generalizing sliceness (Allison Miller)
Abstract: A knot in the 3-sphere is said to be smoothly slice if it bounds a smoothly embedded disc in the 4-ball. Sliceness questions are closely related to interesting phenomena in 4-manifold topology: for example, the existence of a non smoothly slice knot that bounds a flatly embedded disc can be used to give a relatively quick proof of the existence of nonstandard smooth structures on 4-dimensional euclidean space. There are (at least!) two reasonable generalizations of sliceness to arbitrary 4-manifolds: in each of these directions, we will highlight open questions and give some results from joint work with Kjuchokova-Ray-Sakallı and Marengon-Ray-Stipsicz.
Mar. 31: no seminar
Apr. 7:
Title: Instantons on Joyce's G2-manifolds (Langte Ma)
Abstract: As 7-manifolds with special holonomy, examples of compact G2-manifolds were first constructed by Joyce as resolutions of flat G2-orbifolds. Later Walpuski constructed non-trivial G2-instantons over Joyce’s manifolds via gluing techniques. In this talk, I will first explain how to define a deformation invariant of G2-orbifolds by counting flat connections, then describe the moduli space of instantons over certain non-compact G2-manifolds that appeared in Joyce’s construction, with the aim to give a complete description of moduli spaces over some examples in Joyce’s list.
Apr. 14:
Title: Lattice Floer Spectra (Matthew Stoffregen)
Abstract: We calculate the monopole Floer spectra of almost-rational plumbings, including all Seifert-fibered rational homology spheres, following ideas from lattice homology. We'll also talk about a key ingredient, an exact triangle for monopole Floer spectra. This is joint work with Irving Dai and Hirofumi Sasahira.
Apr. 21:
Title: The Dehn surgery characterisation of Whitehead doubles (Laura Wakelin)
Abstract: A slope p/q is said to be characterising for a knot K if the oriented homeomorphism type of the manifold obtained by performing Dehn surgery of slope p/q on K uniquely determines the knot K. In this talk, I will discuss how JSJ decompositions and hyperbolic techniques can be used to find characterising slopes for a special class of satellite knots which includes Whitehead doubles. I will also exhibit a family of pairs of satellite knots sharing a non-characterising slope of the form 1/q.
Apr. 28: no seminar
May. 5:
Cube tilings and alternating links (Joshua Greene)
Consider a planar graph G, and form the lattice of integer-valued flows on G. Is it the case that this lattice embeds into the lattice of integer points in Euclidean space in such a way that each unit cube with integer vertices contains a point of the embedded sublattice?
Consider instead an alternating link L, and form the double-cover of the three-sphere branched along L. Is it the case that this space bounds a smooth four-manifold with trivial rational homology groups?
Under the correspondence that takes L to its Tait graph G, we conjecture that the answers to these two questions are the same. I will explain why a positive answer to the second implies a positive answer to the first using Floer homology. I will then explain why a positive answer to the first implies a positive answer to the second under the added hypothesis that each unit cube contains a *unique* point of the embedded sublattice.
This is joint work, the forward direction with Slaven Jabuka and the partial reverse direction with Brendan Owens.
May. 12: May be cancelled
Fall 2022 Schedule:
Sept 15 (Thursday 3:00-4:00 pm at Science Center Hall B): Zhenkun Li (Stanford University)
Sept 23: Mike Miller Eismeier (Columbia University)
Sept 30: Ian Zemke (Princeton University)
Oct 7: Inanc Baykur (UMass Amherst / Harvard)
Oct 14: Boyu Zhang (University of Maryland)
Oct 21: cancelled
Oct 28: cancelled
Nov 4: Jonathan Hanselman (Princeton University)
Nov 11: Olga Plamenevskaya (Stony Brook University)
Nov 18: Juan Muñoz-Echániz (Columbia University)
Nov 25: cancelled because of Thanksgiving
Abstracts:
Sept 15:
Surgery formulae in instanton theory (Zhenkun Li)
Abstract: Instanton homology was first introduced by Floer in 1980s. It has many important applications in the study of 3-manifolds and knots, especially in studying the SU(2) representations of fundamental groups. It is conjectured by Kronheimer and Mrowka that the framed instanton Floer homology of a 3-manifold is isomorphic to the hat version of Heegaard Floer homology, but not much is known beyond families of computational examples. In this talk, I will present some surgery formulae in instanton theory, which helps us understand the framed instanton Floer homology of Dehn surgeries on knots. These surgery formulae provide some important structural properties for instanton theory and enable us to compute the framed instanton Floer homology of many new families of 3-manifolds that come from Dehn surgeries and splicings. This is a join work with Fan Ye.
Sept 23:
Instantons mod 2 and indefinite 4-manifolds (Mike Miller Eismeier)
Abstract: This talk is on work in preparation with Ali Daemi.
I will explain why Kim Froyshov's mod 2 instanton invariant q_3 gives information about indefinite 4-manifolds: if H_1(W;Z/2) = 0 and W has boundary Y, then -b^+(W) <= q_3(Y) <= b^-(W). This is the first invariant known to enjoy comparable bounds for indefinite manifolds W.
The key observation is that even when b^+(W) > 0, one can define a Donaldson invariant in the (tilde) instanton homology of boundary(W) --- not in the usual instanton tilde complex, but rather a "suspension". This suspension process accounts for the role of obstructed gluing theory, and does not destroy information about q_3 (but does destroy information about all other types of h-invariant).
As a corollary, we show that there exist integer homology spheres with arbitrary integral surgery number S(Y_n) = n. This answers a question of Dave Auckly. Previously, the state of the art was n=2, and further progress was obstructed by the possibility that the linking matrix be indefinite.
Sept 30:
Bordered aspects of the Heegaard Floer surgery formulas (Ian Zemke)
Abstract: In this talk, we will discuss bordered aspects of the Heegaard Floer surgery formulas of Ozsvath--Szabo and Manolescu--Ozsvath. In particular, we will explain how their theories naturally define bordered invariants for manifolds with toroidal boundary components. Time permitting, we will discuss applications of the theory. One application is a proof of the equivalence of lattice homology and Heegaard Floer homology. Another application is a description of the link Floer complexes of algebraic links, which is parallel to Ozsvath and Szabo's description of the knot Floer complex of an L-space link. The latter application is joint with M. Borodzik and B. Liu.
Oct 7:
Exotic 4-manifolds with signature zero (Inanc Baykur)
Abstract: We will talk about our recent construction of the smallest closed exotic 4-manifolds with signature zero known to date. Our novel examples are derived from fairly special small Lefschetz fibrations we build, with spin and non-spin monodromies. This is joint work with N. Hamada.
Oct 14:
The existence of irreducible SU(2) representations of link groups (Boyu Zhang)
Abstract:Representations of 3-manifold groups into groups such as SU(2) and SL(2,C) have been actively studied for decades. Many topological invariants are defined by considering these representations, such as the Casson invariant, the Casson-Lin invariant, and the A polynomial. In 2010, Kronheimer-Mrowka showed that the fundamental group of every non-trivial knot in S^3 admits an irreducible representation in SU(2) such that the image of the meridian is traceless, which answered a conjecture of Cooper. In this talk, I will present a result that generalizes Kronheimer-Mrowka’s theorem to the case of links. We show that for every link L that is not the unknot, the Hopf link, or a connected sum of Hops links, its fundamental group admits an irreducible SU(2) representation such that the image of every meridian is traceless. The proof is based on an excision formula of singular instanton Floer homology. This is joint work with Yi Xie.
Oct 21: No seminar
Oct 28: No seminar
Nov 4:
Title: Immersed curve invariants for knot complements (Jonathan Hanselman)
Abstract: Bordered Floer homology is an extension of Heegaard Floer homology to manifolds with parametrized boundary, and in the case of manifolds with torus boundary knot Floer homology gives another such extension. In earlier joint work with J. Rasmussen and L. Watson, it was shown that in this setting the bordered Floer invariant, which is equivalent to the UV=0 truncation of the knot Floer complex, can be encoded geometrically as a collection of immersed curves in the punctured torus and a pairing theorem recovers HF-hat (the simplest version of Heegaard Floer homology) of a glued manifold via Floer homology of immersed curves. In this talk, we will survey some applications of this result and then discuss a generalization that encodes the full knot Floer complex of a knot as a collection of decorated immersed curves in the torus. When two manifolds with torus boundary are glued, a pairing theorem computes HF^- of the resulting manifold as the Floer homology of certain immersed curves associated with each side. We remark that the curves we describe are invariants of knots, but we expect they are in fact invariants of the knot complements; if this is true, they may be viewed as defining a minus type bordered Floer invariant for manifolds with torus boundary.
Nov 11:
Title: Surface singularities, unexpected fillings, and line arrangements (Olga Plamenevskaya)
Abstract: A link of an isolated complex surface singularity (X, 0) is a 3-manifold Y which is the boundary of the intersection of X with a small ball centered at 0. Smoothings of the singularity give non-singular 4-manifolds, the Milnor fibers, with the same boundary Y. The Milnor fibers carry symplectic (even Stein) structures, and thus provide fillings of the canonical contact structure on Y; another Stein filling comes from the minimal resolution of (X, 0). An important question is whether all Stein fillings of the link come from this algebraic construction: this is true in some simple cases such as lens spaces. However, even in the "next simplest" case, for many rational singularities, we are able to construct "unexpected" Stein fillings that do not arise from Milnor fibers. To this end, we encode Stein fillings via curve arrangements, motivated by T.de Jong-D.van Straten's description of smoothings of certain rational surface singularities in terms of deformations of associated singular plane curves. We then use classical projective geometry to construct unexpected line arrangements and unexpected fillings. This is a topological story, with minimal input from algebraic geometry. Joint work with L. Starkston.
Nov 18:
Title: Monopoles and families of contact structures (Juan Muñoz-Echániz)
Abstract:
Beyond the tight/overtwisted dichotomy, 3-dimensional contact topology would appear to be dominated by flexibility: a central result of Eliashberg and Mishachev says that the contactomorphism group of the standard contact 3-ball has the homotopy type of the diffeomorphism group. In contrast with this, I will discuss how monopole Floer homology imposes constraints on the behaviour of families of contact structures on 3-manifolds. Applications include detecting exotic contactomorphisms given by certain "Dehn twists" on embedded spheres.