This is the website for the weekly Geometry & Topology seminar at MIT. To find more seminars in the Boston area, please find in the shared google calendar. The seminar meets 3:00-4:00 pm, Mondays at 2-449. Fall 2018
**Dec. 10 Genus two Lefschetz fibrations via lantern substitutions** By Anar Akhmedov Abstract: Lefschetz fibrations play very important role in 4-manifold topology. It was shown by Donaldson that, perhaps after some blow-ups, any closed symplectic 4-manifold admit a Lefschetz fibration over two sphere. Conversely, Gompf showed that the total space of a Lefschetz fibration admits a symplectic structure, provided the fibers are non-trivial in homology, generalizing an earlier work of Thurston. In this talk, we will construct a family of genus two Lefschetz fibrations X(n) over two sphere by applying the lantern substitutions to the twisted fiber sums of Matsumoto's genus two Lefschetz fibration. We will compute the fundamental group of X(n) and show that it is isomorphic to the trivial group if n =-3, -1, Z if n=-2, and finite cyclic group Z_|n+2| for all integers n not equal to -3,-2,-1. We also show that the total spaces of these Lefschetz fibrations are symplectically minimal and have the symplectic Kodaira dimension equal to 2. We will also discuss very recent result on indecomposable minimal genus two Lefschetz fibrations. These are joint works with Naoyuki Monden.**Dec. 10 Stable homotopy theory in symplectic geometry** By Thomas Kragh Abstract: I will go through some of the ideas and consequences in applying stable homotopy theory to represent Floer homology. If time permits I will discus how it relates to: Algebraic K-theory of spaces, generating families and/or micro-local sheaf theory.**Dec. 3 Essential Tori in Spaces Of Symplectic Embeddings**by Julian Chaidez Abstract: The problem of when and how one symplectic manifold can be symplectically embedded into another is notoriously subtle, even when the spaces in question are relatively simple. Gromov's non-squeezing theorem and McDuff-Schlenk's Fibonacci staircase are examples of this phenomenon. One can interpret these results as realizing the principle that "variations of quantitative symplectic parameters alter the topology of symplectic embedding spaces". In this talk, we explain recent work (joint with Mihai Munteanu) showing that certain n-torus families of symplectic embeddings between 2n-d ellipsoids become homologically essential if certain quantitative invariants are close enough. We will also discuss work in progress in which we use similar methods to study Lagrangian embeddings.
**Nov. 26****Twisted complexes and Floer theory for Lagrangian cobordisms.** By Baptiste Chantraine Abstract: In a recent work joint with G. Dimitroglou-Rizell, P. Ghiggini and R. Golovko we proved that the Lagrangian cocores generate the wrapped Fukaya category of a Weinstein manifold. The proof uses a result which allows to describe some twisted complexes in this category by geometrical objects. This result of independent interest uses a Floer theory for Lagrangian cobordisms that we have previously developed. In the talk I will explain this result, explain how it relates to some previous works of many other authors and describe how it is applied to prove our generation criterion.**Nov. 19****A simplicial construction of G-equivariant Floer homology** by Kristen Hendricks Abstract: For G a Lie group acting on a symplectic manifold and preserving a pair of Lagrangians, we use techniques from infinity category theory to construct a G-equivariant Floer homology of L0 and L1 without equivariant transversality. We give a sample application to symplectic Khovanov homology. This is joint work with R. Lipshitz and S. Sarkar.**Nov. 5 Foam evaluation for link homology** By Mikhail Khovanov Abstract: A formula of Louis-Hadrien Robert and Emmanuel Wagner allows to evaluate foams to symmetric polynomials. Foams are generic combinatorial 2-dimensional CW-complexes embedded in the 3-space. Their construction has applications to link homology. The latter are bigraded homological invariants of links with the HOMFLYPT polynomials as the Euler characteristics. The talk will center on a variation of the Robert-Wagner construction that leads to a combinatorial homology theory for planar trivalent graphs. This homology theory is closely related to the Kronheimer-Mrowka homology theory for embedded graphs in the 3-space and 3-manifolds that comes from the gauge theory for orbifolds. The talk is based on a joint work with L.-H. Robert.**Oct. 29 The geography problem on 4-manifolds: 10/8 + 4**By Zhouli Xu and Jianfeng Lin**Oct. 29****The Pin(2)-Equivariant Mahowald Invariant** By Zhouli Xu and Jianfeng Lin Abstract: The existence of Pin(2)-equivariant stable maps between representation spheres has deep applications in 4-dimensional topology. We will sketch the proof for our main result on the Pin(2)-equivariant Mahowald invariants. More specifically, we will discuss the Pin(2)-equivariant Mahowald invariants of powers of certain Euler classes in the RO(Pin(2))-graded equivariant stable homotopy groups of spheres. The proof analyzes maps between certain finite spectra arising from BPin(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell-diagrams, known results on the stable homotopy groups of spheres, and the j-based Atiyah-Hirzebruch spectral sequence. This is joint work with Mike Hopkins and XiaoLin Danny Shi. This talk provides the homotopy theory required by the first talk but will not depend upon it.**Oct. 22****Equivariant Floer cohomology and Steenrod squares** By Tim Large Abstract: Given a Z/2-action on a symplectic manifold, one can ask whether there is a Floer-theoretic analogue of Quillen’s “localisation isomorphism” relating the equivariant cohomology to the ordinary cohomology of the fixed point set. As noticed by Seidel-Smith, there are topological obstructions to this coming from the normal bundle to the fixed point set. In this talk we will show how to construct a “twisted” Floer cohomology of the fixed point set, which is then the target for the localisation map. One could then use product structures on Floer cohomology to construct Steenrod-like operations, and compare those to the Steenrod squares on a Floer homotopy type: we will show they coincide.**Oct. 15 Instantons and annular Khovanov homology**By Yi Xie Abstract: The annular Khovanov homology is an invariant for links in a thickened annulus, which generalizes the original Khovanov homology defined for links in a three-sphere. It is a special case of the theory developed by Asaeda, Przytycki and Sikora which works for links in any thickened surface. In this talk, I will introduce an analogue of the annular Khovanov homology using singular instanton Floer theory, called the annular instanton Floer homology. It is related to the annular Khovanov homology by a spectral sequence. As an application of this spectral sequence, I will prove that the annular Khovanov homology detects the unlink in the thickened annulus (assuming all the components are null-homologous). Another application is a new proof of Grigsby and Ni’s result that tangle Khovanov homology distinguishes braids from other tangles.**Oct.1 Involutive Floer Homology and Applications to the Homology Cobordism Group** By Irving Dai Abstract: In this talk, we discuss some recent applications of involutive Heegaard Floer homology (defined by Hendricks and Manolescu) to the homology cobordism group. We establish some non-torsion results and show that the homology cobordism group admits an infinite-rank summand. This is joint work with Jennifer Hom, Matthew Stoffregen, and Linh Truong.**Sep. 24 Equivariant instanton homology and group cohomology** By Mike Miller Abstract: Floer's celebrated instanton homology groups are defined for integer homology spheres, but analagous groups in Heegaard Floer and Monopole Floer homology theories are defined for all 3-manifolds; these latter groups furthermore come in four flavors, and carry extra algebraic structure. Any attempt to extend instanton homology to a larger class of 3-manifolds must be somehow equivariant - respecting a certain SO(3)-action. We explain how ideas from group cohomology and equivariant algebraic topology allow us to define four flavors of instanton homology for rational homology spheres, and how these invariants relate to existing instanton homology theories.**Sep.17: Gauged linear sigma model in geometric phases** By Guangbo Xu Abstract: In 1993 Witten introduced the theory of gauged linear sigma model (GLSM) which provides a way of linearizing Gromov--Witten theory (and all theories of pseudoholomorphic curves). In my talk I will give a brief introduction of Witten's idea and my recent work with Gang Tian, in which we construct a cohomological field theory associated to GLSM in so-called geometric phases.**Sep.10 : Bordered Heegaard Floer homology and immersed curves in the torus** By Jonathan Hanselman Abstract: To a 3-manifold with torus boundary, we can associate an element of the Fukaya category of the punctured torus—that is, a collection of immersed curves in the torus, decorated with local systems—such that when when two such manifolds are glued the Heegaard Floer homology of resulting 3-manifold is recovered from Floer homology of the corresponding curves. These curves are derived from, and indeed equivalent to, the bordered Heegaard Floer defined by Lipshitz, Ozsvath, and Thurston, but their geometric nature makes them more user friendly. We will discuss some properties of these curves and some applications, including invariance of Heegaard Floer homology under genus one mutation and a rank inequality for Heegaard Floer homology of toroidal manifolds. This is joint work with J. Rasmussen and L. Watson.
Abstract: A fundamental problem in 4-dimensional topology is the following geography question: "which simply connected topological 4-manifolds admit a smooth structure?" After the celebrated work of Kirby-Siebenmann, Freedman, and Donaldson, the last uncharted territory of this geography question is the "11/8-Conjecture''. This conjecture, proposed by Matsumoto, states that for any smooth spin 4-manifold, the ratio of its second-Betti number and signature is least 11/8. Furuta proved the ''10/8+2''-Theorem by studying the existence of certain Pin(2)-equivariant stable maps between representation spheres. In this talk, we will present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants. In particular, we improve Furuta's result into a ''10/8+4''-Theorem. Furthermore, we show that within the current existing framework, this is the limit. This is joint work with Mike Hopkins and XiaoLin Danny Shi. |