Differential geometry and symplectic topology seminar (Spring 2024)

Abstracts

• Joseph Breen

  • A fundamental result in 3-dimensional contact topology is the Giroux correspondence between contact structures and open book decompositions. Until recently, the techniques that Giroux used in dimension 3 were not available in higher dimensions; indeed, for reasons I will discuss, contact topology is much more tractable in dimension 3. However, there have been several efforts to improve the situation. In this talk, I will describe joint work with Ko Honda and Yang Huang on extending the Giroux correspondence and its underlying techniques to all dimensions.

• Ko Honda (Tuesday talk)

  • Hecke algebras are ubiquitous in number theory and geometric representation theory. In this talk we describe the appearance of various Hecke algebras such as the affine Hecke algebra and the double affine Hecke algebra (DAHA) in Floer theory, through the higher-dimensional analog of Heegaard Floer homology. This is joint work with Yin Tian and Tianyu Yuan. (This talk is independent of Talks #2.)

• Ko Honda (Friday talk)

  • Around twenty years ago Emmanuel Giroux formulated the equivalence of contact structures and open book decompositions with Weinstein pages up to stabilization. We establish the Giroux correspondence in full generality using the recent developments in convex hypersurface theory, described in the second talk. Time permitting, we also explain their relationship with Lefschetz fibrations for Weinstein domains. This is joint work with Joe Breen and Yang Huang. (This talk is independent of Talks #1.)

• Hao Zhuang

  • In this talk, we will first introduce an invariant Morse-Bott-Smale chain complex for closed $T^l$-manifolds with a special type of $T^l$-invariant Morse-Bott functions. Then, we will establish a chain complex isomorphism between our $T^l$-invariant Morse-Bott-Smale complex and the Witten instanton complex. Finally, if time permits, we will mention the estimates of eigenvalues of the Witten Laplacian, which motivates one of our ongoing projects.

• Roman Krutowski

  • I will define Heegaard Floer symplectic cohomology (HFSH), a novel invariant which serves as a closed string analog of the higher-dimensional Heegaard Floer homology. One may view it as a Floer invariant associated with a problem of Hamiltonian motion of multiple identical particles. This invariant can also be regarded as a deformation of a k-th symmetric version of symplectic cohomology, given by counting curves of higher genus. I will also introduce a multiloop Morse complex of a manifold, which is a Morse-theoretic counterpart of HFSH. At last, I will show that HFSH of a cotangent bundle T^*M is isomorphic to the cohomology of the multiloop complex of M. This result generalizes Viterbo’s isomorphism theorem to the setting of multiple particles.
• Khanh Le
  • Arithmetic subgroups of PSL(2,C) provide a large class of hyperbolic 3-manifolds exhibiting a lot of interactions between geometry, topology and number theory. In this talk, I will give examples of arithmetic hyperbolic manifolds in dimension 2 and 3 with the goal to illustrate the connections between geometrical, topological and number theoretical aspects of these manifolds. I will discuss various invariants and techniques that are useful in distinguishing arithmetic and non-arithmetic manifolds. Finally, I will discuss some observations related to the question of arithmeticity of modular knots.
• Amanda Hirschi
  • In recent work with Luya Wang, we showed that there exist counterexamples to one implication of the Donalson 4-6 question, which inquires about a relation between smooth topology of 4-manifolds and the symplectic geometry of the stabilised 6-manifold. In this talk I will discuss upcoming work with Wang on the other implication of the question. Specifically, I will explain how one can show that the Gromov-Witten (and thus Seiberg-Witten) invariants of two symplectic 4-manifolds agree if their stabilisations are deformation equivalent.
• Hang Yuan
  • The Strominger-Yau-Zaslow (SYZ) conjecture proposes a geometric framework that underlies mirror symmetry for Calabi-Yau manifolds. However, it has been mysterious to define ‘dual’ torus fibrations and comprehend singular fibers within this conjecture. In my talk, I will begin by providing an overview of integrable systems in both symplectic and non-archimedean contexts and propose a toy model of SYZ conjecture. Then, I will explore how to globalize this toy model and establish a mathematically precise statement for the SYZ duality. If time allows, I will also present some concrete examples, such as the conifold, and A_n singularities.
• Nick Meyer
  • In the past 35+ years, many mathematicians have used a variety of cut-and-paste techniques, such as Fintushel-Stern knot surgery, log transforms (aka “torus surgery”), and fiber sums to build a zoo of examples of exotic, simply connected 4-manifolds. In this talk, I show how to construct (and obstruct) non-simply connected (potential) exotica by doing log transforms along $S^1\times K$ in $S^1 \times Y$, where $Y$ is a closed, connected, oriented 3-manifold and $K$ is a fibered knot in $Y$. In particular, I show that for any diffeomorphism $f$ of the 3-torus, the manifold $X_{K, f}$ obtained by doing an f-log transform along $S^1 \times K$ in $S^1\times Y$ is a 3-manifold bundle over the circle. Time permitting, I will also show how to generalize this statement for various other families of gluings.
• Caglar Uyanik
  • Cannon and Thurston showed that a hyperbolic 3-manifold that fibers over the circle gives rise to a sphere filling curve. The universal cover of the fiber surface is quasi-isometric to the hyperbolic plane, whose boundary is a circle, and the universal cover of the 3-manifold is 3-dimensional hyperbolic space, whose boundary is the 2-sphere. Cannon and Thurston showed that the inclusion map between the universal covers extends to a continuous map between their boundaries, whose image is dense. In particular, any measure on the circle pushes forward to a measure on the 2-sphere using this map. We compare several natural measures coming from this construction. This is joint work with Gadre, Haettel, Maher and Pfaff.
• Katherine Maxwell
  • The super Mumford form is a section over the moduli space of super Riemann surfaces, characterized by invariance under the action of the Neveu-Schwarz algebra action. In light of difficulties in performing integrals in superstring theory arising from the super Mumford form, it was suggested in the 80s that the relationship of the moduli space of super Riemann surfaces to the super Sato Grassmannian may be fruitful. Based on joint work with A. Voronov, I will discuss possible approaches to extending the super Mumford form, including our results on the proposed formula by A. Schwarz.
• Shuo Zhang
  • Seidel conjectured that the fix-point-Floer homology of the global monodromy (iterated Dehn twists) of a Lefschetz fibration can be computed from the Lagrangian Floer homologies of the vanishing cycles. In particular there should be a long exact sequence relating them. We construct this long exact sequence by iterating Mak-Wu’s construction of a Lagrangian cobordism of a Lagrangian correspondence associated. Potential applications include homological mirror symmetry for Fano manifolds and dynamics of iterated Dehn twists.
• Mikio Furuta
  • To construct a finite dimensional model for Dirac operator, one method is to use finite direct sum of the eigenspaces. This method is effectively used in some aspects of gauge theory. Another method is the Finite Element Method. In this latter case the naive approximation cannot capture the information of the index of the original Dirac operator. In this talk we explain that if we add “Wilson term” to the naive approximation, then it captures the information, at least for the regular lattice approximation of torus in any dimension. For the integer valued index this was known in physics and mathematically established by D.H. Adams around 2000. We give a new approach which is valid for mod 2 index, family index and equivariant index. Joint work with Hidenori Fukaya, Shinichiroh Matsuo, Tetsuya Onogi, Satoshi Yamaguchi and Mayuko Yamashita.