Our aim here is to explain a new technique for the construction of a smooth Kuranishi structure, which we used in our previous article about the construction of symplectic field theory. We explain this technique in the case of a Morse chain complex.

Using the bulk deformation of Floer cohomology by Schubert classes and non-Archimedean analysis of Fukaya–Oh–Ohta–Ono’s bulk-deformed potential function, we prove that every complete flag manifold $\mathrm{Fl}(n)$ ($n\ge 3$) with a monotone Kirillov–Kostant–Souriau (KKS) symplectic form carries a continuum of nondisplaceable Lagrangian tori which degenerates to a nontorus fiber in the Hausdorff limit. In particular, the Lagrangian $S^3$-fiber in $\mathrm{Fl}(3)$ is nondisplaceable, answering a question raised by Nohara and Ueda who computed its Floer cohomology to be vanishing.

We prove two refinements of the $C^{\mathrm{\infty }}$-closing lemma for 3-dimensional Reeb flows, which was proved by the author as an application of spectral invariants of Embedded Contact Homology (ECH). Specifically, we prove the following two results: (i) for a $C^{\mathrm{\infty }}$-generic contact form on any closed 3-manifold, the union of periodic Reeb orbits representing ECH homology classes is dense; (ii) a certain real-analytic version of the $C^{\mathrm{\infty }}$-closing lemma for 3-dimensional Reeb flows. A few questions and conjectures related to these results are also discussed.

We show that a stabilized convex symplectic (or Liouville) manifold with the homotopy type of a half-dimensional CW-complex is symplectomorphic to a flexible Weinstein manifold.

We first recall Solomon’s relations for Welschinger invariants counting real curves in real symplectic fourfolds and the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV)-style relations for Welschinger invariants counting real curves in real symplectic sixfolds with some symmetry. We then explicitly demonstrate that, in some important cases (projective spaces with standard conjugations, real blowups of the projective plane, and two- and threefold products of the one-dimensional projective space with two involutions each), these relations provide complete recursions determining all Welschinger invariants from basic input. We include extensive tables of Welschinger invariants in low degrees obtained from these recursions with Mathematica. These invariants provide lower bounds for counts of real rational curves, including with curve insertions in smooth algebraic threefolds.

We prove a recent conjecture by Gyenge, Némethi, and Szendrői giving a formula of the generating function of Euler numbers of Hilbert schemes of points ${\mathrm{Hilb}}^N({\mathbb{C}}^2∕\mathrm{\Gamma })$ on a simple singularity ${\mathbb{C}}^2∕\mathrm{\Gamma }$, where Γ is a finite subgroup of $\mathrm{SL}(2)$. We deduce it from the claim that quantum dimensions of standard modules for the quantum affine algebra associated with Γ at $\mathit{\zeta }=exp(\frac{2\pi \sqrt{-1}}{2(h^{\vee }+1)})$ are always 1, which is a special case of an earlier conjecture by Kuniba. Here $h^{\vee }$ is the dual Coxeter number. We also prove the claim, which was not known for $E_7$, $E_8$ before.

We describe a dictionary of geometry ⟷ algebra in Lagrangian topology. As a by-product, we obtain a tautological (in a particular sense which we explain) proof of a folklore conjecture (sometimes attributed to Kontsevich) claiming that the objects and structure of the derived Fukaya category can be represented through immersed Lagrangians. Our construction is based on certain Lagrangian cobordism categories endowed with a structure called surgery models.

We introduce eight “flavors” of cyclic homology of a mixed complex and study their properties. In particular, we determine their behavior with respect to Chen’s iterated integrals.