We define genus zero open Gromov-Witten invariants with boundary and interior constraints for a Lagrangian submanifold of arbitrary even dimension. The definition relies on constructing a canonical family of bounding cochains that satisfy the point-like condition of the second author and Tukachinsky. Since the Lagrangian is even dimensional, the parameter of the family is odd. Thus, to avoid the vanishing of invariants with more than one boundary constraint, the parameter must be non-commutative. The invariants are defined either when the Lagrangian is a rational cohomology sphere or when the Lagrangian is fixed by an anti-symplectic involution, has dimension $2$ modulo $4$, and its cohomology is that of a sphere aside from degree $1$ modulo $4$. In dimension $2$, these invariants recover Welschinger's invariants. We develop an obstruction theory for the existence and uniqueness of bounding cochains in a Fukaya $A_\infty$ algebra with non-commutative coefficients. The obstruction classes belong to twisted cohomology groups of the Lagrangian instead of the de Rham cohomology of the commutative setting. A spectral sequence is constructed to compute the twisted cohomology groups. The extension of scalars of an $A_\infty$ algebra by a non-commutative ring is treated in detail. A theory of pseudo-completeness is introduced to guarantee the convergence of the Maurer-Cartan equation, which defines bounding cochains, even though the non-commutative parameter is given zero filtration.

We define quantum deformations of Adams operations in $K$-theory, in the framework of quasimap quantum $K$-theory. They provide $K$-theoretic analogs of the quantum Steenrod operations from equivariant symplectic Gromov--Witten theory. We verify the compatibility of these operations with the Kahler and equivariant $q$-difference module structures, provide sample computations via $\mathbb{Z}/k$-equivariant localization, and identify them with $p$-curvature operators of the Kahler $q$-difference connections as studied in Koroteev-Smirnov. We also formulate and verify a $K$-theoretic quantum Hikita conjecture at roots of unity, and propose an indirect algebro-geometric definition of quantum Steenrod operations

In this note we review the concept of phase space in classical field theory, discussing several variations on the basic notion, as well as the relation between them. In particular we will focus on the case where the field theory admits local (gauge) symmetry, in which case the physical phase space of the system emerges after a (usually singular) quotient with respect to the action of the symmetry group. We will highlight the symplectic and Poisson underpinnings of the reduction procedure that defines a phase space, and discuss how one can replace quotients with graded smooth objects within classical field theory via cohomological resolutions, a practice that goes under the name of Batalin--Vilkovisky formalism. Special attention is placed on the reduction and resolution of gauge theories on manifolds with corners, which famously depend on a number of arbitrary choices. We phrase these choices in terms of homotopies for the variational bicomplex, and define a homotopy version of Noether's current and conservation theorem.

Reflexive dg categories were introduced by Kuznetsov and Shinder to abstract the duality between bounded and perfect derived categories. In particular this duality relates their Hochschild cohomologies, autoequivalence groups, and semiorthogonal decompositions. We establish reflexivity in a variety of settings including affine schemes, simple-minded collections, chain and cochain dg algebras of topological spaces, Ginzburg dg algebras, and Fukaya categories of cotangent bundles and surfaces as well as the closely related class of graded gentle algebras. Our proofs are based on the interplay of reflexivity with gluings, derived completions, and Koszul duality. In particular we show that for certain (co)connective dg algebras, reflexivity is equivalent to derived completeness.

Moment polytopes of tensors, the study of which is deeply rooted in invariant theory, representation theory and symplectic geometry, have found relevance in numerous places, from quantum information (entanglement polytopes) and algebraic complexity theory (GCT program and the complexity of matrix multiplication) to optimization (scaling algorithms). Towards an open problem in algebraic complexity theory, we prove separations between the moment polytopes of matrix multiplication tensors and unit tensors. As a consequence, we find that matrix multiplication moment polytopes are not maximal, i.e. are strictly contained in the corresponding Kronecker polytope. As another consequence, we obtain a no-go result for a natural operational characterization of moment polytope inclusion in terms of asymptotic restriction. We generalize the separation and non-maximality to moment polytopes of iterated matrix multiplication tensors. Our result implies that tensor networks where multipartite entanglement structures beyond two-party entanglement are allowed can go beyond projected entangled-pair states (PEPS) in terms of expressivity. Our proof characterizes membership of uniform points in moment polytopes of tensors, and establishes a connection to polynomial multiplication tensors via the minrank of matrix subspaces. As a result of independent interest, we extend these techniques to obtain a new proof of the optimal border subrank bound for matrix multiplication.

In this note, we prove that every closed connected oriented odd-dimensional manifold admits a map of non-zero degree (i.e., a domination) from a tight contact manifold of the same dimension. This provides an odd-dimensional counterpart of a symplectic domination result due to Joel Fine and Dmitri Panov. We prove that the dominating contact manifold can be ensured to be Liouville-fillable, but not Weinstein-fillable in general. We discuss an application for contact divisors arising as zero sets of asymptotically contact-holomorphic sections.

We outline a proposal for a $2$-category $\mathrm{Fuet}_M$ associated to a hyperkähler manifold $M$, which categorifies the subcategory of the Fukaya category of $M$ generated by complex Lagrangians. Morphisms in this $2$-category are formally the Fukaya--Seidel categories of holomorphic symplectic action functionals. As such, $\mathrm{Fuet}_M$ is based on counting maps to $M$ satisfying the Fueter equation with boundary values on holomorphic Lagrangians. We make the first step towards constructing this category by establishing some basic analytic results about Fueter maps, such as the energy bound and maximum principle. When $M=T^*X$ is the cotangent bundle of a Kähler manifold $X$ and $(L_0, L_1)$ are the zero section and the graph of the differential of a holomorphic function $F: X \to \mathbb{C}$, we prove that all Fueter maps correspond to the complex gradient trajectories of $F$ in $X$, which relates our proposal to the Fukaya--Seidel category of $F$. This is a complexification of Floer's theorem on pseudo-holomorphic strips in cotangent bundles. Throughout the paper, we suggest problems and research directions for analysts and geometers that may be interested in the subject.

The purpose of this paper is to investigate the definition of symplectic structure on a smooth stratified pseudomanifold in the framework of local $\C^{\infty}$-ringed space theory. We introduce a sheaf-theoretic definition of symplectic form and cohomologically symplectic structure on smooth stratified pseudomanifolds. In particular, we give an indirect definition of symplectic form on the quotient space of a smooth $G$-stratified pseudomanifold. Based on the structure theorem of singular symplectic quotients by Sjamaar--Lerman, we show that the singular reduced space $M_{0}=\mu^{-1}(0)/G$ of a symplectic Hamiltonian $G$-manifold $(M,\omega,G,\mu)$ admits a natural (indirect) symplectic form and a unique cohomologically symplectic structure.

We prove a version of homological mirror symmetry statement for toric Calabi-Yau $3$-orbifolds, thus extending arXiv:1604.06448 to the case of orbifolds under the mirror symmetry setting considered in arXiv:1604.07123. The B-model is the matrix factorization category for the toric Calabi-Yau $3$-orbifold with a superpotential; while the A-model is a topologically defined Fukaya-type category on its mirror curve.

We prove that any symplectic matrix can be factored into no more than 5 unit triangular symplectic matrices, moreover, 5 is the optimal number. This result improves the existing triangular factorization of symplectic matrices which gives proof of 9 factors. We also show the corresponding improved conclusions for structured subsets of symplectic matrices. This factorization further provides an unconstrained optimization method on $2d$-by-$2d$ real symplectic group (a $2d^2+d$-dimensional Lie group) with $2d^2+3d$ parameters.

Given a closed symplectic manifold $X$, we construct Gromov-Witten-type invariants valued both in (complex) $K$-theory and in any complex-oriented cohomology theory $\mathbb{K}$ which is $K_p(n)$-local for some Morava $K$-theory $K_p(n)$. We show that these invariants satisfy a version of the Kontsevich-Manin axioms, extending Givental and Lee's work for the quantum $K$-theory of complex projective algebraic varieties. In particular, we prove a Gromov-Witten type splitting axiom, and hence define quantum $K$-theory and quantum $\mathbb{K}$-theory as commutative deformations of the corresponding (generalised) cohomology rings of $X$; the definition of the quantum product involves the formal group of the underlying cohomology theory. The key geometric input to these results is a construction of global Kuranishi charts for moduli spaces of stable maps of arbitrary genus to $X$. On the algebraic side, in order to establish a common framework covering both ordinary $K$-theory and $K_p(n)$-local theories, we introduce a formalism of `counting theories' for enumerative invariants on a category of global Kuranishi charts.

Brown and Gordon asked whether the Poisson Dixmier-Moeglin equivalence holds for any complex affine Poisson algebra; that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are not Poisson locally closed. These counterexamples also give rise to counterexamples to the classical (noncommutative) Dixmier-Moeglin equivalence in finite $\operatorname{GK}$ dimension. A weaker version of the Poisson Dixmier-Moeglin equivalence is proven for all complex affine Poisson algebras, from which it follows that the full equivalence holds in Krull dimension three or less. Finally, it is shown that everything, except possibly that rationality implies primitivity, can be done over an arbitrary base field of characteristic zero.

Brown and Gordon asked whether the Poisson Dixmier-Moeglin equivalence holds for any complex affine Poisson algebra; that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are not Poisson locally closed. These counterexamples also give rise to counterexamples to the classical (noncommutative) Dixmier-Moeglin equivalence in finite $\operatorname{GK}$ dimension. A weaker version of the Poisson Dixmier-Moeglin equivalence is proven for all complex affine Poisson algebras, from which it follows that the full equivalence holds in Krull dimension three or less. Finally, it is shown that everything, except possibly that rationality implies primitivity, can be done over an arbitrary base field of characteristic zero.

We initiate a systematic study of convex hypersurface theory and generalize the bypass attachment to arbitrary dimensions. We also introduce a new type of overtwisted object called the overtwisted orange which is middle-dimensional and contractible.

We study the dynamics and symplectic topology of energy hypersurfaces of mechanical Hamiltonians on twisted cotangent bundles. We pay particular attention to periodic orbits, displaceability, stability and the contact type property, and the changes that occur at the Mane critical value c. Our main tool is Rabinowitz Floer homology. We show that it is defined for hypersurfaces that are either stable tame or virtually contact, and it is invariant under under homotopies in these classes. If the configuration space admits a metric of negative curvature, then Rabinowitz Floer homology does not vanish for energy levels k>c and, as a consequence, these level sets are not displaceable. We provide a large class of examples in which Rabinowitz Floer homology is non-zero for energy levels k>c but vanishes for k

We study the symplectic topology of certain K3 surfaces (including the "mirror quartic" and "mirror double plane"), equipped with certain K\"ahler forms. In particular, we prove that the symplectic Torelli group may be infinitely generated, and derive new constraints on Lagrangian tori. The key input, via homological mirror symmetry, is a result of Bayer and Bridgeland on the autoequivalence group of the derived category of an algebraic K3 surface of Picard rank one.

We show that every family of isolated hypersurface singularity with constant Milnor number has constant multiplicity. To achieve this, we endow the A'Campo model of "radius zero" monodromy with a symplectic structure. This new approach allows to generalize a spectral sequence of McLean converging to fixed point Floer homology of iterates of the monodromy to a more general setting which is well suited to study $\mu$-constant families.

This paper discusses the cobordism of Lagrangian submanifolds (in the monotone setting) and structures it as a category that is related in a functorial way to an appropriate (derived) Fukaya category. Are also discussed obstructions to cobordism based on properties of Lagrangian quantum homology, relations to Lagrangian surgery, as well as examples of non-isotopic but cobordant Lagrangians. This is a revision of our earlier preprint from September 2011.

In this paper, we study Higgs and co-Higgs bundles on non-K\"ahler elliptic surfaces. We show, in particular, that non-trivial stable Higgs bundles only exist when the base of the elliptic fibration has genus at least two and use this existence result to give explicit topological conditions ensuring the smoothness of moduli spaces of stable rank-2 sheaves on such surfaces. We also show that non-trivial stable co-Higgs bundles only exist when the base of the elliptic fibration has genus 0, in which case the non-K\"ahler elliptic surface is a Hopf surface. We then given a complete description of non-trivial co-Higgs bundles in the rank 2 case; these non-trivial rank-2 co-Higgs bundles are examples of non-trivial holomorphic Poisson structures on $\mathbb{P}^1$-bundles over Hopf surfaces.