Related papers: The contact cut graph and a Weinstein $\mathcal{L}…
This paper introduces the geometry of the open string Floer theory of gauged Landau-Ginzburg model via gauged Witten equations. Given a $G$-invariant Morse-Bott holomorphic function $W$ on a Hamiltonian space $(M,\omega,G),$ Lefschetz…
In this paper, we systematically study the $m$-dimensional sectional category of a fibration, introduced by Schwarz, as an approximating invariant for the sectional category. We develop the basic theory of this invariant, establish its…
We show that positive $S^1$-equivariant symplectic homology is a contact invariant for a subclass of contact manifolds which are boundaries of Liouville domains. In nice cases, when the set of Conley-Zehnder indices of all good periodic…
We develop the Lefschetz fixed-point theory for noncompact manifolds of bounded geometry and uniformly continuous maps. Specifically, we define the uniform Lefschetz class $\mathscr{L}(f)$ of a uniformly continuous map $f\colon M\to M$ of a…
Contact manifolds are odd-dimensional smooth manifolds endowed with a maximally non-integrable field of hyperplanes. They are intimately related to symplectic manifolds, i.e. even-dimensional smooth manifolds endowed with a closed…
The first two authors showed in~\cite{AM1} how the Conley-Zehnder index of any contractible periodic Reeb orbit of a non-degenerate toric contact form on a good toric contact manifold with zero first Chern class, i.e. a Gorenstein toric…
We introduce hyperelliptic simplified (more generally, directed) broken Lefschetz fibrations, which is a generalization of hyperelliptic Lefschetz fibrations. We construct involutions on the total spaces of such fibrations of genus $g\geq…
For a given pair of maps f,g:X->M from an arbitrary topological space to an n-manifold, the Lefschetz homomorphism is a certain graded homomorphism L:H(X)->H(M) of degree (-n). We prove a Lefschetz-type coincidence theorem: if the Lefschetz…
Fintushel and Stern have proved that if S \subset X is a symplectic surface in a symplectic 4-manifold such that S has simply-connected complement and nonnegative self-intersection, then there are infinitely many topologically equivalent…
The Thurston-Bennequin invariant provides one notion of self-linking for any homologically-trivial Legendrian curve in a contact three-manifold. Here we discuss related analytic notions of self-linking for Legendrian knots in Euclidean…
Lipshitz, Ozsv\'ath and Thurston defined a bordered Heegaard Floer invariant CFDA for 3-manifolds with two boundary components, including mapping cylinders for surface diffeomorphisms. We define a related invariant for certain 4-dimensional…
The 2+1 dimensional lattice models of Levin and Wen [PRB 71, 045110 (2005)] provide the most general known microscopic construction of topological phases of matter. Based heavily on the mathematical structure of category theory, many of the…
Feynman diagrams in $\phi^4$ theory have as their underlying structure 4-regular graphs. In particular, any 4-point $\phi^4$ graph can be uniquely derived from a 4-regular graph by deleting a vertex. The Feynman period is a simplified…
We introduce a direct generalization of the Weinstein conjecture to closed, Lichnerowicz exact, locally conformally symplectic manifolds, (for short $\lcs$ manifolds). This conjectures existence of certain 2-curves in the manifold, which we…
We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured monopole Floer homology theory (SHM). Our invariant can be viewed as a generalization of Kronheimer and Mrowka's contact invariant for…
A Riemann-Cartan manifold is a Riemannian manifold endowed with an affine connection which is compatible with the metric tensor. This affine connection is not necessarily torsion free. Under the assumption that the manifold is a homogeneous…
We give invariants of pairs $(M,L)$ consisting of a closed connected oriented three-manifold and an (oriented) framed link $L$ embedded in $M$. This invariant generalizes the Kuperberg and Hennings-Kauffman-Radford (HKR) invariants of…
The L-intersection graphs are the graphs that have a representation as intersection graphs of axis parallel shapes in the plane. A subfamily of these graphs are {L, |, --}-contact graphs which are the contact graphs of axis parallel L, |,…
In the literature, there are two different versions of Hard Lefschetz theorems for a compact Sasakian manifold. The first version, due to Kacimi-Alaoui, asserts that the basic cohomology of a compact Sasakian manifold satisfies the…
The logarithm of the Kontsevich-Kuperberg-Thurston invariant counts embeddings of connected trivalent graphs in an oriented rational homology sphere, using integrals on configuration spaces of points in the given manifold. It is a universal…