arXiv research

Embeds complex 3-manifolds into symplectic space.

Upper bounds for Legendrian links in tight contact 3-manifolds.

Study $G_2$-metrics from non-integrable special Lagrangian fibrations.

Using Morse-Bott techniques adapted to the gauge-theoretic setting, we show that the limiting boundary values of the space of finite energy monopoles on a connected 3-manifold with at least two cylindrical ends provides an immersed Lagrangian submanifold of the vortex moduli space at infinity. By studying the signed in…

The cohomology ring of certain Lagrangian 3-manifolds is analyzed, influencing quantum Reidemeister torsion and superpotential.

New model for Witten-Reshetikhin-Turaev invariants using Lagrangian intersections.

Functorial properties of knot invariants under specific concordances.

The paper extends Gompf's invariant to all compact 3-manifolds and studies its variations.

In this paper, we study the Seiberg-Witten equations on the product R x Y, where Y is a compact 3-manifold with boundary. Following the approach of Salamon and Wehrheim in the instanton case, we impose Lagrangian boundary conditions for the Seiberg- Witten equations. The resulting equations we obtain constitute a nonli…

We prove the existence of a spectral sequence for Lagrangian Floer homology which converges to the Floer homology of the image of a Lagrangian submanifold under multiple fibred Dehn twists. The $E_1$ term of the sequence is given by the hypercube of "resolutions" of the Dehn twists involved. The proof relies on the exa…

We establish an $h$-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact Lagrangian immersion into standard symplectic 6-space $\R^6_\st$ with exactly on…

New knot homology theory from symplectic geometry.

The paper studies special Lagrangian manifolds using algebraic topology.

We examine the $L^2$-topology of the gauge orbits over a closed Riemann surface. We prove a subtle local slice theorem based on the div-curl Lemma of harmonic analysis, and deduce local pathwise connectedness and local uniform quasiconvexity of the gauge orbits. Using these, we generalize compactness results for anti-s…

We derive some restrictions on the topology of a monotone Lagrangian submanifold $L\subset\mathbf{C}^n$ by making observations about the topology of the moduli space of Maslov 2 holomorphic discs with boundary on $L$ and then using Damian's theorem which gives conditions under which the evaluation map from this moduli …

In a pair of papers, we construct invariants for smooth four-manifolds equipped with `broken fibrations' - the singular Lefschetz fibrations of Auroux, Donaldson and Katzarkov - generalising the Donaldson-Smith invariants for Lefschetz fibrations. The `Lagrangian matching invariants' are designed to be comparable with …

We construct infinitely many Legendrian links in the standard contact $\mathbb{R}^3$ with arbitrarily many topologically distinct Lagrangian fillings. The construction is used to find links in $S^3$ that bound topologically distinct pieces of algebraic curves in $B^4 \subset \mathbb{C}^2$, is applied to find contact 3-…

We explain the effect of applying a full Lutz twist along a pre-Lagrangian torus in a contact 3-manifold, on the contact invariant in Heegaard Floer homology with twisted coefficients.

A map from 3-manifold skein to Lagrangian skein via holomorphic curve counting.

Study shows how tangle moduli spaces relate to boundary surfaces.

We extend Perutz's Lagrangian matching invariants to 3-manifolds which are not necessarily fibred using the technology of holomorphic quilts. We prove an isomorphism of these invariants with Ozsvath-Szabo's Heegaard Floer invariants for certain extremal spin^c structures. As applications, we give new calculations of He…

Proves finiteness and holonomicity of skein modules for 3-manifolds.

Formulates a strategy to prove Atiyah-Floer conjecture.

New approach solves problems in gauge theory and symplectic geometry.

We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play important roles in the interaction between the mapping class group and finite-ty…

In fivebrane compactifications on 3-manifolds, we point out the importance of all flat connections in the proper definition of the effective 3d N=2 theory. The Lagrangians of some theories with the desired properties can be constructed with the help of homological knot invariants that categorify colored Jones polynomia…

Given a closed, connected, oriented 3-manifold with positive first Betti number, one can define an instanton Floer group as well as a quilted Lagrangian Floer group. The quilted Atiyah-Floer conjecture states that these cohomology groups are isomorphic. We initiate a program for proving this conjecture.

We use monopole Floer homology for sutured manifolds to construct invariants of Legendrian knots in a contact 3-manifold. These invariants assign to a knot K in Y elements of the monopole knot homology KHM(-Y,K), and they strongly resemble the knot Floer homology invariants of Lisca, Ozsváth, Stipsicz, and Szabó. We pr…

We construct a 2+1 dimensional classical gauge theory on manifolds with spin structure whose action is a refinement of the Atiyah-Patodi- Singer eta-invariant for twisted Dirac operators. We investigate the properties of the Lagrangian field theory for closed, spun 3-manifolds and compact, spun 3-manifolds with boundar…

Develops Floer theory for 3-manifold covers using equivariant structures.

This article introduces a universal moduli space for the set whose archetypal element is a pair that consists of a metric and second fundamental form from a compact, oriented, positive genus minimal surface in some hyperbolic 3-manifold. This moduli space is a smooth, finite dimensional manifold with canonical maps to …

Compactness theorem for Fueter sections yields non-zero harmonic 1-forms.

Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed, oriented, real analytic Riemannian 4-manifold whose bundle of self-dual 2-forms is trivi…

A sheaf-theoretic model connects SL(2,C) Floer homology to 3-manifold invariants.

We study the natural structure on the moduli space of deformations of compact coassociative submanifolds. We show that a G2-manifold with a T^4-action of isomorphisms such that the orbits are coassociative tori is locally equivalent to a minimal 3-manifold in R^{3,3} = H^2(T^4,R) with positive induced metric. By studyi…

In this paper, we study the Seiberg-Witten equations on a compact 3-manifold with boundary. Solutions to these equations are called monopoles. Under some simple topological assumptions, we show that the solution space of all monopoles is a Banach manifold in suitable function space topologies. We then prove that the re…

New symplectic instanton homology constructed via traceless character varieties.

This paper constructs cohomological Hall algebras for 3-Calabi-Yau categories.

This paper uses sheaf theory to constrain knot types in clean intersections.

New theory couples Chern-Simons to matter, topological.

We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds which are symplectically cobordant to something overtwisted, or to…

Characterizes symplectic rational homology ball fillings of Seifert fibered spaces.

Proof that $SU(2)$ character variety of genus 2 surface is ${\mathbb C} P^3$.

Unique minimizing maps from hyperbolic surfaces to quasi-Fuchsian 3-manifolds are studied.

We prove the existence of a unique maximal surface in each anti-de Sitter (AdS) convex Globally Hyperbolic Maximal (GHM) manifold with particles (that is, with conical singularities along time-like lines) for cone angles less than $π$. We interpret this result in terms of Teichmüller theory, and prove the existence of …

In this paper the author discuss the relation between Lagrangian Floer homology and Gauge-theory (Donaldson theory) Floer homology. It can be regarded as a version of Atiyah-Floer type conjecture in the case of $SO(3)$-bundle with non-trivial second Stiefel-Whitney class. This is a first of a series of papers, where we…

We consider a homological enlargement of the mapping class group, defined by homology cylinders over a closed oriented surface (up to homology cobordism). These are important model objects in the recent Goussarov-Habiro theory of finite-type invariants of 3-manifolds. We study the structure of this group from several d…

The paper studies Legendrian mean curvature flow in η-Einstein Sasakian manifolds.