arXiv research

The paper introduces austere and arid submanifolds in Hilbert spaces.

We study transversality for Lipschitz-Fredholm maps in the context of bounded Fréchet manifolds. We show that the set of all Lipschitz-Fredholm maps of a fixed index between Fréchet spaces has the transverse stability property. We give a straightforward extension of the Smale transversality theorem by using the general…

The paper finds many infinite-dimensional weakly reflective PF submanifolds in Hilbert spaces.

In this paper we prove that there does not exists any closed Pre-Lagrangian submanifolds in any closed contact manifolds by using the $J-$holomorphic curves and Gromov's nonlinear Fredholm alternative.

The abstract extends deformation and tangent groupoid constructions to infinite-dimensional manifolds.

In an earlier paper, we showed that the moduli space of deformations of a smooth, compact, orientable special Lagrangian submanifold L in a symplectic manifold X with a non-integrable almost complex structure is a smooth manifold of dimension H^1(L), the space of harmonic 1-forms on L. We proved this first by showing t…

Develops tools for studying intersections of elliptic operators, focusing on $J$-holomorphic maps.

Study axisymmetric ideal fluids on 3-manifolds, proving Fredholm properties.

We show a generic finiteness result for least area planes in 3-dimensional hyperbolic space. Moreover, we prove that the space of minimal immersions of disk into hyperbolic space is a submanifold of a product bundle over a space of immersions of circle into sphere at infinity. The bundle projection map when restricted …

The paper studies curvatures and austere properties of orbits in symmetric spaces.

Moduli spaces of holomorphic disks in a complex manifold Z, with boundaries constrained to lie in a maximal totally real submanifold P, have recently been found to underlie a number of geometrically rich twistor correspondences. The purpose of this paper is to develop a general Fredholm regularity criterion for holomor…

The paper calculates indices for families of Fredholm operators and their extensions.

Let $U_c(H)={u: u is unitary and u-1 is compact}$ stand for the unitary Fredholm group. We prove the following convexity result. Denote by $d_\infty$ the rectifiable distance induced by the Finsler metric given by the operator norm in $U_c(H)$. If $u_0,u_1,u\in U_c(H)$ and the geodesic $β$ joining $u_0$ and $u_1$ in $U…

Researchers create a Fredholm module on fractal shapes like the Cantor set.

Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.

Notes on continuity of discrete-spectrum Fredholm operators.

Extends PF submanifold results and connects Kac-Moody spaces.

Generalizes Novikov conjecture results to infinite-dimensional bundles.

New Fredholm criteria for pseudodifferential operators on manifolds.

Study shows Fredholmness of a Lorentzian Dirac operator on spacetimes.

Parallel transport map over reductive spaces is an affine submersion.

This is a reference volume on polyfold and Fredholm theory.

Study shows solutions to certain equations form smooth manifolds.

Derives Fredholm criteria for isotypical components from a Simonenko principle.

The paper proves homotopy equivalences for spaces of unbounded Fredholm operators.

This article provides a version of scale calculus geared towards a notion of (nonlinear) Fredholm maps between certain types of Frechet spaces, retaining as many as possible of the properties Fredholm maps between Banach spaces enjoy, and the existence of a constant rank theorem for such maps. It does so by extending t…

The spectral flow theorem is applied to operators on finite intervals.

Study perturbs APS boundary conditions for Lorentzian Dirac operators.

We show that the existence of a Fredholm element of the zero calculus of pseudodifferential operators on a compact manifold with boundary with a given elliptic symbol is determined, up to stability, by the vanishing of the Atiyah-Bott obstruction. It follows that, up to small deformations and stability, the same symbol…

We describe a (nonlinear) Fredholm theory for a new class of ambient spaces, as well as for a certain type of categories. The theory is illustrated by an application to the category of stable maps.

We derive simple explicit formulas for the character of a cycle in the Connes' (b,B)-bicomplex of cyclic cohomology and give applications to the Fredholm modules and equivariant characteristic classes.

We describe two topologies on the space of unbounded Fredholm operators and we explain their K-theoretic relevance. In the process we also prove a very general result concerning the continuity of families of first order, elliptic boundary value problems.

In this paper we develop an integration theory for zero sets of polyfold Fredholm sections. The results are needed in the application of the polyfold theory. We use it for example in the construction of symplectic field theory.

We find G2-manifolds with specific asymptotic properties.

We explain the topology of the space, so called, Fredholm-Lagrangian-Grassmannain and the quantity ``Maslov index'' for paths in this space based on the standard theory of Functional Analysis. Our standing point is to define the Maslov index for arbitrary paths in terms of the fundamental spectral property of the Fredh…

Researchers prove Fredholm property for Dirac operator on specific spacetimes.

In an earlier paper, we proved that given an asymptotically cylindrical G_2-manifold M with a Calabi-Yau boundary X, the moduli space of coassociative deformations of an asymptotically cylindrical coassociative 4-fold C in M with a fixed special Lagrangian boundary L in X is a smooth manifold of dimension dim(V_+), whe…

This is the first paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. The theory will be illustrated in upcoming papers by applications to Floer Theory, Gromov-Witten Theory and Symplectic Field Theory.

Fredholm conditions for invariant operators on compact manifolds.

We study the space of holomorphic discs with boundary on a surface in a real 2-dimensional vector bundle over a compact 2-manifold. We prove that, if the ambient 4-manifold admits a fibre-preserving transitive holomorphic action, then a section with a single complex point has $C^{2,α}$-close sections such that any (non…

Study Dirac operators on incomplete cusp edge spaces, proving self-adjointness and Fredholm properties.

Study the exponential map on surfaces using fluid dynamics.

We prove some Fredholm conditions for many algebras of differential operators on particular classes of open manifolds, which include asymptotically Euclidean or asymptotically hyperbolic manifolds. Our typical result is that an operator $P$ is Fredholm if, and only if, it is elliptic and some limit operators $(P_α)_{α\…

The main topic is the development of a Fredholm theory in a new class of spaces called M-polyfolds. In the subsequent Volume II the theory will be generalized to an even larger class of spaces called polyfolds, which can also incorporate local symmetries. The whole package provides a functional analytic framework to de…

This is the revised version of the second paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. The theory will be illustrated in upcoming papers by applications to Floer Theory, Gromov-Witten Theory and Symplectic Field Theory. Some proofs have been improved and …

The study proves curvature rigidity for convex polytopes.

The square root of Fredholm determinants causes numerical instabilities in option pricing models.

Paper defines and proves a new analytic index for Fredholm operators.