arXiv research

New bundles preserve properties under certain maps.

Whyte used the index theory of Dirac operators and Block-Weiberger uniformly finite homology to show that certain infinite connected sums do not carry a metric with nonnegative scalar curvature in their bounded geometry class. His proof uses a coarse version of the $\hat{A}$-class to obstruct such metrics. In this note…

The Lichnerowicz formula yields an index theoretic obstruction to positive scalar curvature metrics on closed spin manifolds. The most general form of this obstruction is due to Rosenberg and takes values in the $K$-theory of the group $C^*$-algebra of the fundamental group of the underlying manifold. We give an overvi…

We use K-area homology to summarize some results about the Novikov conjecture and the Hirzebruch L-class. In fact, we provide necessary and sufficient conditions for closed manifolds to have a homotopy invariant L-class. In order to obtain additional properties we also introduce the cohomology of infinite K--area which…

K-area is an invariant for Riemannian manifolds introduced by Gromov as an obstruction to the existence of positive scalar curvature. However in general it is difficult to determine whether K-area is finite or not. though the definition of K-area is quite natural. In this paper, we study how the invariant changes under…

We use Gromov's K--area to define a generalized homology theory on compact smooth manifolds. In fact, this theory collects obstructions to the enlargeability of the manifold and its nontrivial submanifolds. Moreover, using the K--area homology we can rephrase some classic results about positive scalar curvature.

We show an Uhlenbeck type estimate for closed simply connected manifolds which provides the existence of certain exact sequences in K-area homology. This leads to the behavior of the K-area homology under surgery. Moreover, we give an index theoretic obstruction to positive scalar curvature on compact spin manifolds wi…

In the first part we use Gromov's K--area to define the K--area homology which stabilizes into singular homology on the category of pairs of compact smooth manifolds. The second part treats the questions of certain curvature gaps. For instance, the $L^\infty$--curvature gap of complex vector bundles on a compact manif…

Constructs obstructions and deformation principles for positive scalar curvature metrics with mean convex boundaries.

Extends existence results for scalar curvature on conical manifolds.

Given a closed symplectic manifold $(M,ω)$ we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group ${\hbox{\it Ham}} (M,ω)$ by means of the Hofer metric on ${\hbox{\it Ham}} (M,ω)$. We use pseudo-holomorphic curves involved in the definition of the multiplicative s…

We introduce a new volume definition on normed vector spaces. We show that the induced $k$-area functionals are convex for all $k$. In the particular case $k=2$, our theorem implies that Busemann's 2-volume density is convex, which was recently shown by Burago-Ivanov. We also show how the new volume definition is relat…

We give here some extensions of Gromov's and Polterovich's theorems on $\karea$ of $\mathbb{CP} ^{n}$, particularly in the symplectic and Hamiltonian context. Our main methods involve Gromov-Witten theory, and some connections with Bott periodicity, and loop groups. The argument is closely connected with study of jump…

Extends results on metrics of positive scalar curvature.

There are two themes in the present paper. The first one is spelled out in the title, and is inspired by an attempt to find an analogue of Hersch-Yang-Yau estimate for $lambda_1$ of surfaces in symplectic category. In particular we prove that every split symplectic manifold $T^4 times M$ admits a compatible Riemannian …

Infinite rank surface cluster algebras extend traditional concepts to surfaces with accumulation points.

Veech groups uniformize Teichmüller geodesic curves in Riemann moduli space. Recently, examples of infinitely generated Veech groups have been given. We show that these can even have infinitely many cusps and infinitely many infinite ends. We further show that examples exist for which each direction of an infinite end …

Infinite order loose corks created.

We construct a infinite-dimensional manifold structure adapted to analytic Lie pseudogroups of infinite type. More precisely, we prove that any isotropy subgroup of an analytic Lie pseudogroup of infinite type is a regular infinite-dimensional Lie group, modelled on a locally convex strict inductive limit of Banach spa…

This paper is one step toward infinite energy gauge theory and the geometry of infinite dimensional moduli spaces. We generalize a gluing construction in the usual Yang-Mills gauge theory to an ``infinite energy'' situation. We show that we can glue an infinite number of instantons, and that the resulting instantons ha…

Study on infinite neural networks for generalization.

Book explores infinite translation surfaces, challenging traditional geometry.

Infinite diameter found in compression body graph.

Study examines infinite limits of transformer dynamics, identifying key parameterizations.

Study infinite superelliptic curves and their Veech groups, providing geometric and algebraic insights.

New homology for infinite multi-colored braids, completing previous work.

Study on infinite-dimensional Heisenberg groups using hypoelliptic heat kernels.

Study infinite-depth limits of neural networks with fixed width.

The paper constructs Fuchsian groups for specific infinite genus surfaces.

New findings on asymptotic property C in infinite dimensional spaces.

Paper constructs an infinite surface with one end and infinite genus.

Infinitely many 3D shapes have multiple ways to be filled with special surfaces.

Study infinite genus surfaces and Schottky groups for uniformization.

We define and study analogs of curve graphs for infinite type surfaces. Our definitions use the geometry of a fixed surface and vertices of our graphs are infinite multicurves which are bounded in both a geometric and a topological sense. We show that the graphs we construct are generally connected, infinite diameter a…

Infinite braids' categorification proven using Khovanov homology.

New findings on infinite subgroups in negatively curved spaces.

Study finds infinitely many prime knots with identical Alexander invariants.

Infinite-genus surfaces have many isospectral hyperbolic structures.

Infinite-dimensional contact geometry explored.

VC dimensions of group CNNs are infinite for certain kernels and groups.

New infinite series of hyperbolic polytopes with special growth rates found.

Concrete description of infinite order cork automorphism.

The paper defines infinite Schottky groups and their applications to infinite type surfaces.

New categorification method for infinite braids.

Most Finsler metrics have infinite-dimensional holonomy groups.

Arithmetic 3-manifolds with infinite geodesics are proven.

The Conway knot can't be smoothly tied to any other knot an infinite number of times.

The paper explores non-amenability in infinite-type surfaces and graphs.