arXiv research

We relax indicator matrices to form a manifold for faster optimization.

New gauge invariants from framed 3-manifolds match Hopf algebra indicators.

Real vector bundles are determined by their Dirac indices on specific spin manifolds.

In this paper, we investigate topological aspects of indices of twisted geometric operators on manifolds equipped with fibered boundaries. We define $K$-groups relative to the pushforward for boundary fibration, and show that indices of twisted geometric operators, defined by complete $Φ$ or edge metrics, can be regard…

We prove an off-diagonal expansion for a Toeplitz operator with an indicator function.

New indices defined for manifolds with boundary, generalizing previous results.

Study of invariants on manifolds with boundary involving equivariant spectral flow and η-invariants.

New formulas derived from modular forms for manifold indices.

Methods of parabolic geometries have been recently used to construct a class of elliptic complexes on quaternionic manifolds, the Salamon's complex being the simplest case. The purpose of this paper is to describe an algorithm how to compute their analytical indices in terms of characteristic classes. Using this, we ar…

Novel approach detects early warning indicators in complex systems.

On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no longer work, because they are based on the inverse of the metric, and on related o…

We define two types of local indices of a vector field at an isolated zero on the boundary, and prove Poincare-Hopf-type index theorems for certain vector fields on a compact smooth manifold which have only isolated zeros.

Develops virtual Morse-Bott indices for four-manifolds, proving inequalities.

New cubic forms linked to η-invariants and mod 2 indices.

The study constructs immersions with controlled curvatures between manifolds and identifies obstacles.

M. F. Atiyah proved that the index of a transversally elliptic operator relative to a free action can be computed by using indices of elliptic operators on the orbit manifold. In this paper, we derive an explicit formula for the transversal indices on S^1-bundles over complex projective spaces. Using this explicit form…

Libgober and Wood proved that the Chern number $c_{1}c_{n-1}$ of a $n$-dimensional compact complex manifold can be determined by its Hirzebruch $χ_{y}$-genus. Inspired by the idea of their proof, we show that, for compact, spin, almost-complex manifolds, more Chern numbers can be determined by the indices of some twist…

This paper is a survey on the {\em Zimmer program}. In it's broadest form, this program seeks an understanding of actions of large groups on compact manifolds. The goals of this survey are $(1)$ to put in context the original questions and conjectures of Zimmer and Gromov that motivated the program, $(2)$ to indicate t…

We prove the rigidity and vanishing of several indices of "geometrically natural" twisted Dirac operators on almost even-Clifford Hermitian manifolds admitting circle actions by automorphisms.

The first purpose of this paper is to generalize the well-known Maslov indices of maps of open Riemann surfaces with boundary lying on Lagrangian submanifolds to maps with boundary lying on coisotropic submanifolds in symplectic manifolds. For this purpose, we first define the notion of {\it Maslov loops} of coisotropi…

The paper develops techniques to study entropy and rigidity in RCD-spaces.

We establish a vanishing result for indices of certain twisted Dirac operators on $\text{Spin}^c$-manifolds with non-abelian Lie-group actions. We apply this result to study non-abelian symmetries of quasitoric manifolds. We give upper bounds for the degree of symmetry of these manifolds.

A method of computation of its terms is presented together with some stabilization results. As an application a characterization of symplectic harmonic manifolds is given and a relationship with the C-spectral sequence is indicated.

Let M be a Seifert manifold which belongs to the geometry Flat. In this work we determine all the free involutions τ on M, and the Borsuk-Ulam indice of (M,τ).

Derive K-theoretic Donaldson invariants for various 4-manifolds using path integrals and topological twists.

We compute the rings $H^*(N;\mathbb{F}_2)$ for $N$ a closed $\mathbb{S}ol^3$-manifold and then determine the Borsuk-Ulam indices $BU(N,φ)$ with $φ\not=0$ in $H^1(N;\mathbb{F}_2)$.

We establish a calculus for branched spines of 3-manifolds by means of branched Matveev-Piergallini moves and branched bubble-moves. We briefly indicate some of its possible applications in the study and definition of State-Sum Quantum Invariants.

Let $M$ be a $Spin$-manifold with $S^1$-action and let $σ\in S^1$ be of finite order. We show that the indices of certain twisted Dirac operators vanish if the action of $σ$ has sufficiently large fixed point codimension. These indices occur in the Fourier expansion of the elliptic genus of $M$ in one of its cusps. As …

The paper proves conditions for the existence of holomorphic discs in Kähler manifolds.

Twists agrarian and $\ell^2$-Betti numbers for locally indicable groups.

Gradient-like flows on certain manifolds restrict saddle Morse indices to 1 or n-1.

Study proves certain Zoll manifolds with entire Grauert tubes are isometric.

Introduces partial Jacobi manifolds in convenient spaces.

New spectral invariants distinguish Joyce orbifolds from other manifolds.

Let $M$ be a $G$-manifold and $\om$ a $G$-invariant exact $m$-form on $M$. We indicate when these data allow us to constract a cocycle on a group $G$ with values in the trivial $G$-module $\mathbb R$ and when this cocycle is nontrivial.

The paper studies 3-manifolds with specific Morse-Smale diffeomorphisms and finds they are homeomorphic to lens spaces.

I present a selection of results on locally conformally Kähler geometry published after 1997. The proofs are mainly sketched, some of them are even omitted. Several open problems are indicated in the end.

It is shown that in the multivariate case the orders p, of the AR part, and q, of the MA part, are not invariants of the time series. Thus, it is concluded that it only makes sense to define the class of ARMA(p,p)- irreducible models, where p is the biggest of the system's Kronecker indices. This class is shown not to …

The cobordism invariance of the index on closed manifolds is reproved using the calculus of cusp pseudodifferential operators on a manifold with boundary. More generally, on a compact manifold with corners, the existence of a symmetric cusp differential operator of order 1 and of Dirac type near the boundary implies th…

Study large N oscillations in 3D theories related to black hole physics.

We identify a large class R of three-dimensional N=2 superconformal field theories. This class includes the effective theories T_M of M5-branes wrapped on 3-manifolds M, discussed in previous work by the authors, and more generally comprises theories that admit a UV description as abelian Chern-Simons-matter theories w…

This paper concerns a formula which relates the Lefschetz number L(f) for a map f:M --> M' to the fixed point index I(f) summed with the fixed point index of a derived map on part of the boundary of M. Here M is a compact manifold and M' is M with a collar attached.

Decision forests (Forests), in particular random forests and gradient boosting trees, have demonstrated state-of-the-art accuracy compared to other methods in many supervised learning scenarios. In particular, Forests dominate other methods in tabular data, that is, when the feature space is unstructured, so that the s…

Based on Colombeau's theory of algebras of generalized functions we introduce the concepts of generalized functions taking values in differentiable manifolds as well as of generalized vector bundle homomorphisms. We study their basic properties, in particular with respect to some new point value concepts for generalize…

The paper gives a categorical approach to generalized manifolds such as orbit spaces and leaf spaces of foliations. It is suggested to consider these spaces as sets equipped with some additional structure which generalizes the notion of atlas. The approach is compared with the known ones that use the Grothendieck topos…

We consider manifolds of oriented flags SO(n)/SO(2)xSO(n-3) (n>=4) as 4- and 6-symmetric spaces and indicate characteristic conditions for invariant Riemannian metrics under which the canonical f-structures on these homogeneous $Φ$-spaces belong to the classes Kill f, NKf and G_1f of generalized Hermitian geometry.

For the geodesic flow of an odd dimensional hyperbolic manifold we prove a Lefschetz type formula. The local terms are Fuller indices of the closed orbits. The global "Frobenius operator" is the generator of the flow and its action on tangential cohomology.

We give an overview of some recent results in hypersymplectic and para-quaternionic Kahler geometry, and introduce the notion of split three-Sasakian manifold. In particular, we discuss the twistor spaces and Swann bundles of para-quaternionic Kahler manifolds. These are used to classify examples with a fully homogeneo…