arXiv research

The study tackles realisability of twisted homology classes and introduces new techniques in parametrised homotopy theory.

The paper studies obstructions to homotopy invariance of loop coproducts.

Study cohomotopy classes for 4-manifolds using complex spin structures.

Develops parametrised Poincaré duality for equivariant fixed points.

We show that, for certain families $φ_{\mathbf{s}}$ of diffeomorphisms of high-dimensional spheres, the commutator of the Dehn twist along the zero-section of $T^*S^n$ with the family of pullbacks $φ^*_{\mathbf{s}}$ gives a noncontractible family of compactly-supported symplectomorphisms. In particular, we find example…

In this paper, we use the parametrised strict deformation quantization of C*-bundles obtained in a previous paper, and give more examples and applications of this theory. In particular, it is used here to classify H_3-twisted noncommutative torus bundles over a locally compact space. This is extended to the case of gen…

Finite type for 3-manifolds with boundary.

The abstract proves every knot type can be parametrized by smooth functions and studies limit knot types.

Study on deformations of Einstein and nearly G2 structures in 3-Sasaki manifolds.

The paper proves rigidity for mapping class group actions on metrics of positive scalar curvature.

In this paper, we initiate the study of a parametrised version of Rieffel's strict deformation quantization. We apply it to give a classification of noncommutative principal torus bundles, in terms of parametrised strict deformation quantization of ordinary principal torus bundles. The paper also contains a putative de…

We present a procedure which allows one to integrate explicitly the class of checkerboard IC-nets which has recently been introduced as a generalisation of incircular (IC) nets. The latter class of privileged congruences of lines in the plane is known to admit a great variety of geometric properties which are also pres…

New framework for analyzing games with multi-dimensional singular controls and non-linear jumps.

Surveying ${\mathbb A}^1$-homotopy theory and contractible varieties.

Combines ML and DA to infer unresolved scale parametrisation from noisy data.

V. Turaev introduced the theory of topology of words and phrases in 2005. This is a combinatorialy extension of the theory of virtual knots and links. In this paper we generalize the notion of homotopy of words and phrases and we give geometric meanings of the generalized homotopy of words. Moreover using the generaliz…

A new approach to unsupervised learning using recognition-parametrised models.

Coarse homotopy theory connects Euclidean cones to shape theory of compact spaces.

Paper connects fibre bundles to curvature and homotopy theory.

The paper applies $S^1$-localization to symplectic cohomology.

Classifies colored links and spatial graphs up to colored link-homotopy.

This paper refines homotopy theory for cubical sets and uniform spaces.

Universal geometry organizes heterotic moduli spaces.

Study homotopy types of free racks and quandles, proving analogs of Milnor's theorem.

We explore homotopies in quantum field theory formalism.

New examples of manifolds that are homotopy but not simple homotopy equivalent.

New discrete models for constant mean curvature surfaces and tori.

Proves Poincaré surgery theorem using homotopy theory.

In 2005 V. Turaev introduced the theory of topology of words and phrases. Turaev defined an equivalence relation on generalized words and phrases which is called homotopy. This is suggested by the Reidemeister moves in the knot theory. Then Turaev gave the homotopy classification of generalized words with less than or …

Two approaches study the homotopy of blow ups in algebraic and symplectic geometry.

Conservation laws vanishing along characteristic directions of a given system of PDEs are known as characteristic conservation laws, or characteristic integrals. In 2D, they play an important role in the theory of Darboux-integrable equations. In this paper we discuss characteristic integrals in 3D and demonstrate that…

In this note, we extend the theory of Chern-Cheeger-Simons to construct canonical invariants for a one-parameter family of flat connections on a smooth manifold. These invariants lie in degrees $(2p-2)$-cohomology with $\C/\Z$-cohomology, for $p\geq 2$. Furthermore, they are shown to be rigid in a variation of paths (p…

Generalizes Floer homotopy via Morse-Bott theory.

Paper develops coarse homotopy theory for geometric group theory.

Notes on Khovanov and knot Floer theories' stable homotopy types.

Refines Khovanov homology using signed Burnside categories.

Derived differential manifolds are constructed using the usual homotopy theory of simplicial rings of smooth functions. They are proved to be equivalent to derived differential manifolds of finite type, constructed using homotopy sheaves of homotopy rings (D.Spivak), thus preserving the classical cobordism ring. This r…

We set up foundations of representation theory over $S$, the sphere spectrum, which is the `initial ring' of stable homotopy theory. In particular, we treat $S$-Lie algebras and their representations, characters, $gl_n(S)$-Verma modules and their duals, Harish-Chandra pairs and Zuckermann functors. As an application, w…

Classifies two-dimensional extended homotopy field theories with aspherical targets.

This article constructs the moduli stack of torsionfree $G$-jet-structures in homotopy type theory with one monadic modality. This yields a construction of this moduli stack for any $\infty$-topos equipped with any stable factorization systems. In the intended applications of this theory, the factorization systems are …

Smooth actions of infinite groups linked to homotopy theory.

Morse theory extended to noncompact manifolds with complex geometric data.

We introduce a framework, twisted parametrized stable homotopy theory, for describing semi-infinite homotopy types. A twisted parametrized spectrum is a section of a bundle whose fibre is the category of spectra. We define these bundles in terms of modules over a stack of parametrized spectra and in terms of diagrams o…

Characterizes a specific type of Courant algebroid with a Calabi-Yau structure.

The homotopy theory of gauge groups has received considerable attention in recent decades. In this work, we study the homotopy theory of gauge groups over some high dimensional manifolds. To be more specific, we study gauge groups of bundles over $(n-1)$-connected closed $2n$-manifolds, the classification of which was …

New method for handling multi-dimensional singular controls with jump costs in mean-field problems.

Survey of mathematical developments in gauge theory using thin homotopy.

This is the second of a series of papers which are devoted to a comprehensive theory of maps between orbifolds. In this paper, we develop a basic machinery for studying homotopy classes of such maps. It contains two parts: (1) the construction of a set of algebraic invariants -- the homotopy groups, and (2) an analog o…