arXiv research

Wave maps from circle to manifold controllable if homotopy classes match.

Higher gauge theory via differential nonabelian cohomology

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…

Globally hyperbolic spacetimes admitting infinitely many causal (and timelike) homotopy classes of curves joining two prescribed points, are exhibited and discussed.

New mathematical framework connects M-theory charges to stable homotopy groups.

Constructs Lepage equivalents for arbitrary-order Lagrangians.

Unified classification of equivariant principal bundles using higher homotopy theory.

Global group laws connect equivariant bordism rings to formal group laws.

Homotopy equivalence between formalities with different covariant derivatives.

Global homotopies upgrade classical map in differential geometry.

Classifies defects in ordered media using homotopy theory.

New category theory for complex projective plane sections.

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…

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

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

Developing efficient and guaranteed nonconvex algorithms has been an important challenge in modern machine learning. Algorithms with good empirical performance such as stochastic gradient descent often lack theoretical guarantees. In this paper, we analyze the class of homotopy or continuation methods for global optimi…

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

M5-branes' flux quantization linked to non-abelian cohomology.

We survey some topics in ${\mathbb A}^1$-homotopy theory. Our main goal is to highlight the interplay between ${\mathbb A}^1$-homotopy theory and affine algebraic geometry, focusing on the varieties that are "contractible" from various standpoints.

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.

We consider the reduction along two compact directions of a twisted N=4 gauge theory on a 4-dimensional orientable manifold which is not a global product of two surfaces but contains a non-orientable surface. The low energy theory is a sigma-model on a 2-dimensional worldsheet with a boundary which lives on branes cons…

Introduces a new geometric framework for field theories.

We introduce a class of maps from an affine flat into a Riemannian manifold that solve an elliptic system defined by the natural second order elliptic operator of the affine structure and the nonlinear Riemann geometry of the target. These maps are called affine harmonic. We show an existence result for affine harmonic…

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.

Generalizes Floer homotopy via Morse-Bott 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…

In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Rieman…

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…

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.

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 …

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…

Analyzes string topology operations using Chen's integrals and homotopy transfer.

The variational problem for the functional $F=\frac12\|φ^*ω\|_{L^2}^2$ is considered, where $φ:(M,g)\to (N,ω)$ maps a Riemannian manifold to a symplectic manifold. This functional arises in theoretical physics as the strong coupling limit of the Faddeev-Hopf energy, and may be regarded as a symplectic analogue of the D…

For an oriented manifold $M$ whose dimension is less than $4$, we use the contractibility of certain complexes associated to its submanifolds to cut $M$ into simpler pieces in order to do local to global arguments. In particular, in these dimensions, we give a different proof of a deep theorem of Thurston in foliation …

Classifies compact spaces by shape, finite spaces by weak homotopy.

In "On the homotopy theory of arrangements," published in 1986, the authors gave a comprehensive survey of the subject. This article updates and continues the earlier article, noting some key open problems.

Chern-Simons gauge theories in 3 dimensions and the Poisson Sigma Model (PSM) in 2 dimensions are examples of the same theory, if their field equations are interpreted as morphisms of Lie algebroids and their symmetries (on-shell) as homotopies of such morphisms. We point out that the (off-shell) gauge symmetries of th…

Satellite formula connects knot concordance invariants to surgery.

Study fractional structures on bundle gerbe modules using rational homotopy theory.