arXiv research

This is the second companion paper of arXiv:1601.03586. We consider the morphism from the variety of triples introduced in arXiv:1601.03586 to the affine Grassmannian. The direct image of the dualizing complex is a ring object in the equivariant derived category on the affine Grassmannian (equivariant derived Satake ca…

Abstract: Tangent categories get a Cartan calculus with scalar multiplication by a commutative ring.

Abstract Lie algebroids generalize Lie algebroids to abstract categories.

We determine the structure of the Hodge ring, a natural object encoding the Hodge numbers of all compact Kaehler manifolds. As a consequence of this structure, there are no unexpected relations among the Hodge numbers, and no essential differences between the Hodge numbers of smooth complex projective varieties and tho…

If $X$ is a smooth manifold then the $\mathbb R$-algebra $C^\infty(X)$ of smooth functions $c:X\to\mathbb R$ is a $C^\infty$-$ring$. That is, for each smooth function $f:{\mathbb R}^n\to\mathbb R$ there is an $n$-fold operation $Φ_f:C^\infty(X)^n\to C^\infty(X)$ acting by $Φ_f:(c_1,\ldots,c_n)\mapsto f(c_1,...,c_n)$, a…

We construct certain tensor categories that are dominated by finitely many simple objects. Objects in these categories are modules over rings of algebra integers. We show how to obtain TQFTs defined over algebra integers from these categories.

This is a survey of the author's paper arXiv:1001.0023 on "Algebraic Geometry over C-infinity rings". If X is a smooth manifold then the R-algebra C^\infty(X) of smooth functions c : X --> R is a "C-infinity ring". That is, for each smooth function f : R^n --> R there is an n-fold operation Φ_f : C^\infty(X)^n --> C^\i…

Abstract: Generalizes stability theories over toric varieties and Novikov type rings.

It is proved that the category of simplicial complete bornological spaces over $\mathbb R$ carries a combinatorial monoidal model structure satisfying the monoid axiom. For any commutative monoid in this category the category of modules is also a monoidal model category with all cofibrant objects being flat. In particu…

New Euler characteristic and Burnside group defined for definable groupoids.

After the torch of Anders Kock [Taylor series calculus for ring objects of line type, Journal of Pure and Applied Algebra, 12 (1978), 271-293], we will establish the Baker-Campbell-Hausdorff formula as well as the Zassenhaus formula in the theory of Lie groups.

Defines super projective modules and explores their properties.

A new prior improves generative models' performance.

The paper explores zero-divisors and idempotents in quandle rings, proving their absence in certain cases.

Lie-Rinehart algebras over $C^\infty$-rings defined and studied.

For a fixed parabolic subalgebra p of gl(n,C) we prove that the centre of the principal block O(p) of the parabolic category O is naturally isomorphic to the cohomology ring of the corresponding Springer fibre. We give a diagrammatic description of O(p) for maximal parabolic p and give an explicit isomorphism to Braden…

A new invariant of Poisson manifolds, a Poisson K-ring, is introduced. Hypothetically, this invariant is more tractable than such invariants as Poisson (co)homology. A version of this invariant is also defined for arbitrary algebroids. Basic properties of the Poisson K-ring are proved and the Poisson K-rings are calcul…

Investigates differential smoothness of 3D skew polynomial rings.

We define a notion of stability for chiral ring of four dimensional N=1 theory by introducing test chiral rings and generalized a maximization. We conjecture that a chiral ring is the chiral ring of a superconformal field theory if and only if it is stable. We then study N=1 field theory derived from D3 branes probing …

A classical theorem due to Quillen (1969) identifies the unitary bordism ring with the Lazard ring, which classifies the universal one-dimensional commutative formal group law. We prove an equivariant generalization of this result by identifying the homotopy theoretic $\mathbb{Z}/2$-equivariant unitary bordism ring, in…

The paper examines differential smoothness in skew PBW extensions over polynomial rings.

Criteria for smoothness of ambiskew polynomial rings.

This paper calculates the skein algebra of the Borromean rings complement.

The paper explores idempotents in quandle rings and their connections to quandle coverings.

Defines vector fields and differential forms on local C-infinity-ringed spaces.

Researchers found only one hyperbolic structure for Borromean rings.

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

New argument for 3-manifold cohomology with $\mathbb{F}_2$ coefficients.

Paper computes hyperbolic structure of Borromean rings complement.

New hyperbolic manifolds found with same trace ring.

We calculate the intersection ring of three-dimensional graph manifolds with rational coefficients and give an algebraic characterization of these rings when the manifold's underlying graph is a tree. We are able to use this characterization to show that the intersection ring obstructs arbitrary three-manifolds from be…

Homological algebra used to study local equivalence of complex rings.

In this paper we compute a presentation for the group of ring motions of the split union of a Hopf link with Euclidean components and a Euclidean circle. A key part of this work is the study of a short exact sequence of groups of ring motions of general ring links in $\mathbb{R}^3$. This sequence allowed us to build th…

Study introduces dynamical ideals for non-commutative rings and classifies knots and links.

We show that solutions of Thurston equation on triangulated 3-manifolds in a commutative ring carry topological information. We also introduce a homogeneous Thurston equation and a commutative ring associated to triangulated 3-manifolds.

We build extensions of the arc rings, relate their centers to the cohomology rings of the Springer varieties, and categorify all level two representations of quantum sl(N).

Study Coxeter groups over fusion rings and their geometric realisations.

Differential K-theory gets a $λ$-ring structure.

Division algorithm for surface group rings yields standard complexes and cohomological dimensions.

New Frobenius manifold structures found on Dicyclic group orbits.

We view strict ring spectra as generalized rings. The study of their algebraic K-theory is motivated by its applications to the automorphism groups of compact manifolds. Partial calculations of algebraic K-theory for the sphere spectrum are available at regular primes, but we seek more conceptual answers in terms of lo…

Constructs 2-vector bundles and 2K-theory for Lie groupoids and 2-equivariant settings.

Dual-objective GANs reduce training instabilities with tunable α-loss parameters.

The paper explains the topological origin of the distinction between incidence theorems over division rings and fields.

A C-infinity ring is a set equipped with n-ary operations corresponding to smooth n-ary functions on the real line (satisfying natural axioms). We prove that the cosimplicial abelian group associated to the de Rham complex of Euclidean space has the structure of a cosimplicial C-infinity ring. We also analyse the notio…

This article discusses the twisted adjoint action $\mathrm{Ad}_{g}^κ:G\rightarrow G$, $x\mapsto gxκ(g^{-1})$ given by a Dynkin diagram automorphism $κ\in\mathrm{Aut}(G)$, where $G$ is compact, connected, simply connected and simple. The first aim is to recover the classification of $κ$-twisted conjugacy classes by elem…

Explains the Borromean rings, icosahedron, and Poincaré homology sphere.

Generalizes cohomology ring result for combinatorial line arrangements.