Cohomology
View article: Quantum Information Copy Time: Microscopic Construction, Fixed Points, Gauge Cohomology and Predictive Phenomenology
Quantum Information Copy Time: Microscopic Construction, Fixed Points, Gauge Cohomology and Predictive Phenomenology Open
Title: Information Geometry, Asymptotic Safety, and Standard Model Emergence: Towards a Unified QICT FrameworkThe Quantum Information Copy Time (QICT) program proposes an integrated theoretical framework aimed at solving t…
View article: Quantum Information Copy Time: Microscopic Construction, Fixed Points, Gauge Cohomology and Predictive Phenomenology
Quantum Information Copy Time: Microscopic Construction, Fixed Points, Gauge Cohomology and Predictive Phenomenology Open
Title: Information Geometry, Asymptotic Safety, and Standard Model Emergence: Towards a Unified QICT FrameworkThe Quantum Information Copy Time (QICT) program proposes an integrated theoretical framework aimed at solving the problem of gra…
View article: The G-signature Theorem on Witt spaces
The G-signature Theorem on Witt spaces Open
Let G be a compact Lie group and let X be an oriented Witt G-pseudomanifold. Using intersection cohomology it is possible to define Sign(G,X) in R(G), the G-signature of X. Let g be an element in G. Assuming that the inclusion of the fixed…
View article: Quantum Information Copy Time: Microscopic Construction, Fixed Points, Gauge Cohomology and Predictive Phenomenology
Quantum Information Copy Time: Microscopic Construction, Fixed Points, Gauge Cohomology and Predictive Phenomenology Open
Title: Information Geometry, Asymptotic Safety, and Standard Model Emergence: Towards a Unified QICT FrameworkThe Quantum Information Copy Time (QICT) program proposes an integrated theoretical framework aimed at solving t…
View article: Extendability of group actions on K3 or Enriques surfaces
Extendability of group actions on K3 or Enriques surfaces Open
Let $X$ be a K3 or Enriques surface with good reduction. Let $G$ be a finite group acting (not necessarily linearly) on $X$. We give a criterion for this group action to extend to a smooth model of $X$ in terms of the action of $G$ on the …
View article: Point-like non-commutative families of bounding cochains
Point-like non-commutative families of bounding cochains Open
We define genus zero open Gromov-Witten invariants with boundary and interior constraints for a Lagrangian submanifold of arbitrary even dimension. The definition relies on constructing a canonical family of bounding cochains that satisfy …
View article: A Homotopical Desingularization of the Stable Sphere Spectrum
A Homotopical Desingularization of the Stable Sphere Spectrum Open
The stable sphere spectrum, denoted by S, serves as the fundamental monoidal unit in the stable homotopy category, playing a role analogous to the integers in the category of rings. Despite its foundational importance, the sphere spectrum …
View article: Spectral Realizations of Homotopy Types
Spectral Realizations of Homotopy Types Open
This paper explores the intricate relationship between homotopy types and their spectral realizations. This is a fundamental concept in stable homotopy theory and its applications. Homotopy types classify topological spaces up to continuou…
View article: The $(infty,1)$-Category of Étale Motives
The $(infty,1)$-Category of Étale Motives Open
The concept of motives, pioneered by Alexander Grothendieck, provides a profound and unifying framework for studying algebraic varieties by abstracting their cohomology theories into a universal linear category. While classical motivic the…
View article: Étale Period Maps and Non-commutative Motives
Étale Period Maps and Non-commutative Motives Open
This paper explores the intricate relationship between étale period maps and the emerging theory of non-commutative motives. Étale period maps, central to arithmetic geometry, encode deep information about algebraic varieties over number f…
View article: Hochschild cohomology of Beilinson algebras of graded down-up algebras with weights ($n,m$)
Hochschild cohomology of Beilinson algebras of graded down-up algebras with weights ($n,m$) Open
Let $A=A(α, β)$ be a graded down-up algebra with weights $({\rm deg}\, x, {\rm deg}\, y)=(n,m)$ and $β\neq 0$, and $\nabla A$ the Beilinson algebra of $A$. Note that $A$ is a $3$-dimensional cubic AS-regular algebra. Assume that $\gcd(n, m…
View article: A note on ideals in derived geometries
A note on ideals in derived geometries Open
We develop the basic theory of derived quasi-coherent ideals for stacks relative to a given derived algebraic context. We compare different notions of adic completeness with respect to derived ideals, define and compare formal spectra and …
View article: Spectral Realizations of Homotopy Types
Spectral Realizations of Homotopy Types Open
This paper explores the intricate relationship between homotopy types and their spectral realizations. This is a fundamental concept in stable homotopy theory and its applications. Homotopy types classify topological spaces up to continuou…
View article: Étale Period Maps and Non-commutative Motives
Étale Period Maps and Non-commutative Motives Open
This paper explores the intricate relationship between étale period maps and the emerging theory of non-commutative motives. Étale period maps, central to arithmetic geometry, encode deep information about algebraic varieties over number f…
View article: Associated Sheaf Functors in tt-Geometry
Associated Sheaf Functors in tt-Geometry Open
Given a tensor triangulated category we investigate the geometry of the Balmer spectrum as a locally ringed space by constructing functors assigning to every object in the category a corresponding sheaf of modules over the structure sheaf …
View article: The $(infty,1)$-Category of Étale Motives
The $(infty,1)$-Category of Étale Motives Open
The concept of motives, pioneered by Alexander Grothendieck, provides a profound and unifying framework for studying algebraic varieties by abstracting their cohomology theories into a universal linear category. While classical motivic the…
View article: Hochschild cohomology of Beilinson algebras of graded down-up algebras with weights ($n,m$)
Hochschild cohomology of Beilinson algebras of graded down-up algebras with weights ($n,m$) Open
Let $A=A(α, β)$ be a graded down-up algebra with weights $({\rm deg}\, x, {\rm deg}\, y)=(n,m)$ and $β\neq 0$, and $\nabla A$ the Beilinson algebra of $A$. Note that $A$ is a $3$-dimensional cubic AS-regular algebra. Assume that $\gcd(n, m…
View article: A Homotopical Desingularization of the Stable Sphere Spectrum
A Homotopical Desingularization of the Stable Sphere Spectrum Open
The stable sphere spectrum, denoted by S, serves as the fundamental monoidal unit in the stable homotopy category, playing a role analogous to the integers in the category of rings. Despite its foundational importance, the sphere spectrum …
View article: A note on ideals in derived geometries
A note on ideals in derived geometries Open
We develop the basic theory of derived quasi-coherent ideals for stacks relative to a given derived algebraic context. We compare different notions of adic completeness with respect to derived ideals, define and compare formal spectra and …
View article: Non-commutative Galois Cohomology and Local-Global Principles
Non-commutative Galois Cohomology and Local-Global Principles Open
This paper explores the intricate relationship between non-commutative Galois cohomology and local-global principles, extending classical concepts from commutative algebra and number theory to more general settings. We begin by reviewing t…
View article: An Intrinsic Topology of the Primes and its Cohomological Invariants
An Intrinsic Topology of the Primes and its Cohomological Invariants Open
The prime numbers exhibit a famously complex and seemingly random distribution, yet they underpin the fundamental structure of arithmetic. This paper introduces a topological framework for the set of prime numbers, P, which we term the Div…
View article: Instantons on the Blown-up Surface and the Affine Vertex Algebra
Instantons on the Blown-up Surface and the Affine Vertex Algebra Open
Vafa-Witten observed that Yoshioka's blow-up formula for the Euler characteristics of rank $r$ instantons on an algebraic surface coincides with the character of the Wess-Zumino-Witten model for $\mathrm{SU}(r)$ at level $1$, and raised th…
View article: Floer theory for the variation operator of an isolated singularity
Floer theory for the variation operator of an isolated singularity Open
The variation operator in singularity theory maps relative homology cycles to compact cycles in the Milnor fiber using the monodromy. We construct its symplectic analog for an isolated singularity. We define the monodromy Lagrangian Floer …
View article: Quantum Information Copy Time: Microscopic Construction, Fixed Points, Gauge Cohomology and Predictive Phenomenology
Quantum Information Copy Time: Microscopic Construction, Fixed Points, Gauge Cohomology and Predictive Phenomenology Open
Title: Information Geometry, Asymptotic Safety, and Standard Model Emergence: Towards a Unified QICT FrameworkThe Quantum Information Copy Time (QICT) program proposes an integrated theoretical framework aimed at solving t…
View article: On the de Rham cohomology of cyclic covers
On the de Rham cohomology of cyclic covers Open
We compute explicit bases for the de Rham cohomology of cyclic covers of the projective line defined over an algebraically closed field of characteristic $p\geq 0$. For both Kummer and Artin-Schreier extensions, we describe precise $k$-bas…
View article: Stratified Spaces and Stable Homotopy
Stratified Spaces and Stable Homotopy Open
This paper explores the intricate relationship between stratified spaces and stable homotopy theory, two fundamental yet distinct areas of modern topology. Stratified spaces provide a rigorous framework for studying singular geometric obje…
View article: Three-loop banana integrals with three equal masses
Three-loop banana integrals with three equal masses Open
We obtain and solve the canonical differential equations for the three-loop banana integrals in dimensional regularisation when three of the four masses are equal. The K3 surface associated with the maximal cuts factorises into a product o…
View article: Stratified Spaces and Stable Homotopy
Stratified Spaces and Stable Homotopy Open
This paper explores the intricate relationship between stratified spaces and stable homotopy theory, two fundamental yet distinct areas of modern topology. Stratified spaces provide a rigorous framework for studying singular geometric obje…
View article: Non-commutative Galois Cohomology and Local-Global Principles
Non-commutative Galois Cohomology and Local-Global Principles Open
This paper explores the intricate relationship between non-commutative Galois cohomology and local-global principles, extending classical concepts from commutative algebra and number theory to more general settings. We begin by reviewing t…
View article: An Intrinsic Topology of the Primes and its Cohomological Invariants
An Intrinsic Topology of the Primes and its Cohomological Invariants Open
The prime numbers exhibit a famously complex and seemingly random distribution, yet they underpin the fundamental structure of arithmetic. This paper introduces a topological framework for the set of prime numbers, P, which we term the Div…