arXiv research

The paper develops a Galois theory for cluster algebras and Riemann surfaces.

We enhance the analogy between field extensions and covering spaces by introducing the concept of splitting covering which correspondences to the splitting field in Galois theory. We define semi-topological Galois groups for Weierstrass polynomials and prove the existence of a Galois correspondence. This new tool enabl…

The classical Galois theory deals with certain finite algebraic extensions and establishes a bijective order reversing correspondence between the intermediate fields and the subgroups of a group of permutations called the Galois group of the extension. It has been the dream of many mathematicians at the end of the nine…

We propose an approach to study non-Abelian Iwasawa theory, using the idea of Johnson homomorphisms in low dimensional topology. We introduce arithmetic analogues of Johnson homomorphisms/maps, called the p-Johnson homomorphisms/maps, associated to the Zassenhaus filtration of a pro-p Galois group over a Z_p-extension …

This paper revisits Differential Galois Theory using Hopf algebras for Lie pseudogroups.

Study Galois groupoids of discret Painlevé equations.

Study Galois groupoids of vector fields, proving lower semicontinuity.

We review some ideas of Grothendieck and others on actions of the absolute Galois group Γ Q of Q (the automorphism group of the tower of finite extensions of Q), related to the geometry and topology of surfaces (mapping class groups, Teichm{ü}ller spaces and moduli spaces of Riemann surfaces). Grothendieck's motivation…

We discuss the concept of Galois structure and Galois epimorphism in a general setting. Namely, a Galois structure for an epimorphism $π\colon M\to B$ in some category ${\mathcal C}$ is the action of a group object that gives to $M$ the structure of principal homogeneous space in the relative category ${\mathcal C}_B$.

The paper classifies sextic curves on a Fano 3-fold with rational Galois covers in 3D space.

Second part of a series on higher coverings of racks and quandles.

The purpose of this paper is to connect two subjects: the theory of quantum integrable systems (complete commutative rings of differential operators), and differential Galois theory. We define quantum completely integrable systems (QCIS), algebraically integrable QCIS, the differential Galois group of a QCIS. We show t…

Introduces arithmetic analogues of Orr invariants and spaces for absolute Galois groups.

We present a geometric setting for the differential Galois theory of $G$-invariant connections with parameters. As an application of some classical results on differential algebraic groups and Lie algebra bundles, we see that the Galois group of a connection with parameters with simple structural group $G$ is determine…

Study real line subbundles on curves, extending classical work.

Simply-connected surfaces of general type for n≥5.

Paper constructs connections on curves with specific Galois groups.

We construct a functor from the smooth 4-dimensional manifolds to the hyper-algebraic number fields, i.e. fields with non-commutative multiplication. It is proved that that the simply connected 4-manifolds correspond to the abelian extensions. We recover the Rokhlin and Donaldson's Theorems from the Galois theory of th…

Machine learning predicts properties of number fields with high accuracy.

Characterizes bi-Perron numbers with specific Galois conjugates.

New conjectures link SU(r) Vafa-Witten invariants to Ramanujan's continued fractions.

This paper extends rack and quandle covering theory using higher categorical Galois theory.

New findings on Malgrange-Galois groupoid for Painlevé VI equation parameters.

New field invariant refines real spectrum and relates to absolute Galois group.

The paper calculates Veech groups and Galois invariants for general origamis.

In this paper, we study the Galois conjugates of stretch factors of pseudo-Anosov elements of the mapping class group of a surface. We show that - except in low-complexity cases - these conjugates are dense in the complex plane. For this, we use Penner's construction of pseudo-Anosov mapping classes. As a consequence, …

We study the interplay between the differential Galois group and the Lie algebra of infinitesimal symmetries of systems of linear differential equations. We show that some symmetries can be seen as solutions of a hierarchy of linear differential systems. We show that the existence of rational symmetries constrains the …

Classifies real trivectors in 9D using Galois cohomology.

Our aim of this and subsequent papers is to enlighten (a part of, presumably) arithmetic structures of knots. This paper introduces a notion of profinite knots which extends topological knots and shows its various basic properties. Particularly an action of the absolute Galois group of the rational number field on prof…

The abstract discusses parallels between Galois theory and Stone-Weierstrass theorem in various fields.

Galois action on manifold structures of complex varieties is abelian.

This paper formalizes manifolds in positive characteristic varieties.

The absolute Galois group of 3-manifolds determines their structure up to homeomorphism.

Study shows convergence of Bergman kernels on covering spaces of Kähler manifolds.

Characterizes algebraic integrability and minimality of Lie equations for non-commutative pseudogroups.

The Bauer-Furuta invariants of smooth 4-manifolds are investigated from a functorial point of view. This leads to a definition of equivariant Bauer-Furuta invariants for compact Lie group actions. These are studied in Galois covering situations. We show that the ordinary invariants of all quotients are determined by th…

Let ${\mathcal {B}}$ be a reducible reduced plane curve. We introduce a new point of view to study the topology of $(\PP^2, {\mathcal {B}})$ via Galois covers and Alexander polynomials. We show its effectiveness through examples of Zariski $N$-plets for conic and conic-quartic configurations.

We introduce mod 3 triple Milnor invariants and triple cubic residue symbols for certain primes of the Eisenstein number field $\mathbb{Q}(\sqrt{-3})$, following the analogies between knots and primes. Our triple symbol generalizes both the cubic residue symbol and Rédei's triple symbol, and describes the decomposition…

We prove a generalized version of Kazhdan's theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence $\{S_n \rightarrow S\}$ of finite Galois covers of a hyperbolic Riemann Surface $S$, converging to the universal cover. The theorem states that the sequence of for…

Researchers extend Gamma index theorem to non-compact spacetimes.

Study $L^2$ Hilbert complexes on complex manifolds.

In this article we consider outer Galois actions on a free profinite group of rank two, induced by the étale fundamental group of a projective line minus three points or of a pointed elliptic curve over a number field. Under mild technical assumptions their respective images uniquely determine the curves and the number…

Let $p:Σ'\toΣ$ be a finite Galois cover, possibly branched, with Galois group $G$. We are interested in the structure of the cohomology of $Σ'$ as a module over $G$. We treat the cases of branched and unbranched covers separately. In the case of branched covers, we give a complete classification of possible module stru…

The abstract discusses conjectures about Chern-Simons invariants of 3-manifolds.

The paper explores the geometry of the Spence-Kummer trilogarithm equation and its Galois analogue.

It is proved that the profinite completion of the mapping class group Mod (g,n) of a surface of genus g with n boundary components is isomorphic to such of the arithmetic group GL(6g-6+2n, Z). We establish a relation between the normal subgroups of Mod (g,n) and the absolute Galois group G(K) of a number field K. Using…

We consider a rather special class of translation surfaces (called M-Origamis in this work) that are obtained from dessins by a construction introduced by Martin Möller. We give a new proof with a more combinatorial flavour of Möller's theorem that $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts faithfully on the…

Study group extensions and bundles on manifolds.