Study Galois groupoids of vector fields, proving lower semicontinuity.
problem Computing Galois groupoids for general parameter values of Painlevé equations.
method Prove lower semicontinuity of Galois groupoids of vector fields.
result Results can compute Galois groupoids for general parameter values of Painlevé equations.
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…
New field invariant refines real spectrum and relates to absolute Galois group.
problem Understanding field invariants related to absolute Galois groups.
method Introducing Artin-Schreier quandles and computing their properties for different types of fields.
result Artin-Schreier quandles provide relations between fields and their absolute Galois groups.
The paper develops a Galois theory for cluster algebras and Riemann surfaces.
problem Building a correspondence between cluster subalgebras and automorphism groups.
method Introducing Galois-like extensions and automorphism groups for cluster algebras.
result Conditions for Galois-like extensions and properties of cluster automorphism groups.
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…
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…
Machine learning predicts properties of number fields with high accuracy.
problem Predicting properties of algebraic number fields.
method Training machine learning algorithms on various coefficients or polynomials of number fields.
result Machine learning can distinguish between real quadratic fields with high precision and predict properties of Galois extensions.
The abstract discusses parallels between Galois theory and Stone-Weierstrass theorem in various fields.
problem Connecting distinguishing power and expressive power in different fields.
method Elementary theorem connecting distinguishing power and expressive power.
result Foundational principle in linguistics linking distinguishing power and expressive power.
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 …
Classifies real trivectors in 9D using Galois cohomology.
problem Classifying real trivectors in R^9.
method Galois cohomology, theta-representations, centralizers computation.
result Classification of real trivectors into nilpotent, semisimple, and mixed types.
The paper establishes analogs of Stallings' theorem for group homomorphisms and their nilpotent quotients.
problem Understanding the structure of fundamental groups of geometric objects.
method Develops analogs of Stallings' theorem for group homomorphisms and their nilpotent quotients.
result Derives applications including non-isomorphic number fields and hyperbolic manifolds with isomorphic universal nilpotent quotients.
We finish the proof of the conjecture of F. Bogomolov and F. Pop: Let F1 and F2 be fields finitely-generated and of transcendence degree ≥2 over k1 and k2, respectively, where k1 is either Qˉ or Fˉp, and k2 is algebraically closed. We denote by $G_{…
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…
The Frey--Mazur conjecture states that an elliptic curve over Q is determined up to isogeny by its p-torsion Galois representation for p≥17. We study a geometric analog of this conjecture, and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic multip…
The absolute Galois group of 3-manifolds determines their structure up to homeomorphism.
problem Determining the structure of 3-manifolds using their absolute Galois groups.
method Defined a relative absolute Galois group for 3-manifolds and used Chebotarev density properties and Hilbert ramification theory.
result Two branched covers of the three-sphere over a stably Chebotarev link are homeomorphic if and only if their absolute Galois groups are isomorphic.
Arithmetic topology connects surface and p-adic field studies, enabling new insights into Galois groups.
problem Understanding the relationship between surfaces and p-adic fields through arithmetic topology. method Uniform approach using pro-p groups, graph of groups, and discrete splittings. result Infinite order arithmetic Dehn twists in Galois groups, connecting to classical Dehn twists on surfaces.
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…
This paper revisits Differential Galois Theory using Hopf algebras for Lie pseudogroups.
problem Understanding the structure of algebraic Lie pseudogroups using differential algebra and geometry.
method Mixing differential algebra, differential geometry, and algebraic geometry; using Hopf algebras.
result Reveals confusion between prime differential ideals and maximal ideals in Vessiot's work.
Study Galois groupoids of discret Painlevé equations.
problem Computing Galois groupoids for discret Painlevé equations.
method Using semi-continuity theorem for Galois groupoid in confluence of difference to differential equations.
result Computed Galois groupoids for discret Painlevé equations.
In this paper, complement-equivalent arithmetic Zariski pairs will be exhibited answering in the negative a question by Eyral-Oka on these curves and their groups. A complement-equivalent arithmetic Zariski pair is a pair of complex projective plane curves having Galois-conjugate equations in some number field whose co…
The paper uses Tannakian reconstruction to understand hyperbolic log-orbi curves.
problem Understanding the structure of hyperbolic log-orbi curves.
method Formulates hyperbolic uniformization as a Tannakian reconstruction theorem and constructs a canonical maximal parahoric PSL2-Higgs object.
result Reconstructs the absolute Galois group of a one-variable complex function field as the inverse limit of etale fundamental groups of orbifold models.
We discuss the concept of Galois structure and Galois epimorphism in a general setting. Namely, a Galois structure for an epimorphism π:M→B in some category C is the action of a group object that gives to M the structure of principal homogeneous space in the relative category CB.
Computes the component group of arbitrary real algebraic groups.
problem Computing the component group of arbitrary real algebraic groups.
method Structure results on algebraic groups and Galois cohomology methods.
result The group of connected components π0G(R) is an elementary Abelian 2-group. The paper classifies sextic curves on a Fano 3-fold with rational Galois covers in 3D space.
problem Classifying sextic curves on a Fano 3-fold with rational Galois covers in 3D space.
method Explicit constructions and invariants of Galois groups.
result The moduli space of such sextic curves has complex dimension 2.
We contribute to the arithmetic/topology dictionary by relating asymptotic point counts and arithmetic statistics over finite fields to homological stability and representation stability over $\Cb$ in the example of configuration spaces of n points in smooth varieties. To do this, we import the method of homological …
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.
problem Understanding arithmetic properties of absolute Galois groups through analogies with mapping class groups.
method Introduces arithmetic pro-ℓ Orr invariants and spaces, and investigates their properties and relations. result Determines the rank of the pro-ℓ Orr space as a Zℓ-module. 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…
The paper classifies orbits of semisimple elements in real semisimple Lie algebras.
problem Classifying orbits of semisimple elements in real semisimple Lie algebras.
method Case by case analysis of complex numbers and Galois cohomology for real numbers.
result Characterization of orbits with real representatives.
Let X be a normal, separated and integral scheme of finite type over Z and M a set of closed points of X. To a Galois cover X~ of X unramified over M, we associate a quandle whose underlying set consists of points of X~ lying over M. As the limit of…
Simply-connected surfaces of general type for n≥5.
problem Topological structures of Galois covers of surfaces of minimal degree.
method Investigation of Galois covers of surfaces of minimal degree in complex projective space.
result Galois covers of surfaces of minimal degree are simply-connected for n≥5.
Classifies real trivectors in 9D, following complex classification methods.
problem Classifying real trivectors in 9D space.
method Used Galois cohomology to divide trivectors into nilpotent, semisimple, and mixed groups.
result Classification of real trivectors in 9D space follows the same pattern as complex classification.
Paper constructs connections on curves with specific Galois groups.
problem Constructing connections with prescribed differential Galois groups.
method Restricting to trivial vector bundles and using Lie algebra from regular forms.
result Differential Galois group is a closure of the Lie algebra.
Characterizes bi-Perron numbers with specific Galois conjugates.
problem Understanding bi-Perron numbers with real or unimodular conjugates.
method Characterization through power properties and spectral radii of transformations.
result Bi-Perron numbers with real or unimodular conjugates admit a power as stretch factors or spectral radii.
Proves section conjecture for curves and surface bundles over various fields.
problem Proving Grothendieck's section conjecture for curves and surface bundles.
method Formulated and proved the section conjecture for stable graphs, used Galois cohomology classes to obstruct sections.
result Proved section conjecture for curves and surface bundles over p-adic and number fields.
Study of Bauer-Furuta invariants under Lie group actions and Galois coverings.
problem Investigating invariants of 4-manifolds under group actions and Galois coverings.
method Functorial approach to equivariant invariants and study in Galois covering situations.
result Ordinary invariants of quotients are determined by equivariant invariants of the covering manifold.
New findings on Malgrange-Galois groupoid for Painlevé VI equation parameters.
problem Understanding transformations preserving specific forms for Painlevé VI equation.
method Computed Malgrange-Galois groupoid for Painlevé VI family with all parameters.
result Solutions of Painlevé VI do not satisfy new partial differential equations.
The paper calculates Veech groups and Galois invariants for general origamis.
problem Understanding the structure and symmetries of origamis and their Galois invariants.
method Developed an algorithm to calculate Veech groups and orbits of Galois invariants for general origamis.
result Calculated Veech groups and Galois invariants for all origamis of degree d≤7. Develops Weil bundles over \( p \)-adic manifolds for arithmetic geometry.
problem Connecting differential calculus and arithmetic geometry over \( p \)-adic fields.
method Systematic theory of Weil bundles, developing analytic structures.
result Establishes canonical analytic structures on Weil bundles and their cohomological comparison.
High-performance quantum codes decoded with minimal data.
problem Efficient decoding of linear-rate LDPC quantum codes.
method Tessellations of hyperbolic manifolds, Coxeter groups, and Galois fields.
result Achieved encoding rate of 13/72 with high performance.
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 …
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 introduce a construction of pseudo-Anosov homeomorphisms on n-times punctured spheres and surfaces with higher genus using only sufficiently many positive half-twists. These constructions can produce explicit examples of pseudo-Anosov maps with various number-theoretic properties associated to the stretch factors, i…
Galois action on manifold structures of complex varieties is abelian.
problem Understanding the Galois action on topological manifold structures of complex varieties.
method Definition of profinite normal structure set and Galois action analysis.
result Galois action on manifold structures of simply-connected varieties is abelian.
This paper formalizes manifolds in positive characteristic varieties.
problem Establishing l-adic formal manifold structures on positive characteristic varieties.
method Develops and proves the existence of l-adic formal manifold structures and abelianized Galois symmetries.
result Proves l-adic homotopic equivalence and l-local lifting for simply-connected varieties.
Study shows convergence of Bergman kernels on covering spaces of Kähler manifolds.
problem Analyzing convergence of Bergman kernels on covering spaces of Kähler manifolds.
method Proving convergence of Bergman kernels and L2-Hodge numbers on a tower of coverings. result Sections of canonical line bundles give rise to immersions into projective spaces.
Characterizes algebraic integrability and minimality of Lie equations for non-commutative pseudogroups.
problem Understanding algebraic integrability and minimality of Lie equations for non-commutative pseudogroups.
method Algebraic characterization and differential Galois theory of rational connections.
result Equivalence of algebraic integrability to the triviality of the differential Galois group and demonstration of minimality under certain conditions.