The study extends Kazhdan's theorem to infinite Galois covers of Riemann surfaces.
problem Proving uniform convergence of canonical forms on towers of Riemann surfaces.
method Generalizing Kazhdan's theorem to infinite Galois covers, proving a Gauss--Bonnet type theorem.
result Uniform convergence of canonical forms on towers of Riemann surfaces under infinite Galois covers.
Let p:Σ′→Σ 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 study finds conditions for Kaehler-Einstein and cscK metrics on certain manifold coverings.
problem Existence of Kaehler-Einstein and cscK metrics on ramified coverings.
method Cohomological conditions on Kaehler classes and branching divisors.
result Sufficient conditions for the existence of cscK metrics on ramified coverings.
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…
We show that certain Galois covers of K-semistable Fano varieties are K-stable. We use this to give some new examples of Fano manifolds admitting Kähler-Einstein metrics, including hypersurfaces, double solids and threefolds.
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.
Let Γ be a finitely generated discrete group satisfying the rapid decay condition. We give a new proof of the higher Atiyah-Patodi-Singer theorem on a Galois Γ-coverings, thus providing an explicit formula for the higher index associated to a group cocycle c∈Zk(Γ;C) which is of polynomial growth wit…
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.
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.
Study stabilizes components of Galois cover moduli spaces.
problem Decide equivalence and stable equivalence of monodromy maps.
method Develop algebraic framework to study equivalence classes of monodromy maps.
result Recover a homological invariant that distinguishes equivalence classes.
Note proves index theorem for non-elliptic Heisenberg operators.
problem Proving index theorem for non-elliptic Heisenberg operators.
method Galois covering, Heisenberg elliptic differential operators, Γ-index theorem. result Example of Heisenberg operators with non-trivial Γ-index. 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…
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.
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.
Functoriality proved for higher rho invariants of elliptic operators.
problem Computing higher rho invariants of elliptic operators.
method Functoriality proved through finite-propagation argument.
result Maximal higher rho invariants behave functorially under quotient maps.
New examples of double Kodaira fibrations found using surface braid groups.
problem Finding new examples of double Kodaira fibrations.
method Using finite Galois covers of a product of curves and explicit group epimorphisms from surface braid groups to finite Heisenberg groups.
result Every curve of genus b is the base of a double Kodaira fibration; the number of non-isomorphic surfaces is at least $oldsymbolω(b+1)$. Second part of a series on higher coverings of racks and quandles.
problem Characterizing higher-dimensional centrality conditions in racks and quandles.
method Applying higher categorical Galois theory to racks and quandles.
result Identification and characterization of higher coverings, trivial coverings, and normal coverings.
Let 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.
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.
This paper extends rack and quandle covering theory using higher categorical Galois theory.
problem Developing a higher covering theory of racks and quandles.
method Applying techniques from higher categorical Galois theory to extend and clarify the foundations of rack and quandle coverings.
result Identification of meaningful higher-dimensional centrality conditions defining higher coverings of racks and quandles.
The study characterizes finite vector bundles on specific complex manifolds.
problem Characterizing finite vector bundles on complex manifolds.
method Proves a theorem for compact complex manifolds with Gauduchon astheno-Kahler metrics.
result Establishes a condition for finite vector bundles similar to Nori's theorem.
For any n>1, we construct examples branched Galois coverings from M to the nth projective space Pn where M is one of (P1)n, Cn or (B1)n, and B1 is the 1-ball. In terms of orbifolds, this amounts to giving examples of orbifolds over Pn uniformized by M.…
Researchers extend Gamma index theorem to non-compact spacetimes.
problem Establishing an L2-Gamma index for non-compact spacetimes. method Rewriting L2-Gamma index in terms of spectral flow and connecting to geometric expressions. result Extends Bär and Strohmaier's work to non-compact Cauchy hypersurfaces.
Let T be a complex torus, and X the surface CP^1 x T. If T is embedded in CP^{n-1} then X may be embedded in CP^{2n-1}. Let X_Gal be its Galois cover with respect to a generic projection to CP^2. In this paper we compute the fundamental group of X_Gal, using the degeneration and regeneration techniques, the Moishezon-T…
Globally irreducible nodes (i.e. nodes whose branches belong to the same irreducible component) have mild effects on the most common topological invariants of an algebraic curve. In other words, adding a globally irreducible node (simple nodal degeneration) to a curve should not change them a lot. In this paper we stud…
We show in this paper that the set of irreducible components of the family of Galois coverings of P^1_C with Galois group isomorphic to D_n is in bijection with the set of possible numerical types. In this special case the numerical type is the equivalence class (for automorphisms of D_n) of the function which to each …
Study ramification in knot groups through finite covers and their quotients.
problem Understanding ramification in knot groups and their covers.
method Formalized ramification theory for knot groups, analyzed through finite quotients, profinite completions, and cohomology.
result Characterized ramification and inertia subgroups in knot groups and their covers.
Extends Atiyah-Singer Dirac operator study to non-compact spacetimes.
problem Analyzing Dirac operator on non-compact spacetimes with non-compact Cauchy hypersurface.
method Building on previous works, extends Fredholm result to non-compact Lorentzian spaces, using von Neumann algebras and Galois coverings.
result Γ-Fredholmness of the Dirac operator under APS boundary conditions.
Introduces logarithmic Cartan geometry on complex manifolds with singularities.
problem Holomorphic Cartan geometry with singularities.
method Definition and study of logarithmic Cartan geometry on complex manifolds with polar part supported on a normal crossing divisor.
result Push-forward of a Cartan geometry constructed using a finite Galois ramified covering is a logarithmic Cartan geometry.
This article establishes the algebraic covering theory of quandles. For every connected quandle we explicitly construct a universal covering, which in turn leads us to define the algebraic fundamental group as the automorphism group of the universal covering. We then establish the Galois correspondence between connecte…
New insights into Anosov representations of hyperbolic groups.
problem Understanding Anosov representations of relatively hyperbolic groups.
method Proving representations can be interpreted as restricted Anosov representations over flow spaces and showing stability under deformations.
result Representations of certain types are divergent, extended geometrically finite and stable under small deformations.
Study branched coverings of singular (G,X)-manifolds, solving open questions.
problem Understanding branched coverings of singular (G,X)-manifolds.
method Developed a Galois theory for branched coverings, constructed developping maps for singular manifolds.
result Solved open questions and constructed new examples related to singular (G,X)-manifolds.
Extends canonical measures to metric graphs and proves a generalized Kazhdan's theorem.
problem Understanding limiting measures on metric graphs and their relation to hyperbolic measures.
method Introducing hyperbolic measures on universal covers of metric graphs and proving a generalized Kazhdan's theorem.
result All limiting measures on metric graphs satisfy a Gauss-Bonnet formula, interpreted as a trace formula.
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…
The study finds infinitely many Shimura subvarieties in Jacobian loci for curves of genus 2, 3, and 4.
problem Understanding Shimura subvarieties in Jacobian loci for curves of positive genus.
method Analyzing Galois covers of curves and their Shimura subvarieties under specific numerical conditions.
result The Jacobian locus contains infinitely many Shimura subvarieties of positive dimension for g≤4. 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.
A construction of Kaehler-Einstein metrics using Galois coverings, studied by Arezzo-Ghigi-Pirola, is generalized to orbifolds. By applying it to certain orbifold covers of P^n which are trivial set theoretically, one obtains new Einstein metrics on odd-dimensional spheres. The method also gives Kaehler-Einstein metric…
We study primary and secondary invariants of leafwise Dirac operators on foliated bundles. Given such an operator, we begin by considering the associated regular self-adjoint operator Dm on the maximal Connes-Skandalis Hilbert module and explain how the functional calculus of Dm encodes both the leafwise calculus…
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.
We give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold X which satisfies the Witt condition. This construction is inductive. It is then used to show that the signature operator is essentially self-adjoint and has discrete spectrum of finite multiplicity, so that…
The paper reformulates Hodge index theorem on Kähler manifolds using L2-index theory.
problem Proving the Hodge index theorem on compact Kähler manifolds.
method Using Atiyah's L2-index theory and Galois coverings. result Established the relationship between signature and L2-Hodge numbers. 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.
This paper extends classical Galois theory to differential equations, linking it to geometry and mechanics.
problem Generalizing Galois theory to differential equations and systems.
method Mixing differential algebra, differential geometry, and algebraic geometry.
result Established a new theory of differential Galois theory for algebraic pseudogroups.
Study group extensions and bundles on manifolds.
problem Understanding group extensions and their relation to bundles on manifolds.
method Establish and study a correspondence between extensions and equivariant bundles using non-abelian cohomology.
result A correspondence between G^-bundles and twisted Γ-equivariant bundles on Galois Γ-coverings. Extends methods to study polynomial roots over finite fields.
problem Stability of arithmetic statistics for polynomial roots.
method FI_G-modules, Grothendieck-Lefschetz trace formula, subexponential bounds.
result Average value of Gauss sums stabilizes as polynomial degree increases.
For large genus, precise monodromy groups are calculated for surface covers.
problem Calculating precise monodromy groups for large genus surface covers.
method Hodge-theoretic methods, including a generic Torelli theorem with coefficients.
result Precise connected monodromy groups are calculated for large genus surface covers.
Abstract reviews actions of the absolute Galois group on geometric and topological objects.
problem Understanding the absolute Galois group Γ Q and its actions on geometric/topological structures.
method Exploring Grothendieck's ideas and related works on dessins d'enfant, Teichmüller towers, and nonlinear actions.
result Conjectures and insights into homomorphisms between absolute Galois group and automorphism groups of related objects.
New links are shown to be stably generic based on Chebotarev law.
problem Understanding the relationship between knot analogues and prime ideals.
method Analyzing sequences of knots and their decompositions in 3-manifolds.
result Chebotarev knots form stably generic links.