The document provides tables of prehomogeneous and étale modules for reductive algebraic groups.
problem Classifying and tabulating prehomogeneous and étale modules for reductive algebraic groups.
method Classification and tabulation of prehomogeneous and étale modules based on existing work and the author's determination.
result Tables of prehomogeneous and étale modules for reductive algebraic groups with up to two simple factors.
Study etale modules for reductive groups with 1D center, finding constraints on their irreducible submodules.
problem Characterize etale modules for reductive groups with one-dimensional center.
method Analyze etale modules as prehomogeneous modules, focusing on constraints on irreducible submodules.
result No etale modules exist for certain reductive groups with one-dimensional center.
Given a bundle gerbe on a compact smooth manifold or, more generally, on a compact étale Lie groupoid M, we show that the corresponding category of gerbe modules, if it is non-trivial, is equivalent to the category of finitely generated projective modules over an Azumaya algebra on M. This result can be seen as an …
The classical Serre-Swan's theorem defines a bijective correspondence between vector bundles and finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an et…
We define stacky Lie groups to be group objects in the 2-category of differentiable stacks. We show that every connected and etale stacky Lie group is equivalent to a crossed module of the form (H,G) where H is the fundamental group of the given stacky Lie group and G is the connected and simply connected Lie group int…
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.
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. Paper compares topological and pro-étale fundamental groups.
problem No specific problem stated; comparing two fundamental groups.
method Constructs a comparison map between topological and pro-étale fundamental groups.
result Establishes a map between fundamental groups.
We generalize the notion of a small sheaf of sets over a topological space or manifold to define the notion of a small stack of groupoids over an étale topological or differentiable stack. We then provide a construction analogous to the étalé space construction in this context, establishing an equivalence of 2-categori…
In this article, we derive many properties of étale stacks in various contexts, and prove that étale stacks may be characterized categorically as those stacks that arise as prolongations of stacks on a site of spaces and local homeomorphisms. Moreover, we show that the bicategory of étale differentiable stacks and loca…
Proof shows Chern form is closed on groupoid convolution algebra.
problem Proving the Chern form's closedness on étale groupoids.
method Used bisection for algebraic proof.
result Chern form is closed on étale groupoid convolution algebra.
New framework for symplectic reduction on stacky spaces.
problem Symplectic reduction on stacky spaces.
method Introducing Hamiltonian actions on étale symplectic stacks and proving theorems.
result Established symplectic reduction theorems for stacky spaces.
Any etale Lie groupoid G is completely determined by its associated convolution algebra C_c(G) equipped with the natural Hopf algebroid structure. We extend this result to the generalized morphisms between etale Lie groupoids: we show that any principal H-bundle P over G is uniquely determined by the associated C_c(G)-…
Given two étale groupoids $\Cal G$ and $\Cal G'$, we consider the set of pointed morphisms from $\Cal G$ to $\Cal G'$. Under suitable hypothesis we introduce on this set a structure of Banach manifold which can be considered as the space of objects of an étale groupoid whose space of orbits is the space of morphisms fr…
We give a superconnection proof of Connes' index theorem for proper cocompact actions of etale groupoids. This includes Connes' general foliation index theorem for foliations with Hausdorff holonomy groupoid.
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 …
This paper adapts submersions, immersions, and étale maps to diffeology.
problem Providing suitable analogs for submersions, immersions, and étale maps in diffeology.
method Nonlinear approach to diffeological submersions, immersions, and étale maps.
result Characterization and properties of diffeological embeddings and étale maps.
Constructs explicit nontrivial cycles in Habiro cohomology of smooth varieties.
problem Describing and constructing nontrivial cycles in Habiro cohomology.
method Using either the Picard-Fuchs equation or push-forward of elements of the Habiro ring.
result Explicit classes for 1-parameter Calabi-Yau families and q-holonomic modules.
Finite vector bundles over complex manifolds are trivializable via finite covers.
problem Understanding when holomorphic vector bundles over compact complex manifolds are trivializable.
method Introducing finite bundles and using finite étale covers to trivialize holomorphic vector bundles.
result Holomorphic vector bundles over compact complex manifolds are finite if and only if they admit a flat holomorphic connection with finite monodromy.
Reconstruct Lie structures from functional-analytic data on groupoids.
problem Reconstructing Lie structures from functional-analytic data on groupoids.
method Characterizing smooth structures, introducing Lie twists, and establishing conditions for making twists into Lie twists.
result Conditions for making Renault's Weyl twist into a Lie twist with specified normalizers.
In this paper we establish a duality between etale Lie groupoids and a class of non-necessarily commutative algebras with a Hopf algebroid structure. For any etale Lie groupoid G over a manifold M, the groupoid algebra C_c(G) of smooth functions with compact support on G has a natural coalgebra structure over C_c(M) wh…
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…
Explains model structures for higher orbifolds and applies them to quantum cohomology.
problem Understanding quantum cohomology of higher orbifolds.
method Develops model structures on higher orbifolds and applies them to quantum cohomology.
result Model structures provide insights into quantum cohomology of higher orbifolds.
The study defines differential forms and currents on orbifolds with corners.
problem Defining differential forms and currents on orbifolds with corners.
method Using the formalism of étale proper groupoids with corners, the authors provide constructions and proofs without orbifold charts.
result The Fréchet space of differential forms and the dual space of currents are independent of the chosen groupoid representation.
Homotopy proof for pseudomanifolds via branched covers.
problem Understanding homotopy types of pseudomanifolds.
method Using branched covers and Artin-Mazur's etale homotopy type construction.
result Profinite completion of pseudomanifolds equals etale homotopy type of branched covers.
In this paper we consider a family of Dirac-type operators on fibration P→B equivariant with respect to an action of an etale groupoid. Such a family defines an element in the bivariant K theory. We compute the action of the bivariant Chern character of this element on the image of Connes' map Φ in the cyclic…
The Habiro ring of a number field uses power series to study algebraic K-theory.
problem Analyzing algebraic K-theory of number fields.
method Introduces Habiro ring and modules graded by K3(K). result Establishes connections between Habiro ring and Chern-Simons theory.
Stability of mapping spaces is shown to be related to the D-topology.
problem Understanding the relationship between stability and the D-topology of mapping spaces.
method Reformulation and proof of stability theorems in diffeological étale manifolds.
result Stable classes of mapping spaces are D-open.
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.
The abstract proves a global splitting theorem for Poisson manifolds.
problem Decomposing compact Kähler Poisson manifolds into simpler components.
method Proving a global splitting theorem using symplectic leaves and finite étale covers.
result Compact Kähler Poisson manifolds can be split into simpler components.
Smooth stacks of orbifolds are shown to be infinite-dimensional orbifolds.
problem Understanding the structure of Hom-stacks of orbifolds.
method Using Lie groupoids and Fréchet-Lie groupoids to represent Hom-stacks.
result Hom-stacks of orbifolds are infinite-dimensional orbifolds.
We use a counting argument and surgery theory to show that if D is a sufficiently general algebraic hypersurface in Cn, then any local diffeomorphism F:X→Cn of simply connected manifolds which is a d-sheeted cover away from D has degree d=1 or d=∞ (however all degrees d>1 are poss…
Researchers lift knot coloring polynomial to Habiro ring.
problem Lifting colored Jones polynomial of knots to Habiro ring.
method Introduced new Habiro ring and map, used Alexander polynomial.
result Existence of loop expansion at roots of unity confirmed.
A proper etale Lie groupoid is modelled as a (noncommutative) spectral geometric space. The spectral triple is built on the algebra of smooth functions on the groupoid base which are invariant under the groupoid action. Stiefel-Whitney classes in Lie groupoid cohomology are introduced to measure the orientability of th…
We observe that any regular Lie groupoid G over an manifold M fits into an extension K→G→E of a foliation groupoid E by a bundle of connected Lie groups K. If $\FF$ is the foliation on M given by the orbits of E and T is a complete transversal to $\FF$, this extension restricts to T, as an extension $K_{T}\to…
Study of profinite quandles with constructions and characterizations.
problem Characterizing and constructing profinite quandles.
method Several constructions and characterizations of profinite quandles from profinite groups and other quandles.
result Characterization of algebraically connected profinite quandles in terms of $\widehat{\Inn(Q)}$.
Synthetic theory defines orbifolds as microlinear types with finite identifications.
problem Defining orbifolds in traditional set-level foundations with internal symmetries.
method Synthetic differential cohesive homotopy type theory, microlinearity, finite identifications.
result Proper étale groupoids are orbifolds in synthetic theory.
The study classifies Kähler threefolds with special fiber bundles.
problem Classifying Kähler threefolds with specific fiber bundles.
method Generalized from projective case, using fiber bundles over the circle and étale covers.
result Proves Kotschick's conjecture in dimension 3.
Inverse function theorem and homotopy description for L-infinity bundles.
problem Inverse function theorem and homotopy description for L-infinity bundles.
method Local sections composed of elementary morphisms.
result Simple description of homotopy category of L-infinity bundles.
The paper studies polynomial maps with specific ramification patterns and computes their cohomology.
problem Computing the cohomology of moduli spaces of polynomial maps with prescribed ramification.
method Topological properties of the ramification poset, sheaf-theoretic argument, étale cohomology.
result Cohomology groups are independent of the degree of the polynomial for characteristic 0 or large enough.
Study shows how to separate jets on complex varieties using coverings.
problem Separating jets on complex varieties of maximal Albanese dimension.
method Using finite abelian étale covers and moving Seshadri constants.
result Existence of abelian covers that separate jets.
New algebraic fundamental groups identified for fake projective planes.
problem Characterizing algebraic fundamental groups of fake projective planes.
method Analysis of complex conjugate pairs and explicit finite étale covers.
result Forty-six distinct isomorphism classes of algebraic fundamental groups.
The paper generalizes a theorem and introduces a new characteristic map for foliated manifolds.
problem The challenge is to generalize Bott's vanishing theorem for foliated manifolds.
method The approach involves working with the full holonomy groupoid instead of the Morita equivalent étale groupoid, leading to novel geometric representatives of characteristic classes.
result A characteristic map encoding both primary and secondary characteristic classes is introduced.
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.
Lie groupoids and their orbit spaces are linked through equivalence classes.
problem Understanding the relationship between Lie groupoids and their orbit spaces.
method Introducing lift-complete Lie groupoids and showing equivalence between categories.
result Morita equivalence class of a lift-complete Lie groupoid is determined by its orbit space.
The paper extends arithmetic Chern-Simons invariants to real quadratic fields and calculates mod 2 Dijkgraaf-Witten invariants.
problem Calculating arithmetic Dijkgraaf-Witten invariants for real quadratic number fields.
method Using modified étale cohomology groups and fundamental groups, explicit formulas are derived for real quadratic fields.
result Explicit formulas for mod 2 arithmetic Dijkgraaf-Witten invariants for real quadratic fields are provided.
This paper generalizes wrinkling techniques to Haefliger structures, linking them to foliations.
problem Proving h-principles for partial differential relations with controlled singularities.
method Generalizing wrinkled embeddings to Haefliger structures and interpreting them as holonomic approximations.
result Haefliger structures provide a framework for making general wrinkling statements and imply connectivity results.
We consider the moduli space Rn of pairs of monic, degree n polynomials whose resultant equals 1. We relate the topology of these algebraic varieties to their geometry and arithmetic. In particular, we compute their étale cohomology, the associated eigenvalues of Frobenius, and the cardinality of their…