Study of Tannakian categories for integrable connections on Kaehler manifolds.
problem Understanding Tannakian categories for integrable connections on Kaehler manifolds.
method Analyzing pairs (E, D) where E is a trivial holomorphic vector bundle and D is an integrable holomorphic connection.
result The pro-algebraic affine group scheme uniquely determines the isomorphism class of compact Riemann surfaces.
Geometric models for algebraic suspensions using affine deformation spaces.
problem Analyzing which motivic homotopy types have smooth scheme representatives.
method Affine deformation spaces of embeddings provide models for P1-suspensions of schemes. result Yields many examples of A1-connected and strictly quasi-affine schemes. Anabelian geometry reformulated using Hodge theory for hyperbolic curves.
problem Determining varieties over number fields using their étale fundamental groups.
method Formulating a Hodge-theoretic version of anabelian conjecture, replacing Galois action with Cimes-action. result Proved a Hodge-theoretic analog of Mochizuki's theorem for smooth projective hyperbolic curves over C. Study on curvatures of surfaces in specific Lie groups.
problem Analyzing curvatures of surfaces in 3D contact sub-Riemannian Lie groups.
method Riemannian approximation scheme to derive formulas for curvatures.
result Classification of surfaces with constant horizontal curvatures.
This is the third companion paper of arXiv:1601.03586. When a gauge theory has a flavor symmetry group, we construct a partial resolution of the Coulomb branch as a variant of the definition. We identify the partial resolution with a partial resolution of a generalized slice in the affine Grassmannian, Hilbert scheme o…
Parallel transport is an important step in many discrete algorithms for statistical computing on manifolds. Numerical methods based on Jacobi fields or geodesics parallelograms are currently used in geometric data processing. In this last class, pole ladder is a simplification of Schild's ladder for the parallel transp…
This work deals with the simulation of Wishart processes and affine diffusions on positive semidefinite matrices. To do so, we focus on the splitting of the infinitesimal generator, in order to use composition techniques as Ninomiya and Victoir or Alfonsi. Doing so, we have found a remarkable splitting for Wishart proc…
We study the partial resolutions of singularities related to Hilbert schemes of points on an affine space. Consider a quotient of a vector space V by an action of a finite group G of linear transforms. Under some additional assumptions, we prove that the partial desingularization of Hilbert type is smooth only if t…
The study constructs differential systems on Riemann surfaces and explores their monodromy properties.
problem Constructing holomorphic differential systems with specific monodromy properties.
method Exploring the monodromy of holomorphic differential systems on Riemann surfaces.
result Holomorphic maps from Riemann surfaces to quotient spaces exist without factoring through elliptic curves.
Study of Hilbert schemes and Coulomb branches of hypertoric varieties.
problem Understanding the geometry and topology of Coulomb branches of hypertoric varieties.
method Investigation of transverse equivariant Hilbert schemes and Hamiltonian reductions, proposing new metrics.
result Coulomb branches of hypertoric varieties can be constructed as Hilbert schemes or Hamiltonian reductions.
Improves MCMC performance with adaptive affine transformations.
problem Improving the performance of Markov Chain Monte Carlo samplers.
method Adaptive learning of bijective affine transformations during sampling.
result Adaptive affine transformations improve the quality of samples at low computational cost.
Lectures on link homology and its algebraic/geometric models.
problem Defining and understanding link homology.
method Definition and properties of Khovanov-Rozansky triply graded homology, and three geometric models.
result Three geometric models for link homology: braid varieties, Hilbert schemes, and affine Springer fibers.
Affine cactus groups are CAT(0) and hyperbolic.
problem Characterizing geometric properties of affine cactus groups.
method Analyzing CAT(0) and hyperbolic properties through group theory.
result Affine cactus groups of degree three are hyperbolic.
Generalizes soft noncommutative schemes to flag varieties.
problem Applying soft noncommutative schemes to flag varieties.
method Generalization via toric geometry and distinguished affine charts.
result Soft noncommutative schemes can be applied to flag varieties.
This paper deals essentially with affine or projective transformations of Lie groups endowed with a flat left invariant affine or projective structure. These groups are called flat affine or flat projective Lie groups. Our main results determine Lie groups admitting flat bi-invariant affine or projective structures. Th…
Calculates affine transformations for specific homogeneous spaces.
problem Computing groups of affine transformations on homogeneous spaces.
method Analyzes conditions for affine connections and uses them to establish group isomorphisms.
result Groups of affine transformations are locally isomorphic under specified conditions.
In this paper we exhibit a family of flat left invariant affine structures on the double Lie group of the oscillator Lie group of dimension 4, associated to each solution of classical Yang-Baxter equation given by Boucetta and Medina. On the other hand, using Koszul's method, we prove the existence of an immersion of L…
The affine-additive group is hyperbolic with a non-vanishing 4-capacity.
problem Characterizing the hyperbolicity of the affine-additive group.
method Proving local 4-Ahlfors regularity and hyperbolicity using a left-invariant metric and measure.
result The affine-additive group is hyperbolic with a non-vanishing 4-capacity.
Paper explores properties of slice-matching operators for measure transfer.
problem Efficiently transferring measures in high dimensions.
method Examines an associated slice-matching operator with source, target measures and slicing directions.
result Establishes invariance, equivariance, Lipschitz continuity, and error bounds.
Vector fields on schemes have flows if rings are finitely generated.
problem Understanding vector fields and flows on schemes.
method Analyzing vector fields on affine C∞-schemes with finitely generated rings. result Vector fields on affine C∞-schemes with finitely generated rings have flows and are groupoid internal maps. We give a new characterization of flat affine manifolds in terms of an action of the Lie algebra of classical infinitesimal affine transformations on the bundle of linear frames. We characterize flat affine symplectic Lie groups using symplectic étale affine representations and as a consequence of this, we show that a …
New manifold structures on Weyl group orbit spaces proven.
problem Constructing generalized Frobenius manifold structures.
method Construction on orbit spaces of affine Weyl groups.
result Monodromy groups are parabolic subgroups.
Authors calculate limits of curvatures on surfaces in sub-Riemannian manifolds.
problem Calculating limits of Gaussian and normal curvatures on surfaces in sub-Riemannian manifolds.
method Utilized Riemannian approximations scheme in Heisenberg group to calculate limits of curvatures.
result Obtained Gauss-Bonnet theorem as a limit of theorems in approximations schemes.
This research studies affine invariance in continuous-domain convolutional neural networks.
problem Recognizing patterns and features under affine transformations in continuous domains.
method Introduces a new criterion for assessing affine invariance, embeds images into the affine Lie group, and analyzes convolution over this group.
result Extends the scope of geometrical transformations that deep-learning pipelines can handle.
Formula derived for discrete improper affine spheres.
problem Constructing discrete improper affine spheres.
method Loop group factorizations and Birkhoff decomposition.
result Representation formula for discrete indefinite affine spheres.
We develop the foundations of the deformation theory of compact complete affine space forms and affine crystallographic groups. Using methods from the theory of linear algebraic groups we show that these deformation spaces inherit an algebraic structure from the space of crystallographic homomorphisms. We also study th…
This paper deals with affine connections on real manifolds. We give a new characterization of flat affine connections on real manifolds by means of certain affine representations of the Lie group of automorphisms preserving the connection. Then we specialize the characterization to the case of a left invariant connecti…
Proves nonemptyness of domains for specific group actions.
problem Nonemptyness of domains of proper discontinuity for Anosov groups of affine Lorentzian transformations.
method Proof of nonemptyness of domains of proper discontinuity.
result Proves nonemptyness of domains for Anosov groups of affine Lorentzian transformations.
The paper discusses algorithms for reconstructing curves with given Euclidean or affine curvatures.
problem Reconstructing planar curves with specified Euclidean or affine curvatures.
method The paper presents algorithms for curve reconstruction under the special Euclidean and equi-affine groups.
result The reconstructed curves are close to the original curves in terms of the specified curvatures.
Proof of K(π,1) conjecture for affine Artin groups.
problem Asphericality of complements of affine hyperplane arrangements.
method Combinatorics of noncrossing partition posets, dual Artin groups, and topological models.
result Affine Artin groups are aspherical.
Affine Artin groups have a finite classifying space.
problem Proving the K(π,1) conjecture for affine Artin groups. method Dual Garside structures, Euclidean isometries, and shellability of noncrossing partitions.
result Affine Artin groups have a finite classifying space.
Constructs generalized Frobenius manifolds for specific Weyl groups.
problem Creating structures for orbit spaces of Weyl groups.
method Applying a previously established construction method to specific Weyl groups.
result Generalized Frobenius manifold structures constructed for Aℓ,Bℓ,Cℓ and Dℓ. We give a characterization of flat affine connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the connection. From the infinitesimal point of view, this representation is determined by the 1-connection form and the fundamental for…
Paper generalizes connections between Lie groups and affine connections.
problem Exploring properties of infinitesimal groups and affine connections.
method Introducing second-order infinitesimal groups and using them to define Lie brackets and connections.
result Generalized correspondence between symmetric and non-symmetric affine connections.
New method constructs proper affine actions of groups in higher dimensions.
problem Finding proper affine actions of discrete groups in higher-dimensional spaces.
method Higher strip deformations and Margulis invariant for properness.
result Affine actions of convex cocompact groups and virtually free groups are constructed properly.
We define a fuchsian affine action of a surface group to be such that the linear part factors through a representation of SL(2,R). We prove a fuchsian affine action of a surface group is never proper.
This paper classifies reversible and strongly reversible elements in affine groups.
problem Classifying reversible and strongly reversible elements in affine groups.
method Identifying affine transformations and using conjugacy by involutions.
result Classification of reversible and strongly reversible elements in affine groups.
The study limits the cohomological dimension of certain affine manifolds with partially hyperbolic holonomy groups.
problem Understanding cohomological dimensions of affine manifolds with specific holonomy groups.
method Analyzing the tangent bundle structure and using coarse geometry techniques.
result The cohomological dimension is bounded by the dimension minus the index of the holonomy group.
The study of which mapping class group elements can be realized as affine automorphisms of dilation surfaces.
problem Which elements of the mapping class group can be realized as affine automorphisms of dilation surfaces?
method Investigation into the affine automorphism groups of dilation surfaces, including the construction of dilation surfaces from multicurves.
result Only certain types of mapping class group elements can arise as affine automorphisms of dilation surfaces.
To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N, via such affine transformations. We succeed in translating the existence questio…
The paper calculates curvature limits and Gauss-Bonnet theorems in the Heisenberg group.
problem Computing curvature limits and Gauss-Bonnet theorems in the Heisenberg group.
method Sub-Riemannian limits of Gaussian curvature, Schouten-Van Kampen affine connections, and adapted connections.
result Gauss-Bonnet theorems associated with Schouten-Van Kampen affine connections in the Heisenberg group.
We prove the K(π,1) conjecture for affine Artin groups: the complexified complement of an affine reflection arrangement is a classifying space. This is a long-standing problem, due to Arnol'd, Pham, and Thom. Our proof is based on recent advancements in the theory of dual Coxeter and Artin groups, as well as on sever…
The paper studies Anosov holonomy groups in complete affine manifolds.
problem Characterizing Anosov holonomy groups in complete affine manifolds.
method Representation theory and coarse geometry techniques.
result Complete affine manifolds with Anosov holonomy groups have specific geometric properties.
Paper proposes Sp-GD for sparse max-affine regression with theoretical guarantees.
problem Sparse max-affine regression model selection and estimation.
method Sparse Gradient Descent (Sp-GD) initialization using sparse PCA and covering search.
result Sp-GD provides ε-accurate estimates with optimal number of observations.
The paper derives Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.
problem Computing curvature and geodesic curvature for surfaces and curves in affine and rigid motions groups.
method Defined deformed Schouten-Van Kampen connections, computed Gaussian curvature limits, and signed geodesic curvature.
result Derived Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.
This paper is the first arising from our project announced in math.AG/0211094, "Affine manifolds, log structures, and mirror symmetry." We aim to study mirror symmetry by studying the log structures of Illusie-Fontaine and Kato on degenerations of Calabi-Yau manifolds. The basic idea is that one can associate to certai…
We propose and study a row-and-column affine measurement scheme for low-rank matrix recovery. Each measurement is a linear combination of elements in one row or one column of a matrix X. This setting arises naturally in applications from different domains. However, current algorithms developed for standard matrix rec…
Affine 3-manifolds with centralizing holonomy are complete.
problem Closed flat affine manifolds with parallel volume are not always complete.
method Showed the Markus conjecture holds for 3-manifolds with centralizing holonomy.
result Holonomy centralizing an affine transformation preserves volume completeness.