Proofs for decomposing branched affine surfaces into triangles and cylinders.
problem Decomposing branched affine surfaces into simpler geometric shapes.
method Proof of Veech's theorem and introduction of invariant α.
result Any pair of decompositions can be connected by flips.
Holomorphic projective structures and bundles are studied on surfaces, revealing affine spaces of parameters.
problem Holomorphic projective structures and bundles on surfaces.
method Generalization of principal bundle of projective 2-frames to branched projective structures.
result Affine spaces of branched projective structures with given branching classes.
Classifies orbit closures of mapping class group action on affine character variety.
problem Classifying orbit closures of Mod(Σ)-action on χ(Aff(C)). method Classification up to exceptions of Mod(Σ)-action on affine character variety. result The only obstruction for a non-abelian representation to be the holonomy of a branched affine structure is to be Euclidean and not have positive volume.
Characterizes representations for complex projective structures with specific branch data.
problem Understanding representations of surface groups as holonomy of complex projective structures.
method Computing holonomies for spherical metrics and affine structures with prescribed conical angles.
result Computed holonomies for spherical metrics and affine structures with specific conical angles.
Study on moduli spaces of branched projective structures on surfaces.
problem Characterizing and understanding moduli spaces of branched projective structures.
method Analytic and geometric methods to study the moduli spaces of branched projective structures.
result The moduli space of marked branched projective structures is a complex analytic space with specific dimensions and singular points.
Researchers create a partial resolution of Coulomb branches for gauge theories.
problem Understanding partial resolutions of Coulomb branches in gauge theories.
method Constructing partial resolutions as variants of generalized slices in geometric contexts.
result Identified partial resolutions with specific geometric objects.
Study quantized Coulomb branches of Jordan quiver gauge theories and their connections to Cherednik algebras.
problem Understanding the structure of quantized Coulomb branches in Jordan quiver gauge theories.
method Proved isomorphisms between quantized Coulomb branches and spherical graded Cherednik and cyclotomic rational Cherednik algebras.
result Quantized Coulomb branches are deformations of subquotients of Yangians of affine gl(1). Mathematical definition of Coulomb branches for certain 3D gauge theories.
problem Defining the mathematical structure of Coulomb branches in 3D gauge theories.
method Mathematical definition as an affine algebraic variety with Cimes-action. result Mathematical definition of Coulomb branches for specific cases.
The paper studies Coulomb branches of quiver gauge theories and their connections to Grassmannians.
problem Understanding the structure of Coulomb branches in quiver gauge theories.
method Analyzes the moduli space of rational maps and slices in the affine Grassmannian.
result Identifies the quantized Coulomb branch with the truncated shifted Yangian.
This paper studies ring objects in equivariant derived Satake category from Coulomb branches.
problem Understanding ring objects in equivariant derived Satake category from Coulomb branches.
method Analyzes morphisms from varieties to affine Grassmannian and studies ring objects in equivariant derived category.
result Various constructions in arXiv:1601.03586 work for arbitrary commutative ring objects.
Researchers discover all affinely homogeneous models for surfaces in 4D space.
problem Identifying all affinely homogeneous models for surfaces in 4D space.
method Improved power series method of equivalence, capturing invariants at the origin, creating branches, and infinitesimalizing calculations.
result Find several inequivalent terminal branches yielding each to some nonempty moduli space of homogeneous models.
We prove that in genus bigger than 2, the mapping class group action on Aff(C)-characters is ergodic. This implies that almost every representation π1S⟶Aff(C) is the holonomy of a branched affine structure on S, where S is a closed orientable surface of ge…
First we provide a simple set of sufficient conditions for the weak convergence of scaled affine processes with state space R+×Rd. We specialize our result to one-dimensional continuous state branching processes with immigration. As an application, we study the asymptotic behavior of least squares estimators…
New method proves identifiability of deep generative models with algebraic contrast principles.
problem Proving identifiability of deep generative models in unsupervised learning.
method Introducing three algebraic contrast principles: domain contrast, mechanism contrast, and interaction contrast.
result Proves identifiability of deep generative models with piecewise-affine decoders and Gaussian mixture priors.
Smooth fractal trees via analytic generators, preserving combinatorial and geometric properties.
problem Constructing smooth fractal trees from discrete models.
method Using analytic generator fields to integrate smooth vector fields in an internal state space, generating geometric curves as projections of generator trajectories.
result Analytic generators can represent any discrete tree specification and preserve the asymptotic limit geometry.
Study classifies special Hessian rank 2 hypersurfaces in 4D space.
problem Classifying hypersurfaces with constant Hessian rank 2.
method Power series method of equivalence, Lie's classification spirit.
result 34 inequivalent terminal branches, each with a nonempty moduli space.
Study of tangent cones at infinity for algebraic sets.
problem Characterizing algebraic sets based on their tangent cones at infinity.
method Definition and analysis of tangent cones C4,∞(X) and C5,∞(X), proving properties and relations. result Affine linear subspace characterization based on C5,∞(X)'s dimension. Study on affine surfaces with specific algebraic properties.
problem Characterize homogeneous affine surfaces with Hessian rank 2.
method Investigate algebra of differential invariants under affine transformation group.
result Organize homogeneous models into inequivalent branches.
The paper extends symplectic techniques to generalized complex geometry.
problem Creating stable generalized complex structures on high-dimensional manifolds.
method Introducing generalized Luttinger surgery and generalized Gluck twist.
result Produced stable generalized complex structures with non-homotopy-equivalent components.
Study infinite superelliptic curves and their Veech groups, providing geometric and algebraic insights.
problem Characterize Veech groups of infinite superelliptic curves.
method Analyzing geometric properties, differential equations, and group theory.
result Veech groups of infinite superelliptic curves are all matrices permuting branched points.
Study growth patterns in random networks using i.i.d. perturbations.
problem Understanding the growth of affine regions in random piecewise-linear networks.
method Analyzes a random compositional model with i.i.d. perturbations of the tent map, proving submultiplicative pressure and using finite-state defect process for upper-tail lower bounds.
result Proves the existence of a submultiplicative pressure for \(N_n\) and gives exponential upper bounds for \(n^{-1}\log N_n\).
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.
New connections found between knot invariants and Rozansky-Witten theory.
problem Understanding physical interpretations of knot invariants.
method Studying Rozansky-Witten theory with non-compact target spaces.
result New formulations of knot invariants using affine Grassmannians and q-series.
Study continuous deformations of branched projective structures on surfaces, preserving holonomy and branch points.
problem Continuous deformations of branched projective structures on closed surfaces of genus g≥2. method Schiffer variations and analysis of canonical divisors.
result Branch points are necessarily arranged on a canonical divisor when the underlying complex structure is infinitesimally preserved.
The paper proves conditions for minimal surfaces to be holomorphic and stable.
problem Conditions for stable minimal surfaces to be holomorphic.
method Developed a method of constructing variations to prove the equivalence.
result Holomorphicity and stability conditions for minimal surfaces.
We prove the existence of (branched) conformal immersions F: S^2 -> R^3 with mean curvature H > 0 arbitrarily prescribed up to a 3-dimensional affine indeterminacy. A similar result is proved for the space forms S^3, H^3 and partial results for surfaces of higher genus.
Research characterizes intersection cohomology groups of gauge theories and cotangent bundles.
problem Characterizing intersection cohomology groups of Coulomb branch gauge theories.
method Uses geometric Satake correspondence for Kac-Moody settings.
result Sketches proof of conjecture in affine type A.
Authors prove a contact structure result using branched covers and overtwisted disks.
problem Proving a contact structure result using branched covers and overtwisted disks.
method Explicitly constructing an overtwisted disk in the p-fold cyclic branched cover of S3. result An overtwisted disk is contained in the complement of the branch locus.
We introduce and analyze the characteristic foliation induced by a contact structure on a branched surface, in particular a branched standard spine of a 3-manifold. We extend to (fairly general) singular foliations of branched surfaces the local existence and uniqueness results which hold for genuine surfaces. Moreover…
New model explains low interest rates and large bond market jumps.
problem Explaining recent observations in sovereign bond market.
method Introduces α-CIR model using α-stable Lévy process and branching property. result Unified and parsimonious model for low interest rates and large jumps.
DMTG groups tasks for multi-task learning in one shot.
problem Efficiently grouping and training multiple tasks in machine learning.
method Formulates Multi-Task Grouping as a differentiable pruning problem, training all groups simultaneously.
result Significantly improves training efficiency and mitigates objective bias.
We prove that if S is a closed compact surface of negative Euler characteristic, and if R is a quasi-Fuchsian representation in PSL(2,C), then the deformation space M(k,R) of branched projective structures on S with total branching order k and holonomy R is connected, as soon as k>0. Equivalently, two branched projecti…
We define and study branched shadows of 4-manifolds as a combination of branched spines of 3-manifolds and Turaev's shadows. We use these objects to combinatorially represent 4-manifolds equipped with Spinc-structures and homotopy classes of almost complex structures. We then use branched shadows to study complex 4-…
We propose in this paper a method for studying contact structures in 3-manifolds by means of branched surfaces. We explain what it means for a contact structure to be carried by a branched surface embedded in a 3-manifold. To make the transition from contact structures to branched surfaces, we first define auxiliary ob…
New framework generalizes complex projective structures, proving non-existence of certain structures on Calabi-Yau manifolds.
problem Proving non-existence of certain holomorphic structures on Calabi-Yau manifolds.
method Introducing branched holomorphic Cartan geometries, proving independence of results, and using vector bundle properties.
result Non-projective compact simply connected Kähler Calabi-Yau manifolds do not admit branched holomorphic projective structures.
Study geodesics of meromorphic connections on Riemann surfaces.
problem Understanding the asymptotic behaviors of geodesics in meromorphic connections.
method Use branched affine structure induced by Fuchsian meromorphic connections.
result Examples of geodesics with infinitely many self-intersections and peculiar omega-limit sets.
Affine structures on Lie groupoids are studied, showing rich algebraic properties.
problem Understanding affine structures on Lie groupoids.
method Analyzing affine k-vector fields, k-forms, and (p,q)-tensors, and showing their algebraic properties. result The space of affine structures forms a 2-vector space over multiplicative structures, and affine multivector fields have a Lie 2-algebra structure.
StrTransformer recovers sources without labels by optimizing latent matrices and enforcing structural constraints.
problem Unsupervised blind source recovery in signal processing.
method Source-wise structured Transformer framework with latent source matrix optimization, structural regularization, and branch-specific weights.
result StrTransformer learns distinct temporal-scale structures and recovers source-aligned latent trajectories.
We study affine Jacobi structures on an affine bundle π:A→M, i.e. Jacobi brackets that close on affine functions. We prove that there is a one-to-one correspondence between affine Jacobi structures on A and Lie algebroid structures on the vector bundle A+=⋃p∈MAff(Ap,R) of affine functionals. Som…
New CR structures found for once-punctured torus bundles.
problem No natural CR realisation for hyperbolic once-punctured torus bundles.
method Introduced a new cell decomposition and branched CR structure.
result Every hyperbolic once-punctured torus bundle admits a branched CR structure.
This paper describes integral affine structures on compact 3-manifolds.
problem Understanding integral affine structures on compact 3-manifolds.
method Analyzing complete integral affine structures on compact 3-manifolds up to finite-sheeted coverings.
result A complete list of integral affine structures on the three-dimensional torus and compact three-dimensional nilmanifolds was obtained.
In this paper we study some affine structures on nilpotent Lie algebras endowed with a contact form. These affine structures are constructed from an affine structure on a symplectic Lie algebra by a central extension.
Loi and Piergallini showed that a smooth compact, connected 4-manifold X with boundary admits a Stein structure if and only if X is a simple branched cover of a 4-disk D4 branched along a positive braided surface S in a bidisk D12×D22≈D4. For each integer N≥2, we constr…
We study affine maps between affine manifolds. Even when the fibers are compact and diffeomorphic, two of them can inherit different affine structures from the source space. This leads to a fixed linear holonomy deformation theory of the affine structure of an affine manifold. We found various conditions which make the…
The paper constructs new structures for manifolds using connections and combinations.
problem Understanding smooth manifolds with precise infinitesimal affine structures.
method Constructing new infinitesimal structures for higher-order neighbourhoods of the diagonal.
result Any symmetric affine connection on a manifold extends to a second-order infinitesimally affine structure.
Almost Zoll affine surface found on cylinder.
problem Finding surfaces with special geodesic properties.
method Exhibited an affine structure on a cylinder.
result Affine structure on cylinder is almost Zoll.
Survey on generalized holomorphic Cartan geometries.
problem Classifying holomorphic Cartan geometries on compact Calabi-Yau manifolds.
method Introducing branched holomorphic Cartan geometries and classifying them.
result Classification of branched holomorphic Cartan geometries on compact Calabi-Yau manifolds.
Ideal triangulations of 3-manifolds are shown equivalent up to certain moves.
problem Equivalence of ideal triangulations in 3-manifolds.
method Using branched triangulations and transit equivalences, the paper shows that ideal triangulations are equivalent up to certain moves.
result Ideal triangulations of 3-manifolds are equivalent up to certain moves.