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. In this note we survey recent results on the extrinsic geometry of the Jacobian locus inside Ag. We describe the second fundamental form of the Torelli map as a multiplication map, recall the relation between totally geodesic subvarieties and Hodge loci and survey various results related to totally geodesic…
This paper is devoted to the classification of connected components of Prym eigenform loci in the strata H(2,2)^odd and H(1,1,2) in the Abelian differentials bundle in genus 3. These loci, discovered by McMullen are GL^+(2,R)-invariant submanifolds (of complex dimension 3) that project to the locus of Riemann surfaces …
To any compact Riemann surface of genus g one may assign a principally polarized abelian variety of dimension g, the Jacobian of the Riemann surface. The Jacobian is a complex torus, and a Gram matrix of the lattice of a Jacobian is called a period Gram matrix. This paper provides upper and lower bounds for all the ent…
Geometric regularisation improves statistical models by avoiding degeneracy loci.
problem Non-identifiability, singular information, and moment indeterminacy in statistical models.
method Develops the geometric regularisation of distribution-kernel pairs (T,φ) using Whitney, Thom, and Mather theorems. result Finite-dimensional weak transversality theorem for generic kernels, avoiding degeneracy strata of high codimension.
Researchers create explicit representations for skein algebras of small surfaces, revealing their Azumaya loci.
problem Understanding the structure and properties of skein algebras of small surfaces.
method Constructed finite-dimensional representations at all roots of unity, using explicit formulas and analyzing reducibility.
result Azumaya loci of the surfaces contain the smooth loci of classical shadow varieties, with equality for the one-punctured torus and proper containment for the four-punctured sphere.
In this note, we describe a procedure to construct generalized complex structures with an arbitrarily large number of type change loci on products of the circle with a connected sum of closed 3-manifolds. The loci need not be isotopic.
Study geodesics and shortest arcs on Lie groups with specific metrics.
problem Characterize geodesics and shortest arcs in sub-Riemannian metrics on Lie groups.
method Investigated left-invariant sub-Riemannian metrics on SU(1,1)imesR and SO0(2,1)imesR. result Found geodesics, shortest arcs, cut loci, and conjugate loci.
In this paper, we study Fuchsian loci of PSLn(R)-Hitchin components. In particular, using the Bonahon-Dreyer parametrization of PSLn(R)-Hitchin components, we give an explicit parametrization of Fuchsian loci of a pair of pants.
The Jacobian Conjecture is proven for all Jacobian maps.
problem Proving the Jacobian Conjecture for all Jacobian maps.
method Using the Weyl algebra and holonomic modules, the paper shows that the Jacobian module is 1-generated and has finite length.
result The Jacobian Conjecture is true for all Jacobian maps.
We survey the cohomology jumping loci and the Alexander-type invariants associated to a space, or to its fundamental group. Though most of the material is expository, we provide new examples and applications, which in turn raise several questions and conjectures. The jump loci of a space X come in two basic flavors: th…
This study connects Jacobian regularization to adversarial robustness and improves generalization.
problem Adversarial attacks make deep neural networks vulnerable.
method Developed a connection between Jacobian regularization and adversarial training, and established robust generalization gaps.
result Jacobian norms are related to both standard and robust generalization.
As an increasing number of genome-wide association studies reveal the limitations of attempting to explain phenotypic heritability by single genetic loci, there is growing interest for associating complex phenotypes with sets of genetic loci. While several methods for multi-locus mapping have been proposed, it is often…
Study Riemannian metrics on lens spaces, find cut loci and diameters.
problem Understanding Riemannian metrics on lens spaces and their geometric properties.
method Geometric control theory methods applied to axisymmetric metrics.
result Cut loci and cut times converge to sub-Riemannian structure's values.
Study Riemannian metrics on lens spaces, find cut loci and diameters.
problem Analyzing Riemannian metrics on lens spaces.
method Geometric control theory methods.
result Cut loci and cut times converge to sub-Riemannian structure's cut locus and time.
We introduce the natural and fairly general notion of a subanalytic bundle (with a finite dimensional vector space P of sections) on a subanalytic subset X of a real analytic manifold M, and prove that when M is compact, there is a Baire subset U of sections in P whose zero-loci in X have tubular neighbou…
New method studies discriminantal loci of algebraic varieties.
problem Understanding discriminantal loci of algebraic varieties.
method Efficient use of groupoids to describe monodromy.
result New insights into discriminantal loci of hypersurfaces.
New upper bound for geodesic complexity derived from cut locus decompositions.
problem Understanding geodesic complexity in Riemannian manifolds.
method Study of decompositions of cut loci and their tangent fibers.
result Established a new upper bound for geodesic complexity.
Study curvature loci of 3-manifolds in R^6 and R^5.
problem Characterize curvature loci of 3-manifolds in different dimensions.
method Refine affine classification of real nets of quadrics, study singularities, and analyze systems of ternary cubics.
result Obtain generic curvature loci and singularities of 3-manifolds.
Study shows connections between Jacobian torsors and Fermat curves.
problem Understanding torsors of Jacobian of universal Fermat curves.
method Analyzes torsors of Jacobian of universal family of degree-m Fermat curves. result Every torsor is a connected component of the Picard scheme.
Study elliptic isometries on a matrix manifold with specific metrics.
problem Differential-geometric properties of fixed point loci.
method Explicit description and De Rham decomposition of fixed point loci.
result Explicit description and De Rham decomposition of fixed point loci.
Study geodesics and shortest arcs on Lie groups with specific metrics.
problem Characterize geodesics and shortest paths on Lie groups with sub-Riemannian metrics.
method Analytical and geometric methods to find geodesics and shortest arcs.
result Found geodesics, shortest arcs, distances, and conjugate loci for specified metrics.
Generalizes results on Bieri-Neumann-Strebel-Renz invariants and tropical varieties.
problem Relationship between Bieri-Neumann-Strebel-Renz invariants and homology jump loci.
method Uses tropical varieties to detect components of homology jump loci and generalizes results to integral coefficients.
result Provides a better upper bound for Bieri-Neumann-Strebel-Renz invariants and classifies Kähler groups.
Study geodesics on a special cylinder with arbitrary wind.
problem Global behavior of geodesics on a Randers metric cylinder.
method Solve Zermelo's navigation problem to define the Randers metric; analyze geodesics, conjugate, and cut loci.
result Characterize geodesics and their properties on the base manifold.
New potentials found for sheaves on Calabi-Yau 4-folds.
problem Understanding sheaves on Calabi-Yau 4-folds.
method Derived Quot-stacks and Lagrangian distributions.
result Globally defined −1-shifted potentials on sheaves. Let X⊂PN be a scroll over a smooth curve C and let Ł=OPN(1)∣X denote the hyperplane bundle. The special geometry of X implies that some sheaves related to the principal part bundles of Ł are locally free. The inflectional loci of X can be expressed in terms of these she…
Study Alexander invariants and cohomology jump loci in group extensions with trivial monodromy.
problem Understanding Alexander invariants and cohomology jump loci in group extensions with specific conditions.
method Analyzing integral, rational, and modular Alexander invariants and cohomology jump loci of groups as extensions with trivial algebraic monodromy.
result Established a tight relationship between Alexander invariants, characteristic varieties, and resonance varieties, leading to an inequality between Chen ranks.
This work proves the asymptotic freeness of layerwise Jacobians in MLPs with Haar orthogonal matrices.
problem Proving the asymptotic freeness of layerwise Jacobians in multilayer perceptrons (MLPs).
method Replacing each layer's parameter matrix with itself multiplied by a Haar orthogonal matrix, and using the invariance of the MLP.
result Proves the asymptotic freeness of layerwise Jacobians in MLPs with Haar orthogonal matrices.
Efficiently regularizes deep learning models using Jacobian nuclear norm.
problem Regularizing deep learning models to prevent overfitting and improve generalization.
method Proposes a denoising-style approximation to penalize the Jacobian nuclear norm without computing the Jacobian matrix.
result Demonstrates that penalizing the average squared Frobenius norm of Jg and Jh is equivalent to penalizing the Jacobian nuclear norm for function compositions. We show that some riemannian manifolds diffeomorphic to the sphere have the property that the cut loci of general points are smoothly embedded closed disks of codimension one. Ellipsoids with distinct axes are typical examples of such manifolds.
Study on algebraic fiber spaces and their anti-canonical divisors.
problem Understanding positivity conditions and base loci of algebraic fiber spaces.
method Algebraic and analytic methods for positivity of direct image sheaves.
result Algebraic fiber spaces with semi-ample relative anti-canonical divisor have a product structure.
Abstract: Unknown status of Jacobian Conjecture, proof has a gap.
problem Status of Jacobian Conjecture
method Analysis of proof of theorem 2.1
result Proof of theorem 2.1 contains a gap
Study of Randers metrics on spheres with simple cut loci.
problem Understanding Randers metrics on spheres and their cut loci.
method Analyzing geodesics, conjugate, and cut loci of Finsler metrics of Randers type.
result Found new families of Randers metrics with simple cut loci.
A well-conditioned Jacobian spectrum has a vital role in preventing exploding or vanishing gradients and speeding up learning of deep neural networks. Free probability theory helps us to understand and handle the Jacobian spectrum. We rigorously show almost sure asymptotic freeness of layer-wise Jacobians of deep neura…
Constructs harmonic 1-forms on K3-fibred Calabi-Yau 3-folds.
problem Analyzing harmonic 1-forms on K3-fibred Calabi-Yau 3-folds.
method Analytic construction of nowhere-vanishing harmonic 1-forms.
result Produces examples of compact 7-manifolds with holonomy G2.
The Jacobian conjecture is simplified using polynomial mappings.
problem Simplifying the Jacobian conjecture over the real field.
method Using polynomial mappings to restrict transitions on manifolds.
result An equivalent statement of the Jacobian conjecture.
We prove a Lefschetz hyperplane theorem for the determinantal loci of a morphism between two holomorphic vector bundles E and F over a complex manifold under the condition that $E^*\ox F$ is Griffiths k-positive. We apply this result to find some homotopy groups of the Brill-Noether loci for a generic curve.
Paper tackles Hessian/Jacobian-free stochastic bilevel optimization with O(ε−1.5) complexity.
problem Nonconvex-strongly-convex bilevel optimization problem.
method FdeHBO optimizer with finite-difference Hessian/Jacobian-vector approximation and momentum.
result FdeHBO achieves O(ε−1.5) iterations for ε-accurate stationary point. Study of longest arcs and cut loci in deformed anti de-Sitter spaces.
problem Existence and properties of time-like cycles in deformed Lorentzian manifolds.
method Analysis of universal covering, admissible curves, and Lorentzian geodesics.
result Identification of cut time and cut locus in deformed anti de-Sitter spaces.
Recent work (Pennington et al, 2017) suggests that controlling the entire distribution of Jacobian singular values is an important design consideration in deep learning. Motivated by this, we study the distribution of singular values of the Jacobian of the generator in Generative Adversarial Networks (GANs). We find th…
The paper discusses fractional Sobolev immersions of flat domains into 3D space.
problem Developing C1 regularity and isometric immersions of flat domains with fractional Sobolev regularity. method Analysis of weak Codazzi-Mainardi equations, study of $W^{2, rac2s}$ planar deformations, and properties of the distributional Jacobian determinant.
result Generalization of isometric immersions with local fractional Sobolev regularity.
We derive an analytic formula for the dual Jacobian matrix of a generalised hyperbolic tetrahedron. Two cases are considered: a mildly truncated and a prism truncated tetrahedron. The Jacobian for the latter arises as an analytic continuation of the former, that falls in line with a similar behaviour of the correspondi…
New algorithms estimate Jacobian matrices for large-scale machine learning.
problem Efficiently computing search directions for large nonlinear least squares.
method Exploit low-rank structure in Hessian to estimate Jacobian matrices.
result Two algorithms perform well compared to state-of-the-art methods.
The paper extends infinite-width analysis to neural network Jacobians, revealing convergence to Gaussian processes and linear ODEs.
problem Understanding the training dynamics of neural networks in the infinite-width limit.
method Extending infinite-width analysis to Jacobians, characterizing convergence to Gaussian processes and linear ODEs.
result The evolution of MLPs under robust training in the infinite-width limit is described by a linear ODE.
GrokAlign aligns Jacobians to accelerate grokking in deep networks.
problem Accelerating the training dynamics of deep networks to avoid delayed generalisation and robustness.
method Aligning the Jacobians of a deep network with the training data to ensure grokking under a low-rank assumption.
result GrokAlign regularizes Jacobians to induce grokking sooner than conventional methods.
We provide a characterization for complex analytic curves among two-dimensional minimal graphs in R4 via the Jacobian
The paper studies the locus in the rank 2 Higgs bundle moduli space corresponding to points which are critical for d of the Poisson commuting functions. These correspond to the Higgs field vanishing on a divisor of degree D. The degree D critical locus has an induced integrable system related to K(-D)-twisted Higgs bun…
We extend the well-known result that any f∈W1,n(Ω,Rn), Ω⊂Rn with strictly positive Jacobian is actually continuous: it is also true for fractional Sobolev spaces Ws,sn(Ω) for any s≥n+1n, where the sign condition on the Jacobian is understood in a distr…