arXiv research

The Jacobian Conjecture is proven for all Jacobian maps.

We compute some value of the harmonic volume for the Fermat sextic. Using this computation, we prove that some special algebraic cycle in the Jacobian variety of the Fermat sextic is not algebraically equivalent to zero.

We prove some value of the harmonic volume for the Klein quartic $C$ is nonzero modulo ${1/2}\{mathbb Z}$, using special values of the generalized hypergeometric function ${}_3F_2$. This result tells us the algebraic cycle $C-C^-$ is not algebraically equivalent to zero in the Jacobian variety $J(C)$.

The paper constructs tilting modules for knots using algebraic structures.

New insights into algebraic geometry of a conjecture, leading to origami curves.

A Teichmüller curve is an algebraic and isometric immersion of an algebraic curve into the moduli space of Riemann surfaces. We give the first explicit algebraic models of Teichmüller curves of positive genus. Our methods are based on the study of certain Hilbert modular forms and the use of Ahlfors's variational formu…

Paper discusses conditions for global injectivity of semi-algebraic local diffeomorphisms.

This study connects Jacobian regularization to adversarial robustness and improves generalization.

We prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties. The result is deduced from a generalization of a theorem of Möller. Namely, we prove that the image of a certain twisted Abel-Jacobi map lands in the torsion of a factor of the Jacobians. This statement can be viewed as a split…

Study shows connections between Jacobian torsors and Fermat curves.

Algorithm constructs algebraic curves from translation surfaces.

This work proves the asymptotic freeness of layerwise Jacobians in MLPs with Haar orthogonal matrices.

Efficiently regularizes deep learning models using Jacobian nuclear norm.

We compute the formal Poisson cohomology of a broken Lefschetz fibration by calculating it at fold and Lefschetz singularities. Near a fold singularity the computation reduces to that for a point singularity in 3 dimensions. For the Poisson cohomology around singular points we adapt techniques developed for the Sklyani…

Abstract: Unknown status of Jacobian Conjecture, proof has a gap.

We look into a construction of principal abelian varieties attached to certain spin manifolds, due to Witten and Moore-Witten around 2000 and try to place it in a broader framework. This is related to Weil intermediate Jacobians but it also suggests to associate abelian varieties to polarized even weight Hodge structur…

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…

The Jacobian conjecture is simplified using polynomial mappings.

Paper tackles Hessian/Jacobian-free stochastic bilevel optimization with ${O}(ε^{-1.5})$ complexity.

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.

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.

The paper extends infinite-width analysis to neural network Jacobians, revealing convergence to Gaussian processes and linear ODEs.

GrokAlign aligns Jacobians to accelerate grokking in deep networks.

We provide a characterization for complex analytic curves among two-dimensional minimal graphs in $\mathbb{R}^{4}$ via the Jacobian

We extend the well-known result that any $f \in W^{1,n}(Ω,\mathbb{R}^n)$, $Ω\subset \mathbb{R}^n$ with strictly positive Jacobian is actually continuous: it is also true for fractional Sobolev spaces $W^{s,\frac{n}{s}}(Ω)$ for any $s \geq \frac{n}{n+1}$, where the sign condition on the Jacobian is understood in a distr…

New method reduces deep learning training costs by approximating vector-jacobian products.

Normalizing flows optimize Jacobian determinant for unique likelihood objective.

The Jacobian of Douady-Earle extension equals 1 only for isometries.

The aim here is to continue the investigation in \cite{AB} of Jacobians of a Klein surface and also to correct an error in \cite{AB}.

To a compact Riemann surface of genus g can be assigned a principally polarized abelian variety (PPAV) of dimension g, the Jacobian of the Riemann surface. The Schottky problem is to discern the Jacobians among the PPAVs. Buser and Sarnak showed, that the square of the first successive minimum, the squared norm of the …

We show that the Goldman flows preserve the holomorphic structure on the moduli space of homomorphisms of the fundamental group of a Riemann surface into U(1), in other words the Jacobian.

Design of reliable systems must guarantee stability against input perturbations. In machine learning, such guarantee entails preventing overfitting and ensuring robustness of models against corruption of input data. In order to maximize stability, we analyze and develop a computationally efficient implementation of Jac…

The paper identifies magnetic ground states and their role in determining the conformal class of a surface.

Generative adversarial networks (GANs) are notoriously difficult to train and the reasons underlying their (non-)convergence behaviors are still not completely understood. By first considering a simple yet representative GAN example, we mathematically analyze its local convergence behavior in a non-asymptotic way. Furt…

New approach approximates c-space geometry of multi-loop linkages.

Recovering hidden influence networks from cascade data using Jacobian-based machine learning.

The paper analyzes privacy leakage in federated learning using linear algebra and optimization theory.

Geometrically represents path integral reduction Jacobian for interacting systems.

The paper analyzes the observability of relative pose estimation using dual quaternions.

This work relaxes energy constraints in self-attention layers for a more general analysis.

JacNet learns Jacobians to enforce structure on derivatives for invertibility and Lipschitz functions.

The paper studies global invertibility of maps on Finsler manifolds.

The Jacobian matrix (or the gradient for single-output networks) is directly related to many important properties of neural networks, such as the function landscape, stationary points, (local) Lipschitz constants and robustness to adversarial attacks. In this paper, we propose a recursive algorithm, RecurJac, to comput…

We show that there are separated nets in the Euclidean plane which are not biLipschitz equivalent to the integer lattice. The argument is based on the construction of a continuous function which is not the Jacobian of a biLipschitz map.

We conjecture an expression for the dimensions of the Khovanov-Rozansky HOMFLY homology groups of the link of a plane curve singularity in terms of the weight polynomials of Hilbert schemes of points scheme-theoretically supported on the singularity. The conjecture specializes to our previous conjecture relating the HO…

We study families of Galois covers of curves of positive genus. It is known that under a numerical condition these families yield Shimura subvarieties generically contained in the Jacobian locus. We prove that there are only 6 families satisfying this condition, all of them in genus 2,3 or 4. We also show that these fa…