arXiv research

Study of uncountable family of finitely generated groups.

Generalization bounds derived for neural ODEs and deep residual networks.

We study conformal $Spin$-subgeometry of submanifolds in a semi-Riemannian $Spin$-manifold, focusing on conformal $Spin$-manifolds $(M,[h])$ and their Poincaré-Einstein metrics $(X,g_+)$. Our approach is based on the spectral theory of Dirac operator in the ambient $Spin$-manifold, and associated spinor valued meromorp…

Researchers define residue families and use them to solve singular Yamabe problems.

Protein contacts contain important information for protein structure and functional study, but contact prediction from sequence remains very challenging. Both evolutionary coupling (EC) analysis and supervised machine learning methods are developed to predict contacts, making use of different types of information, resp…

We provide local expressions for Chern-Weil type forms built from superconnections associated with families of Dirac operators previously investigated in work by S. Scott and later work by S. Scott and the second author. When the underlying fibration of manifolds is trivial, the even degree forms can be interpreted as …

Graph neural networks suffer from oversmoothing, but adding residual connections helps.

An analytic approach and description are presented for the moduli cotangent sheaf for suitable stable curve families including noded fibers. For sections of the square of the relative dualizing sheaf, the residue map at a node gives rise to an exact sequence. The residue kernel defines the vanishing residue subsheaf. F…

The fundamental n-quandles of links are residually finite for n ≥ 2.

Residual networks with block width max(d_x, d_y) approximate all functions.

Flow-based generative models parameterize probability distributions through an invertible transformation and can be trained by maximum likelihood. Invertible residual networks provide a flexible family of transformations where only Lipschitz conditions rather than strict architectural constraints are needed for enforci…

New knot groups found to be bi-orderable using pretzel knots.

Deep residual networks implicitly converge to neural ODEs.

Deviance-style normalization for sparse, jointly overdispersed count matrices

The paper improves random forest models for non-Gaussian responses.

Global analysis of Dixmier traces and Wodzicki residues on compact Lie groups.

Study of $η$ invariants on lens spaces detects distinctions invisible to ordinary $η$.

Paper improves bike-sharing demand prediction by adapting to changing patterns.

Abstract: Generalizes modular forms to family case and finds new anomaly cancellation formulas.

Residual networks' depth is mathematically equivalent to expanding an implicit ensemble size.

In this paper we introduce a new family of operator-valued distributions on Euclidian space acting by convolution on differential forms. It provides a natural generalization of the important Riesz distributions acting on functions, where the corresponding operators are $(-Δ)^{-α/2}$, and we develop basic analogous prop…

Improves CNN stability by translating classical signal denoising methods.

We introduce new aspects in conformal geometry of some very natural second-order differential operators. These operators are termed shift operators. In the flat space, they are intertwining operators which are closely related to symmetry breaking differential operators. In the curved case, they are closely connected wi…

Using the Wodzicki residue, we build Wodzicki-Chern-Simons (WCS) classes in $H^{2k-1}(LM)$ associated to the residue Chern character on the loop space $LM$ of a Riemannian manifold $M^{2k-1}$. These WCS classes are associated to the $L^2$ connection and the Sobolev $s=1$ connections on $LM.$ The WCS classes detect seve…

In this paper, we study the interplay between modules and sub-objects in holomorphic Poisson geometry. In particular, we define a new notion of "residue" for a Poisson module, analogous to the Poincaré residue of a meromorphic volume form. Of particular interest is the interaction between the residues of the canonical …

We show that for a $C^1$ residual subset of diffeomorphisms far away from tangency, every non-trivial chain recurrent class that is accumulated by sources ia a homoclinic class contains periodic points with index 1 and it's the Hausdorff limit of a family of sources.

The paper establishes a Poisson Poincaré-Dulac theorem for Poisson-flat connections.

Study of resonances and residue operators for hyperbolic spaces.

Artificial neural networks (ANNs) have gained a well-deserved popularity among machine learning tools upon their recent successful applications in image- and sound processing and classification problems. ANNs have also been applied for predicting the family or function of a protein, knowing its residue sequence. Here w…

For a holomorphic family of classical pseudodifferential operators on a closed manifold we give exact formulae for all coefficients in the Laurent expansion of its Kontsevich-Vishik canonical trace. This generalizes a known result identifying the Wodzicki residue with the pole at zero to all higher order terms.

We consider an analytic family of Riemannian metrics on a compact smooth manifold $M$. We assume the Dirichlet boundary condition for the $η$-Laplacian and obtain Hadamard type variation formulas for analytic curves of eigenfunctions and eigenvalues. As an application, we show that for a subset of all $C^r$ Riemannian …

We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via towers of finite coverings of surfaces for which we need to develop a relative …

Study compact Willmore surfaces without complex structure convergence, computing energy loss and geodesic lengths.

Proves small cancellation free products have geometric actions on CAT(0) cube complexes.

The paper proves geometric and spectral alignment for deep neural networks.

An econometric analysis of CRIX family indices.

This paper shows how to construct Abelian differentials with any prescribed singularities.

We present in this article a family of new combinatorial identities via purely differential/complex geometry methods, which include as a speical case a unified and explicit formula for Chern numbers of all complex flag manifolds. Our strategy is to construct concrete circle actions with isolated fixed points on these m…

The paper classifies constraint mappings in optimization problems.

We formulate certain sufficient conditions for the symplectic monodromy of an isolated quasihomogeneous singularity to be of infinite order in the relative symplectic mapping class group of the Milnor fibre and give a proof using Maslov classes, stability theory for Lagrangian folds resp. stable Morse theory for genera…

Study Eisenstein metrics on modular group representations.

We obtain in arbitrary codimension a removability result on the order of singularity of weak limits and bubbles of Willmore immersions measured by the second residue. This permits to reduce significantly the number of possible bubbling scenarii. As a consequence, out of the twelve families of non-planar minimal surface…

We describe GJMS-operators as linear combinations of compositions of natural second-order differential operators. These are defined in terms of Poincaré-Einstein metrics and renormalized volume coefficients. As special cases, we find explicit formulas for conformally covariant third and fourth powers of the Laplacian. …

Abstract: Non-residually finite hyperbolic groups imply non-residually finite rigid hyperbolic groups.

Residual finiteness is known to be an important property of groups appearing in combinatorial group theory and low dimensional topology. In a recent work [2] residual finiteness of quandles was introduced, and it was proved that free quandles and knot quandles are residually finite. In this paper, we extend these resul…

In this note, residual finiteness of quandles is defined and investigated. It is proved that free quandles and knot quandles of tame knots are residually finite and Hopfian. Residual finiteness of quandles arising from residually finite groups (conjugation, core and Alexander quandles) is established. Further, residual…

Graph-structured data such as social networks, functional brain networks, gene regulatory networks, communications networks have brought the interest in generalizing deep learning techniques to graph domains. In this paper, we are interested to design neural networks for graphs with variable length in order to solve le…

We extend the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator $δ$, constructed from an elliptic family of operators indexed by $S^1$. We show that the regularized values $η(δ_t,0)$ and $tζ(δ_t,0)$ are smooth functions of …