arXiv research

Proves conjecture linking WRT invariants and homological blocks for plumbed 3-manifolds.

Knots and 4-manifolds linked via matrix kinking.

New metrics defined on SPD matrices link to divergences and curvature.

This paper derives radial fields on manifolds of symmetric positive definite matrices.

In this note, we complete the classification of quasi-alternating Montesinos links. We show that the quasi-alternating Montesinos links are precisely those identified independently by Qazaqzeh-Chbili-Qublan and Champanerkar-Ording. A consequence of our proof is that a Montesinos link $L$ is quasi-alternating if and onl…

Study of metrics on positive-definite matrices from power potential, linking to power means.

Proves a conjecture about 3-manifold invariants using trees and asymptotic formulas.

The class of Schoenberg transformations, embedding Euclidean distances into higher dimensional Euclidean spaces, is presented, and derived from theorems on positive definite and conditionally negative definite matrices. Original results on the arc lengths, angles and curvature of the transformations are proposed, and v…

We exhibit the first examples of links which are homologically thin but not quasi-alternating. To show that they are not quasi-alternating, we argue that none of their branched double-covers bounds a negative definite 4-manifold with non-torsion H_1. Using this method, we also complete the determination of the quasi-al…

Let $D$ be an oriented link diagram with the set of regions $\operatorname{r}_{D}$. We define a symmetric map (or matrix) $\operatornameτ_{D}\colon\operatorname{r}_{D}\times \operatorname{r}_{D} \to \mathbb{Z}[x]$ that gives rise to an invariant of oriented links, based on a slightly modified $S$-equivalence of Trotter…

Consider a random vector with finite second moments. If its precision matrix is an M-matrix, then all partial correlations are non-negative. If that random vector is additionally Gaussian, the corresponding Markov random field (GMRF) is called attractive. We study estimation of M-matrices taking the role of inverse sec…

New $k$-means method clusters radar image sequences using SPD matrices.

The main goal of the present article is the computation of the Heegaard Floer homology introduced by Ozsvath and Szabo for a family of plumbed rational homology 3-spheres. The main motivation is the study of the Seiberg-Witten type invariants of links of normal surface singularities.

We provide a mathematical definition of fragility and antifragility as negative or positive sensitivity to a semi-measure of dispersion and volatility (a variant of negative or positive "vega") and examine the link to nonlinear effects. We integrate model error (and biases) into the fragile or antifragile context. Unli…

Paper proposes a new algorithm for graph learning with covariance constraints.

We show that the link cobordism maps defined by the author are graded and satisfy a grading change formula. Using the grading change formula, we prove a new bound for $Υ_K(t)$ for knot cobordisms in negative definite 4-manifolds. As another application, we show that the link cobordism maps associated to a connected, cl…

Algorithm transforms weakly negative plumbing trees to negative definite ones.

The paper explores Cholesky decompositions for symmetric matrices and their geometric properties.

We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weights. In particular, it is shown th…

Improved method for computing Fréchet means on SPD matrices.

Study negative definite spin fillings of knot covers.

R-PLS improves analysis of brain functional connectivity matrices.

Study of J-Hermitian matrices and geometric mean definition.

The (stochastic) gradient descent and the multiplicative update method are probably the most popular algorithms in machine learning. We introduce and study a new regularization which provides a unification of the additive and multiplicative updates. This regularization is derived from an hyperbolic analogue of the entr…

Classifies contact structures on negative-definite Seifert fibred spaces.

We find an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices.

Paper derives explicit expression of Alekseev-Meinrenken diffeomorphism.

Develops log-Euclidean Lie groups for SPD and correlation matrices.

Proposes a new Sliced-Wasserstein distance for covariance matrices in M/EEG signals.

Algorithm calculates Seifert matrices for colored links.

Bootstrapping regularizes singular correlation matrices, reducing the need for complex regularization.

Counterexample disproves conjecture on flat metrics and fiber bundles.

For any negative definite plumbed 3-manifold M we construct from its plumbed graph a graded Z[U]-module. This, for rational homology spheres, conjecturally equals the Heegaard-Floer homology of Ozsvath and Szabo, but it has even more structure. If M is a complex singularity link then the normalized Euler-characteristic…

Laundry surfaces for closed braid diagrams are presented. It is shown that braid diagrams are characterized by linking matrices obtained by lifting cycles from these surfaces. Oriented link types are then characterized by equivalence classes of linking matrices. Similar equivalence classes can be composed of Gordon and…

We study the differential-geometric properties of the loci of fixed points of the elliptic isometries of the manifold of definite positive real matrices with the trace metric. We also give an explicit description of such loci and in particular we find their De Rham decomposition.

Paper introduces a generalized Bures-Wasserstein geometry for SPD matrices.

New method classifies manifold-valued data using Riemannian geometry.

This work improves understanding of symmetrizing Bregman divergences on positive definite matrices.

Efficiently reduces rank of non-negative matrices with quadratic time complexity.

We study the differential geometric properties of the manifold of non-singular symmetric real matrices endowed with the trace metric; in case of positive definite matrices we describe the full group of isometries

A new way to describe correlation matrices makes modeling easier.

Positive definite matrices abound in a dazzling variety of applications. This ubiquity can be in part attributed to their rich geometric structure: positive definite matrices form a self-dual convex cone whose strict interior is a Riemannian manifold. The manifold view is endowed with a "natural" distance function whil…

Study of strictly accretive matrices using Finsler geometry.

This paper classifies minimal fillings of lens spaces.

A non-singular sesquilinear form is constructed that is preserved by the Lawrence-Krammer representation. It is shown that if the polynomial variables q and t of the Lawrence-Krammer representation are chosen to be appropriate algebraically independant unit complex numbers, then the form is negative-definite Hermitian.…

Study uses graph Laplacians to analyze surface links.

The study examines entropy and pressure at infinity in negatively curved manifolds, linking them to strong positive recurrence.

We provide in this note two relevant examples of Lagrangian cobordisms. The first one gives an example of two exact Lagrangian submanifolds which cannot be composed in an exact fashion. The second one is an example of an exact Lagrangian cobordism on which all primitive of the Liouville form is not constant on the nega…