arXiv research

We pursue the study of holomorphic Cartan geometry with singularities. We introduce the notion of logarithmic Cartan geometry on a complex manifold, with polar part supported on a normal crossing divisor. In particular, we show that the push-forward of a Cartan geometry constructed using a finite Galois ramified coveri…

New framework for logarithmically divergent integrals on manifolds with corners.

Logarithmic connections on complex manifolds with trivial tangent bundle.

The paper constructs a Saito basis for a specific class of divisors and applies it to logarithmic Poisson geometry.

Finite-time queue peaks in stochastic networks have logarithmic scaling after geometric thresholds.

Study flat connections with logarithmic singularities on complex plane curves.

Study of foliations' geometric and topological structures.

We show that Masur's logarithmic law of geodesics in the moduli space of translation surfaces does not imply unique ergodicity of the translation flow, but that a similar law involving the flat systole of a Teichmüller geodesic does imply unique ergodicity. It shows that the flat geometry has a better control on ergodi…

Paper studies statistical manifolds with logarithmic divergences.

The paper explores the geometric structure of cost functions in multiple dimensions.

New aesthetic curves in equiaffine geometry include the quadratic and logarithmic spiral.

We study the logarithmic $L^{(α)}$-divergence which extrapolates the Bregman divergence and corresponds to solutions to novel optimal transport problems. We show that this logarithmic divergence is equivalent to a conformal transformation of the Bregman divergence, and, via an explicit affine immersion, is equivalent t…

We present a new method to solve certain $\bar{\partial}$-equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a $\bar{\partial}$-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness…

New uncertainty principle for Schrödinger equations on hyperbolic manifolds.

Uniform heat kernel and diffusion bridge asymptotics for sub-Riemannian geometry.

This paper constructs a non-Archimedean Teichmüller space using tropical geometry.

Paper explores geometry of covariance matrices using associated bundles.

New MD algorithms using Tempesta logarithms for machine learning.

We develop a general theory of log spaces, in which one can make sense of the basic notions of logarithmic geometry, in the sense of Fontaine-Illusie-Kato. Many of our general constructions with log spaces are new, even in the algebraic setting. In the differentiable setting, our theory yields a framework for treating …

We investigate three-dimensional surfaces where the normal vector forms a constant angle with the radius vector. These surfaces naturally extend equiangular (logarithmic) spirals in the plane.

Characterizes the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.

We generalize Cartan's logarithmic derivative of a smooth map from a manifold into a Lie group $G$ to smooth maps into a homogeneous space $M=G/H$, and determine the global monodromy obstruction to reconstructing such maps from infinitesimal data. The logarithmic derivative of the embedding of a submanifold $Σ\subset M…

New formulas for geodesics on Stiefel and flag manifolds using trust-region method.

A relation between the conformal anomaly and the logarithmic term in the entanglement entropy is known to exist for CFT's in even dimensions. In odd dimensions the local anomaly and the logarithmic term in the entropy are absent. As was observed recently, there exists a non-trivial integrated anomaly if an odd-dimensio…

The length of shortest non-simple geodesics grows logarithmically with surface genus.

In the previous article we derived a detailed asymptotic expansion of the heat trace for the Laplace-Beltrami operator on functions on manifolds with conic singularities. In this article we investigate how the terms in the expansion reflect the geometry of the manifold. Since the general expansion contains a logarithmi…

We prove the sharp local L^1 - L^\infty smoothing estimate for the logarithmic fast diffusion equation, or equivalently, for the Ricci flow on surfaces. Our estimate almost instantly implies an improvement of the known L^p - L^\infty estimate for p larger than 1. It also has several applications in geometry, providing …

Researchers map the fundamental group of polynomial strata to a braid group.

This paper consists of two parts. In the first part we show that in odd dimension, as well as in even dimension below the critical weight (i.e. half the dimension), the logarithmic singularities of Schwartz kernels and Green kernels of conformal invariant pseudodifferential operators are linear combinations of Weyl con…

Paper develops Riemannian geometry for SPSD matrices with DA applications.

Green functions play an important role in conformal geometry. In this paper, we explain how to compute explicitly the logarithmic singularities of the Green functions of the conformal powers of the Laplacian. These operators include the Yamabe and Paneitz operators, as well as the conformal fractional powers of the Lap…

The Milnor fiber conjecture is proven for splice type singularities.

New method tackles bilevel optimization with polyhedral constraints.

This survey consists of two parts. Part 1 is devoted to amoebas. These are images of algebraic subvarieties in the complex torus under the logarithmic moment map. The amoebas have essentially piecewise-linear shape if viewed at large. Furthermore, they degenerate to certain piecewise-linear objects called tropical vari…

We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding soltions with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with low…

We give a geometric derivation of Branson's Q-curvature in terms of the ambient metric associated with conformal structures; it naturally follows from the ambient metric construction of conformally invariant operators and can be applied to a large class of invariant operators. This procedure can be also applied to CR g…

Geodesics in Sol geometry described with invariant k and spiral properties.

Study explores warped geometries of tensor manifolds, finding non-geodesic connections for some parameters.

We investigate the infinitesimal invariants of an immersed submanifold $Σ$ of a Klein geometry $M\cong G/H$, and in particular an invariant filtration of Lie algebroids over $Σ$. The invariants are derived from the logarithmic derivative of the immersion of $Σ$ into $M$, a complete invariant introduced in the companion…

Ancient Ricci flows with asymptotic solitons have uniform bounds and inequalities.

This paper has four main parts. In the first part, we construct a noncommutative residue for the hypoelliptic calculus on Heisenberg manifolds, that is, for the class of Heisenberg PsiDOs introduced by Beals-Greiner and Taylor. This noncommutative residue appears as the residual trace on integer order Heisenberg PsiDOs…

Develops a new geometric framework for quantum metrics.

In the framework of Abstract Differential Geometry, we show that to a given principal sheaf and a representation of its stuctural sheaf in $A^n$, where A is a sheaf of associative, commutative, unital algebras (over R or C), we associate a vector sheaf. Moreover, under some natural assumptions on the compatibility of t…

We consider reinforcement learning in parameterized Markov Decision Processes (MDPs), where the parameterization may induce correlation across transition probabilities or rewards. Consequently, observing a particular state transition might yield useful information about other, unobserved, parts of the MDP. We present a…

This paper describes the connection between scattering matrices on conformally compact asymptotically Einstein manifolds and conformally invariant objects on their boundaries at infinity. The conformally invariant powers of the Laplacian arise as residues of the scattering matrix and Branson's Q-curvature in even dimen…

We investigate the geometry of the Kodaira moduli space $M$ of sections of $π:Z\to {\mathbb P}^1$, the normal bundle of which is allowed to jump from ${\mathcal O}(1)^{n}$ to ${\mathcal O}(1)^{n-2m}\oplus {\mathcal O}(2)^{m}\oplus {\mathcal O}^{m}$. In particular, we identify the natural assumptions which guarantee tha…

By introducing an invariant of loops on a compact oriented surface with one boundary component, we give an explicit formula for the action of Dehn twists on the completed group ring of the fundamental group of the surface. This invariant can be considered as ``the logarithms" of Dehn twists. The formula generalizes the…

We consider the volume expansion of the Blaschke metric, which is a projectively invariant metric on a strictly convex domain in a locally flat projective manifold. When the boundary is even dimensional, we express the logarithmic coefficient L as the integral of affine invariants over the boundary. We also formulate a…