Extended logarithm for solvable elements in mapping class groups.
problem Logarithm of Johnson map extension to solvable elements.
method Extension to exponential solvable elements in mapping class groups using solvable Lie groups.
result Solvability of extended logarithm.
New formulas for geodesics on Stiefel and flag manifolds using trust-region method.
problem Computing geodesics and logarithms on Stiefel and flag manifolds.
method Closed-form geodesic formulas, trust-region solver, Fréchet derivatives.
result Efficient computation of geodesic distance and logarithm map.
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…
Study of foliations' geometric and topological structures.
problem Analyzing the geometric and topological properties of transversely affine foliations.
method Attach holonomy group and quotient stack, identify reparametrisations, classify them, and study the Kato-Nakayama space.
result Holonomy group controls the geometric part, while the Kato-Nakayama space captures the topological and dynamical aspects.
A new method maps value estimates to logarithmic space to enable lower discount factors in reinforcement learning.
problem The poor performance of low discount factors in reinforcement learning.
method Introducing a logarithmic mapping to value estimates.
result The method enables lower discount factors, solving challenging reinforcement learning problems.
Researchers map the fundamental group of polynomial strata to a braid group.
problem Understanding the fundamental group of polynomial strata.
method Analyzing the logarithmic derivative of polynomials to determine the map to a braid group.
result The map from the fundamental group of a stratum to a braid group is characterized by the geometry of the translation surface structure.
Paper studies invariants of knots using logarithmic Gauss maps and character varieties.
problem Understanding invariants of knots using logarithmic Gauss maps and character varieties.
method Develops a homological point of view on the slope using non-abelian representations.
result Defines a rational function on the character variety that unifies various known invariants.
Study flat connections with logarithmic singularities on complex plane curves.
problem Modeling flat connections with logarithmic singularities.
method Explicit finite-dimensional model construction and detailed investigation of specific cases.
result Construction of shifted Poisson structure on moduli spaces.
New MD algorithms using Tempesta logarithms for machine learning.
problem Optimization in machine learning with tailored hyperparameters.
method Developed Mirror Descent algorithms using Tempesta multi-parametric logarithms.
result Wide and flexible family of Mirror Descent and mirror-less updates.
Abstract: New geometric incarnation of isomonodromy functors.
problem Classical isomonodromic deformations.
method Functorial upgrade of isomonodromic deformations using Lie groupoids.
result Geometric incarnation of isomonodromy functors as Morita equivalences.
Computes the decomposition of rank-three bundles over the projective line with three marked points.
problem Decomposing rank-three bundles over the projective line with three marked points.
method Using the monodromy derivative to compute the roots of the bundles.
result Computes the exact decomposition of rank-three bundles for m=3. Infinitesimal calculations link fundamental groups to Lie algebras.
problem Calculating logarithm maps in fundamental groups.
method Hopf invariants defined by Harrison cohomology of commutative cochains.
result Zeroth Harrison cohomology is a universal dual to Malcev Lie algebra.
Characterizes the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.
problem Characterizing the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.
method Introduce the diffeomorphic logarithm of special orthogonal matrices and an efficient algorithm.
result The region containing the principal logarithm has a special multiplicity structure.
In this article it is shown that the study of harmonic diffeomorphisms, with nonvanishing Hopf differential, reduces to the study of the Beltrami equation of a certain type: the imaginary part of the logarithm of the Beltrami function coincides with the imaginary part of the logarithm of the Hopf differential, therefor…
Random walks on mapping class groups identified with geodesic laminations.
problem Understanding random walks on mapping class groups.
method Electrification of curve graph, identifying Poisson boundary, using geodesic laminations.
result Random walk on mapping class group identified with geodesic laminations.
Kim-Milman flow map stable under regular target measures
problem Stability of Kim-Milman flow map under target measure variations
method Stability in relative entropy and 2-Wasserstein distance result Lipschitz stability up to logarithmic factor
We show that there is no bi-Lipschitz homeomorphism of R2 that maps a spiral with a sub-exponential decay of winding radii to an unwinded arc. This result is sharp as shows an example of a logarithmic spiral.
Logarithmic regret achieved in RL with linear function approximation.
problem Achieving logarithmic regret in reinforcement learning with linear function approximation.
method LSVI-UCB for linear MDP assumption, UCRL-VTR for linear mixture MDP assumption.
result Logarithmic regret bounds established for RL with linear function approximation.
Torically maximal curves (known also as simple Harnack curves) are real algebraic curves in the projective plane such that their logarithmic Gauß map is totally real. In this paper we show that hyperplanes in projective spaces are the only torically maximal hypersurfaces of higher dimensions.
Optimal unimodal fitting for linear loss functions in a sequential, efficient manner.
problem Optimal unimodal transformation of univariate model scores under linear loss functions.
method Proposes a sequential approach to estimate the optimal rectangular fit for observed samples with each new sample.
result Sequential approach achieves optimal efficiency with logarithmic time complexity per iteration.
We investigate the connections between the differential-geometric properties of the exponential map from the space of real skew symmetric matrices onto the group of real special orthogonal matrices and the manifold of real orthogonal matrices equipped with the Riemannian structure induced by the Frobenius metric.
Study shows a universal local obstruction to the Samuelson condition for tangent Lagrangian 2-webs.
problem Obstruction to the Samuelson condition for tangent Lagrangian 2-webs.
method Local analysis of tangent lines and their intersection maps.
result A universal local phenomenon produces a nonzero mixed derivative, obstructing the Samuelson condition.
Method constructs spirals with given tangents and curvatures.
problem Constructing a spiral with specified tangents and curvatures.
method Inversion of the involute of a circle to find the spiral.
result Spiral construction method using linear-fractional map.
We show that the largest subsurface projection distance between a marking and its image under the nth step of a random walk grows logarithmically in n, with probability approaching 1 as n tends to infinity. Our setup is general and also applies to (relatively) hyperbolic groups and to Out(Fn). We then use t…
For Riemannian metrics of constant positive curvature on a punctured sphere with conic singularities at the punctures and co-axial monodromy of the developing map, possible angles at the singularities are completely described. This completes the recent result of Mondello and Panov. The related problem of describing pos…
This paper describes a family of pseudo-Anosov braids with small dilatation. The smallest dilatations occurring for braids with 3, 4 and 5 strands appear in this family. A pseudo-Anosov braid with 2g+1 strands determines a hyperelliptic mapping class with the same dilatation on a genus-g surface. Penner showed that log…
P. Buser and P. Sarnak showed in 1994 that the maximum, over the moduli space of Riemann surfaces of genus s, of the least conformal length of a nonseparating loop, is logarithmic in s. We present an application of (polynomially) dense Euclidean packings, to estimates for an analogous 2-dimensional conformal systolic i…
Deviation inequalities and limit laws for random walks on metric spaces.
problem Understanding random walks on metric spaces with contracting isometries.
method Adapting Gouëzel's pivotal time construction to establish deviation inequalities.
result Exponential bounds and limit laws for random walks on mapping class groups and CAT(0) spaces.
Inspired by results of Eskin and Mirzakhani counting closed geodesics of length ≤L in the moduli space of a fixed closed surface, we consider a similar question in the Out(Fr) setting. The Eskin-Mirzakhani result can be equivalently stated in terms of counting the number of conjugacy classes (within the mapping…
In this paper, we introduce the notions of logarithmic Poisson structure and logarithmic principal Poisson structure; we prove that the latter induces a representation by logarithmic derivation of the module of logarithmic Kahler differentials; therefore, it induces a differential complex from which we derive the notio…
Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective threefold Y that admits a no…
We consider the flows generated by generic gradients of Morse maps of a closed connected manifold M to a circle. To each such flow we associate an invariant counting the closed orbits of the flow. Each closed orbit is counted with the weight derived from its index and homotopy class. The resulting invariant is called…
Maps complex varieties into buildings with harmonic properties.
problem Harmonic maps from quasi-projective varieties into Bruhat-Tits buildings.
method Constructs equivariant pluriharmonic maps with asymptotic behavior.
result Quasi-projective varieties have nonzero global logarithmic symmetric differentials.
Maps between acute triangles with minimal stretch found and studied.
problem Finding the minimal stretch between acute triangles.
method Formula for the smallest Lipschitz constant and analysis of the metric space.
result Metric space of pairs of acute triangles with fixed area is Finsler and geodesics determined.
New uncertainty principle for Schrödinger equations on hyperbolic manifolds.
problem Uncertainty principle for Schrödinger equations on hyperbolic manifolds.
method General strategy of Escauriaza-Kenig-Ponce-Vega, new Carleman estimates, logarithmic convexity, new mollifier and weight function.
result Similar rigidity phenomenon as in Euclidean space persists in hyperbolic geometry.
Study rigidity by logarithmic capacity and related functions.
problem Rigidity phenomena in kernel functions and capacities.
method Exploration of Bergman kernel, logarithmic capacity, Green's function, and Euclidean distance/volume.
result Established rigidity theorems by logarithmic capacity.
Study excess logarithmic residues for foliations to bound invariant hypersurfaces and test log canonicity.
problem Bounding invariant hypersurfaces and testing log canonicity of singularities.
method Introduce excess logarithmic residues, prove residue formula, derive Poincaré-type bound, and use them to recover log discrepancies.
result Componentwise logarithmic residues of a lifted foliation along the exceptional divisor recover log discrepancies of singularities.
Study real logarithms of semi-simple matrices, focusing on differential structure.
problem Understanding the differential structure of real logarithms of semi-simple matrices.
method Examines the differential structure of real logarithms of semi-simple matrices under specific matrix types.
result Characterizes the differential structure of real logarithms of semi-simple matrices.
Logarithmic connections on principal bundles over normal varieties are studied.
problem Existence and properties of logarithmic connections on principal bundles over normal varieties.
method Introducing logarithmic connections, showing equivalence to covariant derivatives, and proving existence conditions.
result Existence of logarithmic connections on principal bundles over normal varieties is equivalent to certain conditions on the associated vector bundles and adjoint bundles.
Introduces logarithmic Cartan geometry on complex manifolds with singularities.
problem Holomorphic Cartan geometry with singularities.
method Definition and study of logarithmic Cartan geometry on complex manifolds with polar part supported on a normal crossing divisor.
result Push-forward of a Cartan geometry constructed using a finite Galois ramified covering is a logarithmic Cartan geometry.
We present a new method to solve certain ∂ˉ-equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a ∂ˉ-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness…
Intertwining curvature bounds for graphs and quantum Markov semigroups verified.
problem Intertwining curvature bounds for graphs and quantum Markov semigroups.
method Introducing and verifying curvature bounds in various examples.
result Improved entropic curvature bounds for depolarizing semigroups and qubits.
Local logarithmic Brunn-Minkowski holds for zonoids.
problem Logarithmic Brunn-Minkowski conjecture for zonoids
method Bochner method variant
result Local form of conjecture proven for zonoids
Algorithm approximates functions into manifolds with curvature bounds.
problem Approximating functions into manifolds with lower curvature bounds.
method Algorithm using manifold exponential and logarithm, with error bounds based on sectional curvature.
result Error bounds for nonnegative sectional curvature are similar to linear space approximations.
The coamoeba of any complex algebraic plane curve V is its image in the real torus under the argument map. The area counted with multiplicity of the coamoeba of any algebraic curve in (C∗)2 is bounded in terms of the degree of the curve. We show in this Note that up to multiplication by a constant in $(\…
The germ of the universal isomonodromic deformation of a logarithmic connection on a stable n-pointed genus g curve always exists in the analytic category. The first part of this paper investigates under which conditions it is the analytic germification of an algebraic isomonodromic deformation. Up to some minor techni…
New framework for logarithmically divergent integrals on manifolds with corners.
problem Logarithmically divergent integrals on manifolds with corners.
method Introduces new geometric framework and morphisms in logarithmic geometry.
result Functorial characterization of regularized integration.
Directly proves logarithmic systolic growth for all hyperbolic surfaces.
problem Proving logarithmic systolic growth for all hyperbolic surfaces.
method Using original Brooks/Buser-Sarnak surfaces through a direct approach.
result Directly proves logarithmic systolic growth for all hyperbolic surfaces.