Study shows unbounded Pontryagin numbers on curved manifolds.
problem Understanding unbounded Pontryagin numbers on curved manifolds.
method Analyzing rational linear combinations of Pontryagin numbers and their relation to the universal elliptic genus.
result Proves existence of unbounded Pontryagin numbers on nonnegatively curved spin manifolds.
We prove that any rational linear combination of Pontryagin numbers that is not a multiple of the signature is unbounded on connected closed oriented manifolds of nonnegative sectional curvature. Combining our result with Gromov's finiteness result for the signature yields a new characterization of the L-genus.
Generalizes Thorpe's inequality for 4k-manifolds.
problem Euler characteristic and Pontryagin numbers of 4k-manifolds.
method Generalization and correction of Thorpe's original work.
result Corrected and generalized Thorpe's inequality.
A rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least three only multiples of the top Chern number, which is the Euler characteristic, are invarian…
Study of characteristic numbers in 24-dimensional String manifolds.
problem Characterizing and understanding characteristic numbers of 24-dimensional String manifolds.
method Using Pontryagin numbers, integral basis of String cobordism group, and divisibility results.
result Established 2- and 3-primary divisibilities of characteristic numbers.
Topological Pontryagin classes are algebraically independent in high-dimensional spaces.
problem Algebraic independence of topological Pontryagin classes.
method Analyzing rationalised cohomology of BTop(d).
result Topological Pontryagin classes are algebraically independent.
Enhances Pontryagin-Thom theorem for manifold maps.
problem Identifying map spaces with moduli spaces of submanifolds.
method Space-level enhancement of Pontryagin-Thom theorem.
result Maps from manifolds to Thom spaces identified with moduli spaces of submanifolds.
Develops method to compute Chern-Simons potentials from higher-dimensional Pontryagin densities.
problem Computing Chern-Simons potentials from higher-dimensional Pontryagin densities.
method Systematic approach using a generic affine connection with non-vanishing torsion and non-metricity.
result Algorithm and code for determining Chern-Simons potential from Pontryagin density in arbitrary even dimensions.
In this note we compute low degree rational Pontryagin classes for every closed locally symmetric manifold of noncompact type. In particular, we answer the question: Which locally symmetric M have at least one nonzero Pontryagin class?
Generalizes Pontryagin's construction for proper maps in stable dimensions.
problem Mapping submanifolds to homotopy classes of proper maps.
method Introduces a new bijection between cobordism sets and homotopy classes for proper maps.
result Provides a bijection for cobordism sets of submanifolds embedded in WimesRn. The paper shows examples of geodesics switching infinitely often on certain manifolds.
problem Understanding geodesics with infinitely many switches on Finsler and sub-Finsler manifolds.
method Provided examples and explicit structures on Carnot groups, presented a sufficient condition for chattering.
result Geodesics on certain manifolds can exhibit a countable number of switches in arbitrarily small time intervals.
To each three-component link in the 3-sphere, we associate a geometrically natural characteristic map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its charact…
Unified approach to Merton's portfolio problem using Pontryagin's principles.
problem Optimizing consumption and investment strategies in financial portfolios.
method PG-DPO framework combining neural networks with Pontryagin's maximum principle.
result Locally optimal policies closely tied to classical stochastic control.
The Pontryagin forms on 1-jet bundle of Riemannian metrics, are shown to provide, in a natural way, diffeomorphism-invariant pre-symplectic structures on the space of Riemannian metrics for dimensions n=4r−2. The equivariant Pontryagin forms provide canonical moment maps for these structures. In dimension two, the sy…
Study on Heisenberg group's Lorentzian problems using Pontryagin's principle.
problem Lorentzian problems on the Heisenberg group.
method Applied Pontryagin's maximum principle to obtain extremal trajectories.
result Parameterization of abnormal and normal extremal trajectories, investigation of reachability sets and existence of optimal trajectories.
A 4-manifold is parallelizable if its Stiefel-Whitney and Pontryagin classes vanish.
problem Characterizing parallelizable 4-manifolds.
method Classification of SO(4)-bundles over the 4-sphere using Euler and first Pontryagin classes. result A closed orientable 4-manifold is parallelizable if and only if its second Stiefel-Whitney class, first Pontryagin class, and Euler characteristic vanish.
Study extremals on Lie groups with asymmetric polyhedral Finsler structures using Pontryagin's Maximal Principle.
problem Finding extremals on Lie groups with asymmetric polyhedral Finsler structures.
method Using Pontryagin's Maximal Principle and control systems of Euler-Arnold type to find extremals on the cotangent bundle of the group.
result Uniqueness of the control u(t) can be studied through the asymptotic curvature of the vertical part of the Pontryagin extremal. Revisits Pontryagin's proof of stable stems 0, 1, and 2.
problem Computing stable stems in dimensions 0, 1, and 2.
method Introduction of homotopy theory concepts, framed cobordism, and Pontryagin-Thom construction.
result Corrects the result of \(\pi_2^S\) from \cite{Pont2} and computes stable stems in dimensions 0, 1, and 2.
The purpose of this paper is to define the concept of multi-Dirac structures and to describe their role in the description of classical field theories. We begin by outlining a variational principle for field theories, referred to as the Hamilton-Pontryagin principle, and we show that the resulting field equations are t…
The study determines fiber homotopy trivial bundles and their impact on curvature.
problem Understanding fiber homotopy trivial bundles and their effect on curvature.
method Classical approach via block bundles and surgery theory.
result Existence of elements of infinite order in homotopy groups of spaces of positive curvature.
Researchers calculate alpha invariant for certain complex projective spaces.
problem Computing alpha invariant for specific types of complex projective spaces.
method Computed alpha invariant using total degree and Pontryagin classes.
result Alpha invariant depends only on total degree and Pontryagin classes.
We discuss the cobordism type of spin manifolds with nonnegative sectional curvature. We show that in each dimension 4k≥12, there are infinitely many cobordism types of simply connected and nonnegatively curved spin manifolds. Moreover, we raise and analyze a question about possible cobordism obstructions to non…
We give a Pontryagin-Thom-Szucs type construction for non-positive codimensional singular maps, and obtain results about cobordism and bordism groups of -1 codimensional stable maps with prescribed singular fibers.
In this paper, first we give a detailed study on the structure of a transitive Lie 2-algebroid and describe a transitive Lie 2-algebroid using a morphism from the tangent Lie algebroid TM to a strict Lie 3-algebroid constructed from derivations. Then we introduce the notion of a quadratic Lie 2-algebroid and define its…
Two groups with specific limit sets in hyperbolic spaces are identified.
problem Identifying convex cocompact subgroups with specific limit sets in real hyperbolic spaces.
method Examples of subgroups generated by reflections and rotations with limit sets as Pontryagin spheres and Menger curves.
result Examples of convex cocompact subgroups with limit sets as Pontryagin spheres and Menger curves are found.
We prove that a rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least three we prove that only multiples of the top Chern number, which is the Euler …
Introduces new geodesic fields for Finsler manifolds.
problem Finding optimal paths in Finsler manifolds.
method Introduced Pontryagin type C0-Finsler structures and defined geodesic fields. result Extended geodesic field E provides more natural geodesics. We express characteristic numbers of compact hyperkähler manifolds in graph-theoretical form, considering them as a special case of the curvature invariants introduced by Rozansky and Witten. The appropriate graphs are generated by ``wheels'' and we use the recently proved Wheeling Theorem to give a formula for the L2 …
In this paper, we study the existence of high-dimensional, closed, smooth manifolds whose rational homotopy type resembles that of a projective plane. Applying rational surgery, the problem can be reduced to finding possible Pontryagin numbers satisfying the Hirzebruch signature formula and a set of congruence relation…
Study cobordisms of nested manifolds and their invariants.
problem Understanding cobordisms of nested manifolds and their invariants.
method Identify a nested analog of the Pontryagin-Thom construction and find spaces homotopy equivalent to nested Pontryagin-Thom spaces.
result Discover nested cobordism invariants and provide an alternative proof of Wall's splitting result.
In this work, we use the Sternberg phase space (which may be considered as the classical phase space of particles in gauge fields) in order to explore the dynamics of such particles in the context of Hamilton-Dirac systems and their associated Hamilton-Pontryagin variational principles. For this, we develop an analogue…
Researchers find optimal paths on a specific geometric group.
problem Finding optimal paths on a Cartan group with a sub-Finsler quasimetric.
method Using the Pontryagin Maximum Principle in coordinates of the first kind.
result They found extremals for arbitrary left-invariant sub-Finsler quasimetrics.
We present a new geometric unfolding of a prototype problem of optimal control theory, the Mayer problem. This approach is crucially based on the Stokes Theorem and yields to a necessary and sufficient condition that characterizes the optimal solutions, from which the classical Pontryagin Maximum Principle is derived i…
New framework optimizes multi-asset portfolio choice for high dimensions.
problem Optimizing high-dimensional continuous-time portfolio choice.
method Combines Pontryagin's Maximum Principle with BPTT for neural network policy learning.
result Achieves near-optimal policies with improved efficiency and precision.
The paper generalizes Sperner's lemma to higher dimensions and calculates a new invariant.
problem Generalizing Sperner's lemma to higher dimensions and quantifying its outcomes.
method Using triangulations of (m+1)-discs and simplicial mappings, the authors define a new invariant and prove a theorem about fully colored simplices. result The number of fully colored simplices is not less than the new invariant μ([f]).
Novikov theorem extended to rational Pontryagin classes for cyclic group C4.
problem Classifying stable Cp-smoothings of high-dimensional manifolds. method Computing equivariant homotopy groups and applying to C4. result Novikov's theorem extended to rational Pontryagin classes for C4. Maps self-duality in little disks operad to framed manifolds.
problem Self-duality of little disks operad.
method Configuration space level Pontryagin--Thom constructions.
result Existence of compatible self-duality map for framed manifolds.
By adapting the Cheeger-Simons approach to differential cohomology, we establish a notion of differential cohomology with compact support. We show that it is functorial with respect to open embeddings and that it fits into a natural diagram of exact sequences which compare it to compactly supported singular cohomology …
The study finds obstructions for certain Weyl curvature tensors on manifolds.
problem Can manifolds admit metrics with purely electric or magnetic Weyl tensors?
method Analyzes algebraic curvature tensors and their Pontryagin classes on scalar product spaces.
result Obstructions to the existence of metrics with PE or PM Weyl tensors in top-degree cohomology.
A method for dynamic portfolio choice with uncertain parameters using Pontryagin projection.
problem Continuous-time CRRA portfolio choice in markets with estimated and uncertain coefficients.
method Simulation-based two-stage solver (DPO + Pontryagin projection) to maximize ex-ante objective.
result Projection stabilizes learning and accurately recovers analytic decisions, improving over model-free PPO.
A scalable framework optimizes multi-asset portfolios with constraints.
problem Optimizing multi-asset portfolios with inequality constraints.
method Integrates neural policies with Pontryagin's Maximum Principle, enforcing feasibility via log-barrier regularization.
result Recover KKT-optimal policies in high-dimensional problems without violating constraints.
The paper studies Hodge Laplacians from point clouds, proving spectral convergence and harmonic form consistency.
problem Analyzing Riemannian submanifolds from point cloud data.
method Constructing deformed Hodge Laplacians and empirical operators from point clouds, proving convergence properties.
result Empirical spectral cluster contains the k-th Betti number and converges to harmonic k-forms. On the ground of origins of the theory of Lie groups and Lie algebras, their (co)adjoint representations, and the Pontryagin maximum principle for the time-optimal problem are given an independent foundation for methods of geodesic vector field to search for normal geodesics of left-invariant (sub-)Finsler metrics on L…
Paper shows string cobordism at 24 dims can be determined by elliptic genus.
problem Determining string cobordism at dimension 24.
method Using elliptic genus as an invariant.
result Shows string cobordism at 24 dims can be determined by elliptic genus.
This paper considers the Pontryagin characters of graded vector bundles of finite rank, in the cohomology vector spaces of a Lie algebroid over the same base. These Pontryagin characters vanish if the graded vector bundle carries a representation up to homotopy of the Lie algebroid. As a consequence, this gives a stron…
Proves Massey's theorems on complex structure obstructions.
problem Finding complex structures on real vector bundles.
method Fractional Stiefel-Whitney classes and combinations of Pontryagin, Chern, and Euler classes.
result Determines the second obstruction for rank six bundles.
Geometric model for Hodge filtered complex cobordism constructed.
problem Constructing a geometric model for Hodge filtered complex cobordism.
method Refinement of Pontryagin-Thom construction to create an explicit isomorphism.
result Explicit isomorphism between geometric and abstract models for complex manifolds.
Researchers found optimal paths on a specific geometric group.
problem Finding optimal paths in a geometric group with a sub-Finsler metric.
method Used Pontryagin Maximum Principle for time-optimal control problem.
result Found extremals for left-invariant sub-Finsler metric.