Study homotopy types of 4-manifolds, finding decompositions and conditions for desuspension.
problem Determine homotopy types of double suspensions of 4-manifolds with 2-torsion.
method Use Postnikov square and analyze homology groups to find decompositions and conditions for desuspension.
result Homotopy decompositions of double suspensions as wedge sums of specific complexes.
The thesis examines the signature modulo 8 of fibre bundles and identifies an obstruction to multiplicativity.
problem Determining the divisibility of the signature of fibre bundles by 8.
method Using the Z_8-valued Brown-Kervaire invariant and Arf invariant of Z_2-valued quadratic forms.
result The divisibility of the signature by 8 is detected by the Arf invariant of a Z_2-valued quadratic form.
This paper identifies a new invariant for signature non-multiplicativity modulo 8.
problem Identifying the obstruction to signature non-multiplicativity modulo 8.
method Identifying the obstruction to signature non-multiplicativity modulo 8 with the Arf-Kervaire invariant of a Pontryagin squaring operation.
result The signature of a fibre bundle is multiplicative modulo 8 under certain conditions.
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.
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.
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?
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 …
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 paper defines and studies the first Pontryagin class for quadratic Lie 2-algebroids.
problem Defining and studying the first Pontryagin class for quadratic Lie 2-algebroids.
method Detailed study of transitive Lie 2-algebroids, introduction of quadratic Lie 2-algebroids, definition of first Pontryagin class, construction of quadratic Lie 2-algebroids.
result The first Pontryagin class is the obstruction class for the existence of a CLWX-extension and trivial for certain quadratic Lie 2-algebroids.
Optimal control theory improves machine teaching efficiency.
problem Finding the shortest training sequence for a sequential learning algorithm to reach a target model.
method Formulated as a time-optimal control problem, leveraging optimal control theory and computational tools.
result Optimal training sequences can vastly outperform existing heuristics.
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.
New compact support differential cohomology theory with Pontryagin duality proof.
problem Developing a new mathematical framework for differential cohomology with compact support.
method Adapting Cheeger-Simons approach to introduce differential cohomology with compact support, proving functoriality, excision theorem, and using Pontryagin duality.
result Pontryagin duality for differential cohomology, showing isomorphism between ordinary differential cohomology and the smooth Pontryagin dual of compactly supported differential cohomology.
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. 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.
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.
New geometric approach to optimal control theory using Stokes Theorem.
problem Optimal control theory, specifically the Mayer problem.
method Geometric unfolding based on the Stokes Theorem.
result Derives a new necessary and sufficient condition for optimal solutions.
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 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.
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.
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 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.
Abstract: New methods for finding optimal controls in geometric problems on Lie groups.
problem Finding optimal controls for geodesics on Lie groups.
method Pontryagin maximum principle, (co)adjoint representation, geodesic vector field.
result Developed methods to find normal geodesics and locally optimal controls.
Proves stable properties of proper maps between manifolds.
problem Stability of homotopy classes of proper maps and Pontryagin-Thom construction.
method Explicit construction and proof of bijection in a stable range.
result Stabilization of homotopy classes of proper maps and Pontryagin-Thom type bijection.
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.
Explores contact geometry for optimal control problems.
problem Optimal control problems and sub-Riemannian geometry.
method Contact geometry and Pontryagin Maximum Principle.
result Simple and elegant characterizations of extremals.
The paper finds obstructions to Lie algebroid representations up to homotopy.
problem Obstacles to representations up to homotopy of Lie algebroids.
method Analyzes Pontryagin characters and vector bundles over Lie algebroids.
result Vanishing of Pontryagin characters implies no representation up to homotopy.
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…
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.
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.
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. 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. Expressing L-polynomials coefficients in terms of multiple zeta values.
problem Expressing L-polynomials coefficients in terms of multiple zeta values.
method Expressing L-polynomials coefficients in terms of multiple zeta values.
result Every monomial in the Pontryagin classes appears with a non-zero coefficient, with the expected sign.
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.
Researchers confirm a 15-vertex triangulation of the quaternionic projective plane is minimal.
problem Prove the quaternionic projective plane has a minimal triangulation with 15 vertices.
method Implemented Gaifullin's algorithm to compute the first rational Pontryagin class of the combinatorial manifold.
result The 15-vertex triangulation is indeed a minimal triangulation of the quaternionic projective plane.
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.
Geometric model for K0 pairing with analytic and topological terms.
problem Analytic duality pairing between K0 and its Pontryagin dual. method Geometric model, analytic terms involving eta-invariant, topological terms involving differential forms.
result Robust R/Z-valued invariant derived from pairing. Generalizes Routh's reduction method for Lagrangian systems with symmetry.
problem Generalizing Routh's reduction method for Lagrangian systems with symmetry.
method Using implicit Lagrange-Routh equations and Dirac structures, the paper generalizes Routh's reduction method for Lagrangian systems with symmetry.
result The reduced implicit Lagrange-Routh equations can be described in the context of dynamical systems associated to Dirac structures.
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.
This paper is on homotopy classification of maps of (n+1)-dimensional manifolds into the n-dimensional sphere. For a continuous map f of an (n+1)-manifold into the n-sphere define the degree deg f to be the class dual to f^*[S^n], where [S^n] is the fundamental class. We present a short and direct proof of the followin…
The fundamental theorem of the theory of optimal control, the Pontryagin maximum principle (PMP), is extended to the setting of almost Lie (AL) algebroids, geometrical objects generalizing Lie algebroids. This formulation of the PMP yields, in particular, a scheme comprising reductions of optimal control problems simil…
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…