arXiv subjects

Ruizhi Huang

Publications and source records attributed to Ruizhi Huang.

At least 19 recordsLinked to original sources

arXiv2026

A Classification of Self-Maps of Generalized Grassmannians

Generalized Grassmannians form a fundamental class of flag manifolds associated with Lie groups. The purpose of this paper is to classify self-maps of generalized Grassmannians of nonzero degree in terms of their induced actions on cohomology. We prove a rigidity theorem showing that, despite the rich and intricate structure of their cohomology rings, the induced cohomology endomorphisms fall into only two natural types: Adams operations determined by the degree and Dynkin symmetries arising from automorphisms of the Dynkin diagram. Combining the geometry of root and weight systems, the actions of Weyl and Dynkin symmetries on Schubert classes, and Bott--Samelson desingularizations of Schubert varieties, our approach applies uniformly to generalized Grassmannians of all Lie types.

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arXiv2026

Local Inertness of Poincaré duality complexes

We prove that, under certain homological conditions, the attaching map of the top cell of a Poincaré duality complex is inert when localised away from a finite set of primes. This improves on a result of Félix and Tanré in these cases. As an additional application of the methods, we give a loop space decomposition of simply-connected $6$-dimensional Poincaré duality complexes satisfying certain hypotheses. We also show that, under the hypotheses of the inertness theorem, the $(n-1)$-skeleton of an $n$-dimensional Poincaré duality complex satisfies the hyperbolic form of Moore's Conjecture after localising away from an explicit finite set of primes, and use this to obtain new examples of $p$-local maps between spheres that are not inert.

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arXiv2026

On the degrees of equivariant maps from spheres to complex Stiefel manifolds

We study the set of degrees of $\mathbb{Z}/m$-equivariant maps from spheres to complex Stiefel manifolds, motivated by the work of Astey--Gitler--Micha--Pastor. Under a suitable arithmetic condition, this set is determined using results of James, Atiyah--Todd, and Adams--Walker. Our approach is homotopy-theoretic.

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arXiv2026

Potential Matching Optimal Transport: Continuous Normalizing Flows for Exact $p$-Wasserstein Dynamics

We introduce Potential Matching Optimal Transport (PMOT), a potential-flow framework for general $p$-cost optimal transport with $c_p(x,y)=\|x-y\|^p$. PMOT parameterizes the CNF velocity field with a scalar potential in the generalized Benamou--Brenier form for the chosen exponent $p$. It trains the potential gradient with a self-induced matching loss along straight bridges determined by the model's own endpoints, while allowing flexible terminal distribution matching. Our main result establishes zero-loss exactness: under the stated regularity, exact terminal matching, and uniqueness assumptions, any zero-loss solution satisfies the generalized Benamou--Brenier optimality system and recovers the corresponding $p$-optimal transport map and dynamics. On synthetic benchmarks, PMOT learns $p$-specific maps that agree with the corresponding $p$-matched OT references. It also remains competitive as a likelihood-based density model on high-dimensional tabular data, and an MMD-based color transformation experiment demonstrates flexible sample-based terminal matching.

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arXiv2026

Local hyperbolicity, inert maps and Moore's conjecture

We show that the base space of a homotopy cofibration is locally hyperbolic under various conditions. In particular, if these manifolds admit a rationally elliptic closure, then almost all punctured manifolds and almost all manifolds with rationally spherical boundary are $\mathbb{Z}/p^r$-hyperbolic for almost all primes $p$ and all integers $r \geq 1$, and satisfy Moore's conjecture at sufficiently large primes.

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arXiv2026

Homotopy of Simply Connected Complexes with a Spherical Pair

We establish a loop space decomposition for certain $CW$-complexes with a single top cell in the presence of a spherical pair, thereby generalizing several known decompositions of Poincaré duality complexes in which a loop of a product of spheres appears as a direct summand. This decomposition is further applied to derive results on local hyperbolicity, on inertness and non-inertness, on the gaps between rational inertness and local or integral inertness, and on the homotopy theory of smooth manifolds with transversally embedded spheres. In particular, in every dimension greater than three, there exist infinitely many finite $CW$-complexes, pairwise non-homotopy-equivalent, whose loop spaces retract off the loops of their lower skeletons rationally but not locally, and whose top cell attachments produce infinitely many new torsion homotopy groups with exponentially growing ranks.

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arXiv2025

An Almost Flat Spin$^c$ Manifold Bounds

We prove that every almost flat spin^$c$ manifold bounds a compact orientable manifold, thereby settling, in the spin^$c$ case, a long-standing conjecture of Farrell--Zdravkovska and S. T. Yau.

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arXiv2025

Suspension Splitting and Cohomotopy Sets of Simply Connected $7$-manifolds

Let $M$ be a closed simply connected $7$-manifold. In this paper we establish homotopy decompositions of the reduced suspension space $ΣM$ into a wedge sum of simpler spaces when localized at a set of primes. These decompositions are applied to study the cohomotopy sets $π^k(M)$ and the $p$-local cohomotopy sets $π^4(M;\mathbb{Z}_{(p)})$.

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arXiv2025

Homotopy of blow ups after looping

The homotopy theory of the blow up construction in algebraic and symplectic geometry is investigated via two approaches. The first approach introduces and develops fibrewise surgery theory, for which the fibrewise framing is characterized by the homotopy groups of a certain gauge group. This is used to obtain a homotopy decomposition of the based loop space on a blow up that holds $p$-locally for all but finitely many primes $p$ and holds rationally. The second approach is purely homotopy theoretic and obtains a homotopy decomposition of the based loop space on a blow up that holds integrally provided a certain condition is satisfied by an associated homotopy action. As applications, we obtain $p$-local homotopy decompositions of the based loop space of a focal genus $2$ manifold, improve an earlier result of the authors on the homotopy of manifolds stabilized by projective spaces, and obtain for blow-ups a refinement of the rational dichotomy.

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arXiv2025

Sphere bundles over $4$-manifolds are trivial after looping

We show that except two special cases, the sphere bundle of a vector bundle over a simply connected $4$-manifold splits after looping. In particular, this implies that though there are infinitely many inequivalent sphere bundles of a given rank over a $4$-manifold, the loop spaces of their total manifolds are all homotopy equivalent.

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arXiv2025

Stabilization of Poincaré duality complexes and homotopy gyrations

Stabilization of manifolds by a product of spheres or a projective space is important in geometry. There has been considerable recent work that studies the homotopy theory of stabilization for connected manifolds. This paper generalizes that work by developing new methods that allow for a generalization to stabilization of Poincaré Duality complexes. This includes the systematic study of a homotopy theoretic generalization of a gyration, obtained from a type of surgery in the manifold case. In particular, for a fixed Poincaré Duality complex, a criterion is given for the possible homotopy types of gyrations and shows there are only finitely many.

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arXiv2024

Homotopy theoretic properties of open books

We study the homotopy groups of open books in terms of those of their pages and bindings. Under homotopy theoretic conditions on the monodromy we prove an integral decomposition result for the based loop space on an open book, and under more relaxed conditions prove a rational loop space decomposition. The latter case allows for a rational dichotomy theorem for open books, as an extension of the classical dichotomy in rational homotopy theory. As a direct application, we show that for Milnor's open book decomposition of an odd sphere with monodromy of finite order the induced action of the monodromy on the homology groups of its page cannot be nilpotent.

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arXiv2024

Comparison techniques on inert top cell attachments

We establish various criteria for the inertness of the top cell attachments of Poincaré duality complexes through nonzero degree maps, algebraic intersection theory and various types of homotopy fibrations. Many examples are provided, including specific surgery, homogeneous spaces and low dimensional manifolds. Additionally, we propose eight open problems.

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arXiv2023

Homotopy of manifolds stabilized by projective spaces

We study the homotopy of the connected sum of a manifold with a projective space, viewed as a typical way to stabilize manifolds. In particular, we show a loop homotopy decomposition of a manifold after stabilization by a projective space, and provide concrete examples. To do this, we trace the effect in homotopy theory of surgery on certain product manifolds by showing a loop homotopy decomposition after localization away from the order of the image of the classical $J$-homomorphism.

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arXiv2023

A deep transfer learning network for structural condition identification with limited real-world training data

Structural condition identification based on monitoring data is important for automatic civil infrastructure asset management. Nevertheless, the monitoring data is almost always insufficient, because the real-time monitoring data of a structure only reflects a limited number of structural conditions, while the number of possible structural conditions is infinite. With insufficient monitoring data, the identification performance may significantly degrade. This study aims to tackle this challenge by proposing a deep transfer learning (TL) approach for structural condition identification. It effectively integrates physics-based and data-driven methods, by generating various training data based on the calibrated finite element (FE) model, pretraining a deep learning (DL) network, and transferring its embedded knowledge to the real monitoring/testing domain. Its performance is demonstrated in a challenging case, vibration-based condition identification of steel frame structures with bolted connection damage. The results show that even though the training data are from a different domain and with different types of labels, intrinsic physics can be learned through the pretraining process, and the TL results can be clearly improved, with the identification accuracy increasing from 81.8% to 89.1%. The comparative studies show that SHMnet with three convolutional layers stands out as the pretraining DL architecture, with 21.8% and 25.5% higher identification accuracy values over the other two networks, VGGnet-16 and ResNet-18. The findings of this study advance the potential application of the proposed approach towards expert-level condition identification based on limited real-world training data.

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arXiv2023

Some asymptotic formulae for torsion in homotopy groups

Inspired by a remarkable work of Félix, Halperin and Thomas on the asymptotic estimation of the ranks of rational homotopy groups, and more recent works of Wu and the authors on local hyperbolicity, we prove two asymptotic formulae for torsion rank of homotopy groups, one using ordinary homology and one using $K$-theory. We use these to obtain explicit quantitative asymptotic lower bounds on the torsion rank of the homotopy groups for many interesting spaces after suspension, including Moore spaces, Eilenberg-MacLane spaces, complex projective spaces, complex Grassmannians, Milnor hypersurfaces and unitary groups.

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arXiv2022

Exponential growth in the rational homology of free loop spaces and in torsion homotopy groups

Using integral methods we recover and generalize some results by Félix, Halperin and Thomas on the growth of the rational homology groups of free loop spaces, and obtain a new family of spaces whose $p$-torsion in homotopy groups grows exponentially and satisfies Moore's Conjecture for all but finitely many primes. In view of the results, we conjecture that there should be a strong connection between exponential growth in the rational homotopy groups and the $p$-torsion homotopy groups for any prime $p$.

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arXiv2022

Fractional structures on bundle gerbe modules and fractional classifying spaces

We study the homotopy aspects of the twisted Chern classes of torsion bundle gerbe modules. Using Sullivan's rational homotopy theory, we realize the twisted Chern classes at the level of classifying spaces. The construction suggests a notion, which we call fractional U-structure serving as a universal framework to study the twisted Chern classes of torsion bundle gerbe modules from the perspective of classifying spaces. Based on this, we introduce and study higher fractional structures on torsion bundle gerbe modules parallel to the higher structures on ordinary vector bundles.

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