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Qiaochu Ma

Publications and source records attributed to Qiaochu Ma.

9 recordsLinked to original sources
arXiv2026

Gromov's Simplicial Volume Vanishing Conjecture for Positive Scalar Curvature

In this paper, we prove Gromov's simplicial volume vanishing conjecture for closed manifolds with spin universal cover. More precisely, we show that if a closed oriented manifold admits a metric of nonnegative scalar curvature and its universal cover is spin, then its simplicial volume vanishes. In particular, a closed oriented aspherical manifold with nonzero simplicial volume admits no metric of nonnegative scalar curvature.

math.DG↗
arXiv2026

Arithmetic unique ergodicity for flat vector bundles

In this paper, we prove a uniform version of quantum unique ergodicity for highfrequency eigensections of Pauli-Schrödinger spin operators on a certain series of unitary flat bundles over arithmetic surfaces.

math.DS↗
arXiv2026

Discrete Mixed Quantization

In this paper, we develop a mixed quantization technique for graph vector bundles and apply it to several asymptotic spectral problems, including the Alon-Boppana bound, the Kesten-McKay law, asymptotic determinant, quantum ergodicity, zero divisor convergence, and Ramanujan vector bundles.

math.SP↗
arXiv2025

Superconnection and Orbifold Chern character

We use flat antiholomorphic superconnections to study orbifold Chern character following the method introduced by Bismut, Shen, and Wei. We show the uniqueness of orbifold Chern character by proving a Riemann-Roch-Grothendieck theorem for orbifold embeddings.

math.DG↗
arXiv2025

Small scale index theory, scalar curvature, and Gromov's simplicial norms

In this article, we study the topological complexity of manifolds with a lower scalar curvature bound. We introduce a small scale index theorem to establish an upper bound for Gromov's simplicial norm of the Poincaré dual of the A-hat class for manifolds with spin universal covering, in terms of a scalar curvature lower bound, volume upper bound, and injectivity radius lower bound of the universal covering. This result can be viewed both as a generalization of Lichnerowicz vanishing theorem and as a scalar curvature analogue to Cheeger finiteness theorem.

math.DG↗
arXiv2025

Mixed quantization and partial hyperbolicity

We establish stable quantum ergodicity for spin Hamiltonians, also known as Pauli-Schrödinger operators. Our approach combines new analytic techniques of mixed quantization, inspired by local index theory, with stable ergodicity results for partially hyperbolic systems.

math.DS↗
arXiv2024

Bounding the A-hat genus using scalar curvature lower bounds and isoperimetric constants

In this paper, we prove an upper bound on the $\widehat{A}$ genus of a smooth, closed, spin Riemannian manifold using its scalar curvature lower bound, Neumann isoperimetric constant, and volume. The proof of this result relies on spectral analysis of the Dirac operator. We also construct an example to show that the Neumann isoperimetric constant in this bound is necessary. Our result partially answers a question of Gromov on bounding characteristic numbers using scalar curvature lower bound.

math.DG↗
arXiv2023

Toeplitz operators and the full asymptotic torsion forms

This paper aims to study the asymptotic expansion of analytic torsion forms associated with a certain series of flat bundles. We prove the existence of the full expansion and give a formula for the sub-leading term, while Bismut-Ma-Zhang have studied the first-order expansion and expressed the leading term as the integral of a locally computable differential form.

math.DG↗