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Dingxin Zhang

Publications and source records attributed to Dingxin Zhang.

29 records · Page 2Linked to original sources
arXiv2024

Revisiting Dwork cohomology: Visibility and divisibility of Frobenius eigenvalues in rigid cohomology

We study Frobenius eigenvalues of the compactly supported rigid cohomology of a variety defined over a finite field of $q$ elements via Dwork's method. A couple of arithmetic consequences will be drawn from this study. As the first application, we show that the zeta functions for finitely many related affine varieties are capable of witnessing all Frobenius eigenvalues of the rigid cohomology of the variety up to Tate twist. This result does not seem to be known for $\ell$-adic cohomology. As the second application, we prove several $q$-divisibility lower bounds for Frobenius eigenvalues of the rigid cohomology of the variety in terms of the multi-degrees of the defining equations. These divisibility bounds for rigid cohomology are generally better than what is suggested from the best known divisibility bounds in $\ell$-adic cohomology, both before and after the middle cohomological dimension.

math.AG↗
arXiv2023

On a conjecture of Huang--Lian--Yau--Yu

We verify a formula on the solution rank of the tautological system arising from ample complete intersections in a projective homogeneous space of a semisimple group conjectured by Huang--Lian--Yau--Yu arXiv:1801.01194. As an application, we prove the existence of the rank one point for such a system, where mirror symmetry is expected.

math.AG↗
arXiv2023

Twisted GKZ hypergeometric functions and relative cohomology

We investigate the GKZ $A$-hypergeometric $\mathscr{D}$-modules, introduced by Gel'fand, Kapranov, and Zelevinskii, arising from cyclic covers of toric varieties and find its Riemann--Hilbert partner. This extends our earlier results in arXiv:1902.01536.

math.AG↗
arXiv2023

PaRot: Patch-Wise Rotation-Invariant Network via Feature Disentanglement and Pose Restoration

Recent interest in point cloud analysis has led rapid progress in designing deep learning methods for 3D models. However, state-of-the-art models are not robust to rotations, which remains an unknown prior to real applications and harms the model performance. In this work, we introduce a novel Patch-wise Rotation-invariant network (PaRot), which achieves rotation invariance via feature disentanglement and produces consistent predictions for samples with arbitrary rotations. Specifically, we design a siamese training module which disentangles rotation invariance and equivariance from patches defined over different scales, e.g., the local geometry and global shape, via a pair of rotations. However, our disentangled invariant feature loses the intrinsic pose information of each patch. To solve this problem, we propose a rotation-invariant geometric relation to restore the relative pose with equivariant information for patches defined over different scales. Utilising the pose information, we propose a hierarchical module which implements intra-scale and inter-scale feature aggregation for 3D shape learning. Moreover, we introduce a pose-aware feature propagation process with the rotation-invariant relative pose information embedded. Experiments show that our disentanglement module extracts high-quality rotation-robust features and the proposed lightweight model achieves competitive results in rotated 3D object classification and part segmentation tasks. Our project page is released at: https://patchrot.github.io/.

cs.CV↗
arXiv2021

3D Medical Point Transformer: Introducing Convolution to Attention Networks for Medical Point Cloud Analysis

General point clouds have been increasingly investigated for different tasks, and recently Transformer-based networks are proposed for point cloud analysis. However, there are barely related works for medical point clouds, which are important for disease detection and treatment. In this work, we propose an attention-based model specifically for medical point clouds, namely 3D medical point Transformer (3DMedPT), to examine the complex biological structures. By augmenting contextual information and summarizing local responses at query, our attention module can capture both local context and global content feature interactions. However, the insufficient training samples of medical data may lead to poor feature learning, so we apply position embeddings to learn accurate local geometry and Multi-Graph Reasoning (MGR) to examine global knowledge propagation over channel graphs to enrich feature representations. Experiments conducted on IntrA dataset proves the superiority of 3DMedPT, where we achieve the best classification and segmentation results. Furthermore, the promising generalization ability of our method is validated on general 3D point cloud benchmarks: ModelNet40 and ShapeNetPart. Code is released.

eess.IV↗
arXiv2021

Mixed Hodge structure and $\mathcal{N}=2$ Coulomb branch solution

The Coulomb branch of four dimensional $\mathcal{N}=2$ theories can be solved by finding a Seiberg-Witten (SW) geometry and a SW differential. While lots of SW geometries are found, the extraction of low energy theory out of it is limited due to following reasons: (a) the difficulty of distinguishing electric-magnetic and flavor charges; (b) the difficulty of determining the low energy theory at singular point; (c) the lack of SW differential. We show that the mixed Hodge structure (MHS) can be used to fully solve the low energy physics of Coulomb branch at every vacuum. The MHS can be used to solve above three problems as follows: (a) The smooth fiber of SW geometry carries a Mixed Hodge Structure structure with two weights: one weight describes electric-magnetic charge and the other for the flavor charge; (b) for the singular fiber, there are three MHS which can be used to determine the IR theory; (c) the MHS at $\infty$ is used to find the SW differential.

hep-th↗
arXiv2021

$\ell$-adic Tautological Systems

Tautological systems was introduced in Lian-Yau as the system of differential equations satisfied by period integrals of hyperplane sections of some complex projective homogenous varieties. We introduce the $\ell$-adic tautological systems for the case where the ground field is of characteristic $p$.

math.AG↗
arXiv2020

$A$-hypergeometric systems and relative cohomology

We investigate the space of solutions to certain $A$-hypergeometric $\mathscr{D}$-modules, which were defined and studied by Gelfand, Kapranov, and Zelevinsky. We show that the solution space can be identified with certain relative cohomology group of the toric variety determined by $A$, which generalizes the results of Huang, Lian, Yau, and Zhu. As a corollary, we also prove the existence of rank one points for Calabi--Yau complete intersections in toric varieties.

math.AG↗
arXiv2020

On Vlasenko's formal group laws

Given a Laurent polynomial over a ring flat over \(\mathbb{Z}\), Vlasenko defines a formal group law. We identify this formal group law with a coordinate system of a formal group functor, prove its integrality. When the Hasse--Witt matrix of the Laurent polynomial is invertible, Vlasenko defines a matrix by taking a certain \(p\)-adic limit. We show that this matrix is the Frobenius of the Dieudonné module of this formal group modulo \(p\).

math.NT↗
arXiv2019

The Monsky--Washnitzer topos

Mimicking Ogus's construction, we define a site, for varieties over a field of char. p > 0, using Monsky--Washnitzer algebras/weak formal schemes. We prove a comparison theorem between the MW cohomology and a certain analytic cohomology.

math.AG↗