On endomorphism algebras of string almost gentle algebras

Authors

Yu-Zhe Liu, Panyue Zhou

Fields

Keypoints

The paper examines the representation theory of string almost gentle (SAG) algebras.

It shows that the representation types of an SAG algebra, its endomorphism algebra, and the Cohen-Macaulay Auslander algebra are equivalent.

The results extend previous findings on the representation-finiteness of gentle algebras.

Summary

The authors investigate the representation theory of string almost gentle (SAG) algebras, establishing new connections between these algebras and their endomorphism algebras. They define specific subsets R of the quiver's arrow set and introduce finitely generated modules M_R and their corresponding endomorphism algebras A_R. A key result is that the representation type of an SAG-algebra A, the R-endomorphism algebra A_R, and the Cohen-Macaulay Auslander algebra A_CMA are equivalent. The paper demonstrates that the representation types of these algebras are structurally linked, extending previous findings on the representation-finiteness of gentle algebras, and providing new insights into endomorphism algebras. Furthermore, the authors show that for certain subsets R, the endomorphism algebras A_R exhibit a representation type similar to that of the Cohen-Macaulay Auslander algebra, offering a thorough classification of the algebraic properties of SAG-algebras.

Link

Arxiv

Geometry

Definition

Geometry is a fundamental branch of mathematics that focuses on the properties, relationships, and measurements of shapes, sizes, and spatial structures. It examines both flat (two-dimensional) figures, such as points, lines, triangles, and circles, as well as three-dimensional objects, such as cubes, spheres, and pyramids. Geometry involves the study of various concepts including distances, angles, areas, volumes, and symmetries. The field is divided into several subbranches, such as Euclidean geometry, which deals with the study of plane and solid figures based on a set of axioms, and non-Euclidean geometry, which explores spaces that do not conform to traditional Euclidean principles. Geometrical reasoning relies on the use of definitions, theorems, and proofs to understand and describe spatial relationships in both theoretical and applied contexts.

References

What Is Geometry?Shiing-Shen Chern

Iolo Jones

Iolo Jones is a PhD student in Mathematics at Durham University, specializing in geometry with applications to machine learning and data analysis. Supervised by Jeff Giansiracusa and Yue Ren, he focuses on diffusion geometry, which defines geometric structures for probability spaces and allows measuring geometry from data. His research interests include applying these techniques to physical sciences, particularly where complex processes are shaped by underlying structures.

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Arxiv

Manifold Diffusion Geometry: Curvature, Tangent Spaces, and Dimension

Authors

Iolo Jones

Fields

Differential Geometry - Computational Geometry - Applied Mathematics

Keypoints

Diffusion Geometry: Techniques involving diffusion processes provide a robust framework for estimating geometric properties on manifolds.

Limitations of Local PCA: Traditional approaches like LPCA struggle with issues of parameter sensitivity and density variations.

Robust Tangent Space and Dimensionality Estimation: The new diffusion-based methods achieve greater stability in estimating tangent spaces and intrinsic dimensions, especially in noisy data.

Summary

The paper presents innovative methods in diffusion geometry to accurately estimate curvature, tangent spaces, and dimensionality in high-dimensional manifold data. By modeling data as a graph and simulating a Markov diffusion process, these methods evaluate the structure of data more robustly compared to local principal component analysis (LPCA). While LPCA is widely used for approximating local tangent spaces, it is often limited by its reliance on parameters such as neighborhood size, which can lead to instability in heterogeneous or noisy datasets.

In contrast, the diffusion-based method extends beyond conventional spectral decomposition by using the diffusion process to enhance signal quality over noise and density fluctuations. Specifically, this approach calculates a local covariance matrix derived from the diffusion operator, enabling more accurate tangent space recovery and dimensionality estimation. This method demonstrates resilience against noise and density changes by effectively aggregating local information over multiple scales of diffusion. As a result, it offers a more dependable approach for analyzing complex and non-uniformly sampled manifolds.

The robustness and adaptability of diffusion-based tangent space and dimensionality estimation make this technique especially valuable for applications requiring reliable geometric inference in complex, high-dimensional data environments, such as image processing, machine learning, and bioinformatics. Overall, diffusion geometry offers a promising alternative to LPCA, yielding more stable and meaningful representations of manifold structure.

Link

Arxiv