代数与拓扑中的同调剩余域计算

通过计算拓扑与范畴论实现P与NP的同调分离

A Homological Separation of $\mathbf{P}$ from $\mathbf{NP}$ via Computational Topology and Category Theory

本文通过计算拓扑与范畴论方法，首次利用同调理论证明P≠NP，揭示了计算过程的拓扑不变量。

A framework for torsion theory computations on elliptic threefolds

We give a list of statements on the geometry of elliptic threefolds phrased only in the language of topology and homological algebra. Using only notions from topology and homological algebra, we recover existing results and prove new results on torsion pairs in the category of coherent sheaves on an elliptic threefold.

From Homological Algebra to Topology via Type B Zigzag Algebra and Heisenberg Algebra

We construct a faithful categorical action of the type $B$ braid group on the bounded homotopy category of finitely generated projective modules over a finite dimensional algebra which we call the type $B$ zigzag algebra. This categorical action is closely related to the action of the type $B$ braid group on curves on the disc. Thus, our exposition can be seen as a type $B$ analogue of the work of Khovanov-Seidel in arXiv:math/0006056. Moreover, we relate our topological (respectively categorical) action of the type $B$ Artin braid group to their topological (respectively categorical) action o…

Basic Set Theory and Algebra: Hints on Representation, Topology, Geometry, Analysis

In these self-contained low prerequisite introductory notes we first present (in part 1) basic concepts of set theory and algebra without explicit category theory. We then present (in part 2) basic category theory involving a somewhat detailed discussion of system limits and the exact imbedding of abelian categories. This is followed (in part 3) by a discussion of localization, homological algebra, and generalizations of additive and abelian categories such as triangulated and derived categories. Based on the concepts of category theory from parts 2 and 3, (in part 4) we provide hints for cons…

Non-Archimedean analytic geometry as relative algebraic geometry

We show that Berkovich analytic geometry can be viewed as relative algebraic geometry in the sense of Toën--Vaquié--Vezzosi over the category of non-Archimedean Banach spaces. For any closed symmetric monoidal quasi-abelian category we can define a topology on certain subcategories of the of the category of affine schemes with respect to this category. By examining this topology for the category of Banach spaces we recover the G-topology or the topology of admissible subsets on affinoids which is used in analytic geometry. This gives a functor of points approach to non-Archimedean analytic geo…

Isomorphisms between left and right adjoints

There are many contexts in algebraic geometry, algebraic topology, and homological algebra where one encounters a functor that has both a left and right adjoint, with the right adjoint being isomorphic to a shift of the left adjoint specified by an appropriate ``dualizing object''. Typically the left adjoint is well understood while the right adjoint is more mysterious, and the result identifies the right adjoint in familiar terms. We give a categorical discussion of such results. One essential point is to differentiate between the classical framework that arises in algebraic geometry and a de…

Matroids and Canonical Forms: Theory and Applications

This thesis proposes a combinatorial generalization of a nilpotent operator on a vector space. The resulting object is highly natural, with basic connections to a variety of fields in pure mathematics, engineering, and the sciences. For the purpose of exposition we focus the discussion of applications on homological algebra and computation, with additional remarks in lattice theory, linear algebra, and abelian categories. For motivation, we recall that the methods of algebraic topology have driven remarkable progress in the qualitative study of large, noisy bodies of data over the past 15 year…

孤立的重要性

The importance of being isolated

本文研究了交换环的导出范畴由其余数域生成的条件，指出Foxby的小支持分类局部化子类别的例子，并证明多项式环在无限多个变量下的余数域不生成，最后给出了导出范畴满足局部到全局原理的充要条件。