Related papers: Invitation to higher local fields, Part I, section…
We discuss the homological algebra of representation theory of finite dimensional algebras and finite groups. We present various methods for the construction and the study of equivalences of derived categories: local group theory, geometry…
We give a rigorous formulation of the intuitive idea that a differentiable map should be thesame thing as a locally, or infinitesimally, linear map: just as a linear map respects the operations of addition and multiplication by scalars ina…
We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed.…
Using valuation rings and valued fields as examples, we discuss in which ways the notions of "topological IFS attractor" and "fractal space" can be generalized to cover more general settings.
Recent evidence indicates that the abundance of recurring elementary interaction patterns in complex networks, often called subgraphs or motifs, carry significant information about their function and overall organization. Yet, the…
These third-year lecture notes are designed for a 1-semester course in topological quantum field theory (TQFT). Assumed background in mathematics and physics are only standard second-year subjects: multivariable calculus, introduction to…
We categorify the adjunction between locales and topological spaces, this amounts to an adjunction between (generalized) bounded ionads and topoi. We show that the adjunction is idempotent. We relate this adjunction to the Scott adjunction,…
In this paper we define a topology with sufficiently good properties on the set of prime ideals of a number field.
Topological excitations are prominent candidates for explaining nonperturbative effects in QCD like confinement. In these lectures, I cover both formal treatments and applications of topological objects. The typical phenomena like BPS…
By introducing various topologies on the homotopy groups of a topological space, some researchers make these well known notions in algebraic topology more useful and powerful. In this paper, first we recall and review some known topologies…
We show that any complex (respectively real) representation of finite group naturally generates a open-closed (respectively Klein) topological field theory over complex numbers. We relate the 1-point correlator for the projective plane in…
In this work, we Extend Pawlak approximation spaces by topological spaces. Also, Rough Membership, equality and inclusion relations are extended using topological near open sets. In addition, new extended measures of accuracy and quality of…
Higher order anisotropic superspaces are constructed as generalized vector superbundles provided with compatible nonlinear connection, distinguished connection and metric structures.
Topological landscape is introduced for networks with functions defined on the nodes. By extending the notion of gradient flows to the network setting, critical nodes of different indices are defined. This leads to a concise and…
This is a survey of the theory of complex projective (CP^1) structures on compact surfaces. After some preliminary discussion and definitions, we concentrate on three main topics: (1) Using the Schwarzian derivative to parameterize the…
Topological transforms have been very useful in statistical analysis of shapes or surfaces without restrictions that the shapes are diffeomorphic and requiring the estimation of correspondence maps. In this paper we introduce two…
Invited contribution to the Encyclopedia of Mathematical Physics. We give an introduction to the homotopical theory of higher categories, focused on motivating the definitions of the basic objects, namely $\infty$-categories and…
We give a Hodge theoretic criterion to characterize locally the period domains of certain projective manifolds to be complex balls.
The force fields of interior boundaries of matters are calculated as superposition of fields of modified Sierpinski carpet. The long-range effect interaction in irradiated crystals is introduced as various manifestations of second order…
Directional fields, including unit vector, line, and cross fields, are essential tools in the geometry processing toolkit. The topology of directional fields is characterized by their singularities. While singularities play an important…