This manuscript will be published as Chapter 5 in Wiley’s textbook Mathematical Tools for Physicists, 2nd edition, edited by Michael Grinfeld from the University of Strathclyde. The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebr...

The homology of Lie algebras is closely related to the cyclic homology of associative algebras [LQ]. In [L] the first author constructed a "noncommutative" analog of Lie algebra homology which is, similarly, related to Hochschild homology [C, L]. For a Lie algebra g this new theory is the homology of the complex C,(g) ... ~ ~| g|-+ ... ~1 ~ k, whose boundary map d is given by the formula d(gl|1...

A very general recursive graph invariant, introduced by E.J. Farrell, is revisited and extended to graphs embedded into surfaces, by use of the surface homology.

We present a new multistage method to study the N-Methyl-D-Aspartate (NMDA) neuroreceptor starting from reconstruction of its crystallographic structure. Thanks combination Homology Modelling, Molecular Dynamics and Lattice Boltzmann simulations, we analyse allosteric transition NDMA upon ligand binding compute receptor response ionic passage across membrane.

Given an oriented link diagram we construct an Ω -spectrum whose homotopy type is a link invariant and whose homotopy groups are the Khovanov homology of the link. A key ingredient is the identification of unnormalised Khovanov homology with the derived functors of the inverse limit. Motivation and introduction Quantum link invariants are so called because their definition may be formulated usi...

Let E pq be the vertical homology Hg(E 0 p∗) of this complex. In other words, if Z pq = ker(d v pq), B pq = im(d v p,q+1), then E pq = Z 1 pq/B 1 pq. We note that d h induces maps E pq → E p−1,q as follows. The condition dd = −dd shows that d(Z pq) ⊆ Z p−1,q and d(B pq) ⊆ B p−1,q; therefore, there is a homomorphism d̃pq : E 1 pq → E p−1,q. Let E pq be the horizontal homology Hp(E ∗q) of this com...

Generalizing a concept of Lipshitz, Ozsváth and Thurston from Bordered Floer homology, we define D-structures on algebras of unital operads. This construction gives rise to an equivalence of derived categories, which can be thought of as a unital version of Koszul duality using non-unital Quillen homology, even though the non-unital Quillen homology of unital objects is zero.

We consider chains of generalized submanifolds, as defined by Gualtieri in the context of generalized complex geometry, and define a boundary operator that acts on them. This allows us to define generalized cycles and the corresponding homology theory. Gauge invariance demands that D-brane networks on flux vacua must wrap these generalized cycles, while deformations of generalized cycles inside...

Persistent homology has proven to be a useful tool in a variety of contexts, including the recognition and measurement of shape characteristics of surfaces in R. Persistence pairs homology classes that are born and die in a filtration of a topological space, but does not pair its actual homology classes. For the sublevelset filtration of a surface in R persistence has been extended to a pairing...