Hosted at one of the major centres for

algebraic topology in the world and supervised by a world leader in homotopical group theory, The project will have a substantial impact on the er~s career by deepening and broadening his research expertise, Especially concerning p-compact groups and cohomology of finite groups; By significantly broadening his research network; And by solidifying his position as a trailblazer and leader in applying techniques from string topology to tackle difficult problems in group homology and cohomology.

He assumes readers to have a basic knowledge of complex analysis, differential equations,

algebraic topology, and algebraic geometry.

Jean-Pierre Serre's mathematical contributions, leading to a Fields Medal in 1954, were largely in the field of

algebraic topology, but his later work ranged widely--within algebraic geometry, group theory, and especially number theory.

The Hodge conjecture is a major unsolved problem in the field of algebraic geometry that relates the

algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety.

A few years later, Poincare stated the Poincare duality and provided a preliminary proof (2) in his 1895 seminal paper Analysis situs, which marked the formal establishment of

algebraic topology (Ferreiros, 2010b).

Many researchers, such as Rosenfeld, Kong, Kopperman, Kovalevsky, Boxer, Karaca, Han and others, have studied the properties of digital images using topology and

algebraic topology.

He assumes students are familiar with homological algebra,

algebraic topology based on different forms, and de Rham cohomology.

Proved by the work of French mathematician Jean-Pierre Serre (who has made fundamental contributions to

algebraic topology, algebraic geometry, and algebraic number theory) and American mathematician John Torrence Tate, Jr.

Within the discipline of mathematics, Gray argues that Poincare's chief contribution was the formulation of

algebraic topology.

But the soul of change was made up of a few youths who started their studies during the Second World War, and finished them a short time before or after its end: Gheorghe Galbura, who introduced the teaching of Algebraic Geometry; Tudor Ganea, who initiated and taught the

Algebraic Topology course; Iulian Petrescu, Barbilian's assistant at Axiomatic Algebra; but more than all, C.

We assume familiarity with the basic

algebraic topology of simplicial complexes; see, e.

In Algebra,

Algebraic Topology and Their Interactions (Roos, J.