Algebraic Topology
The idea of algebraic topology is to study slippery geometric objects (topology) using rigid discrete objects (algebra). We look at algebraic shadows of the geometric objects, and see what we can learn. If we are lucky, the algebraic shadows will suggest geometric insights that can then be proved.
Topology −→ Algebra
Algebraic topology is one of the most important creations in mathematics which uses algebraic tools to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism (though usually classify up to homotopy equivalence). The most important of these invariants are homotopy groups, homology groups, and cohomology groups (rings).
Furthermore, Algebraic topology can be roughly defined as the study of techniques for forming algebraic images of topological spaces. Most often these algebraic images are groups, but more elaborate structures such as rings, modules, and algebras also arise. The mechanisms that create these images.
Also, one of the main objectives of topology is to classify spaces up to continuous deformations, i.e. up to homeomorphism. Unfortunately, in general it is a quite difficult problem to show when two spaces are not homeomorphic. A classification which is usually easier to obtain is based on the rougher notion of homotopy.
we define an algebraic group called the fundamental group of a space. This group consists of different equivalence classes of loops in that space.
We use the fundamental group as a tool to tell different spaces apart. For example, what makes a torus different from a double torus, or an n-fold torus? Or how is a coffee cup like a donut?
The main purpose of this course to presentation to the basic materials of algebraic topology
Topology −→ Algebra
Algebraic topology is one of the most important creations in mathematics which uses algebraic tools to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism (though usually classify up to homotopy equivalence). The most important of these invariants are homotopy groups, homology groups, and cohomology groups (rings).
Furthermore, Algebraic topology can be roughly defined as the study of techniques for forming algebraic images of topological spaces. Most often these algebraic images are groups, but more elaborate structures such as rings, modules, and algebras also arise. The mechanisms that create these images.
Also, one of the main objectives of topology is to classify spaces up to continuous deformations, i.e. up to homeomorphism. Unfortunately, in general it is a quite difficult problem to show when two spaces are not homeomorphic. A classification which is usually easier to obtain is based on the rougher notion of homotopy.
we define an algebraic group called the fundamental group of a space. This group consists of different equivalence classes of loops in that space.
We use the fundamental group as a tool to tell different spaces apart. For example, what makes a torus different from a double torus, or an n-fold torus? Or how is a coffee cup like a donut?
The main purpose of this course to presentation to the basic materials of algebraic topology
Teacher: Shadman Rahman
Integro-Differential Equations
This course deals with one of the most applied problems in the engineering sciences.
It is concerned with the integro-differential equations where both differential and integral operators will appear in the same equation. This type of equations was introduced by Volterra for the first time in the early 1900. Volterra investigated the population growth, focussing his study on the hereditary influences, where through his research work the topic of integro-differential equations was
established.
Scientists and engineers come across the integro-differential equations through their research work in heat and mass diffusion processes, electric circuit problems, neutron diffusion, and biological species coexisting together with increasing and decreasing rates of generating. Applications of the integro-differential equations in electromagnetic theory and dispersive waves and ocean circulations are enormous.
More details about the sources where these equations arise can be found in physics, biology, and engineering applications as well as in advanced integral equations literatures.
It is concerned with the integro-differential equations where both differential and integral operators will appear in the same equation. This type of equations was introduced by Volterra for the first time in the early 1900. Volterra investigated the population growth, focussing his study on the hereditary influences, where through his research work the topic of integro-differential equations was
established.
Scientists and engineers come across the integro-differential equations through their research work in heat and mass diffusion processes, electric circuit problems, neutron diffusion, and biological species coexisting together with increasing and decreasing rates of generating. Applications of the integro-differential equations in electromagnetic theory and dispersive waves and ocean circulations are enormous.
More details about the sources where these equations arise can be found in physics, biology, and engineering applications as well as in advanced integral equations literatures.
Teacher: Mohammed Shami