THE RESEARCH OF ALGEBRAIC STRUCTURESWITH OR WITHOUT TOPOLOGY

What is an algebraic structure?

What is an algebraic structure?

It is a nonempty set, where there are defined some algebraic operations,which might resemble operations like ”addition”, ”multiplication”,”taking inverses”, etc.

What is an algebraic structure?

It is a nonempty set, where there are defined some algebraic operations,which might resemble operations like ”addition”, ”multiplication”,”taking inverses”, etc.

The easiest examples might be the set R of real numbers or the set Z ofintegers, both equipped with usual addition, multiplication etc.

What is an algebraic structure?

It is a nonempty set, where there are defined some algebraic operations,which might resemble operations like ”addition”, ”multiplication”,”taking inverses”, etc.

The easiest examples might be the set R of real numbers or the set Z ofintegers, both equipped with usual addition, multiplication etc.Another example is vectors in 3-dimensional space equipped with the”triangle rule” for addition of vectors and for exampe the cross-productas a multiplication.

What is an algebraic structure?

It is a nonempty set, where there are defined some algebraic operations,which might resemble operations like ”addition”, ”multiplication”,”taking inverses”, etc.

The easiest examples might be the set R of real numbers or the set Z ofintegers, both equipped with usual addition, multiplication etc.Another example is vectors in 3-dimensional space equipped with the”triangle rule” for addition of vectors and for exampe the cross-productas a multiplication. One can also think about the set of matrices ofsome fixed dimension with matrix addition and multiplication etc.

What is topology and why to use it?

What is topology and why to use it?

Topology could be considered as an extra structural property of someset X , determining one special collection of subsets of a set X , whichwill are usually called ”open sets”.

What is topology and why to use it?

Topology could be considered as an extra structural property of someset X , determining one special collection of subsets of a set X , whichwill are usually called ”open sets”.

Using these ”open sets”, one defines for every element a of a set Xsomething, which is called ”a neighbourhood of a” and which is, in aprincipal, a generalisation of notions like interval (a− b, a + b) fornumbers or a circle with center in a and with some radius on the plane.

What is topology and why to use it?

Topology could be considered as an extra structural property of someset X , determining one special collection of subsets of a set X , whichwill are usually called ”open sets”.

Using these ”open sets”, one defines for every element a of a set Xsomething, which is called ”a neighbourhood of a” and which is, in aprincipal, a generalisation of notions like interval (a− b, a + b) fornumbers or a circle with center in a and with some radius on the plane.

These neighbourhoods have entered the mathematics because of theneeds of practical problems, where people want to know, for example,how exactly should we measure some data in order to be able to claimthat some function, depending on this data, does not differ from somewanted value by no more than some fixed number.

What is topology and why to use it?

Topology could be considered as an extra structural property of someset X , determining one special collection of subsets of a set X , whichwill are usually called ”open sets”.

Using these ”open sets”, one defines for every element a of a set Xsomething, which is called ”a neighbourhood of a” and which is, in aprincipal, a generalisation of notions like interval (a− b, a + b) fornumbers or a circle with center in a and with some radius on the plane.

These neighbourhoods have entered the mathematics because of theneeds of practical problems, where people want to know, for example,how exactly should we measure some data in order to be able to claimthat some function, depending on this data, does not differ from somewanted value by no more than some fixed number.

It appears that taking an algebraic structure without considering anytopology on it, there are several algebraic problems, which can not besolved by algebraic methods only.

What is topology and why to use it?

Topology could be considered as an extra structural property of someset X , determining one special collection of subsets of a set X , whichwill are usually called ”open sets”.

Using these ”open sets”, one defines for every element a of a set Xsomething, which is called ”a neighbourhood of a” and which is, in aprincipal, a generalisation of notions like interval (a− b, a + b) fornumbers or a circle with center in a and with some radius on the plane.

These neighbourhoods have entered the mathematics because of theneeds of practical problems, where people want to know, for example,how exactly should we measure some data in order to be able to claimthat some function, depending on this data, does not differ from somewanted value by no more than some fixed number.

It appears that taking an algebraic structure without considering anytopology on it, there are several algebraic problems, which can not besolved by algebraic methods only.Topology can make it much easier to solve the algebraic problems.

Research topics of Mart Abel (employeed by Tallinn Universitysince 2012.)

Research topics of Mart Abel (employeed by Tallinn Universitysince 2012.)

Research field: various problems connected with (complex) algebras andrings, mainly with topology, but sometimes also without topology.

Research topics of Mart Abel (employeed by Tallinn Universitysince 2012.)

Research field: various problems connected with (complex) algebras andrings, mainly with topology, but sometimes also without topology.

For example: problems with A-modules, galbed algebras, sheaves;descriptions of ideals in topological algebras; derivations on topologicalrings and topological algebras; algebraic and topological K -theory;representations of topological algebras; bundles; properties of thespectrum and spectral radius of elements in unital (topological orarbitrary) algebras.

Research topics of Mart Abel (employeed by Tallinn Universitysince 2012.)

Research field: various problems connected with (complex) algebras andrings, mainly with topology, but sometimes also without topology.

For example: problems with A-modules, galbed algebras, sheaves;descriptions of ideals in topological algebras; derivations on topologicalrings and topological algebras; algebraic and topological K -theory;representations of topological algebras; bundles; properties of thespectrum and spectral radius of elements in unital (topological orarbitrary) algebras.

In my research my main interest is to find out in how general situationssome results, which are known for some special topological algebras,hold.