u/Short-Cheek2650

I am looking for European unis which have english undergrad courses in maths and are not so expensive the upper bound for the annual fees should be around 15000 euros. If you have suggestions other than europe feel free to share those as well,but I do want a good math department.

I am writing this to give the general math interested audience a brief idea about topology.

Warning: I am trying to make this accessible to everyone so if you haven't done topology formally it's highly likely that you might get a wrong idea of some concept so don't take everything I say literally and look into these things deeper and more rigorously if you want clarity.

So let's begin with what even is topology? To answer this let's try to relate topology to something we probably have some idea about which is geometry. Topology is both related and independent of geometry in some way. If you consider geometry to be the study of shapes then topology becomes a subfield of geometry because we also study shapes in topology. But if you take a more rigid definition that geometry studies properties of shapes like length,angle,area, volume,etc then topology becomes independent of geometry because in topology we study properties of shapes which remain same even if we twist ,strech and bend the shape and angles,area, volume obviously don't stay the same under these transformations so they are not topological properties. Modern mathematicians usually consider geometry to be simply study of shapes and not put too many rigid conditions on the kind of properties we study cuz most modern geometric studies like differential geometry, algebraic geometry study more qualitative properties like the dimension of a shape,if it can be embedded in some other shape or not ,etc rather than more quantitative properties like length, angles,etc even though these concepts are still of importance but the focus shifts from these specific quantities to more general stuff. So according to the more modern loose formulation of geometry, topology is a subfield of it which studies shapes and properties of shapes which don't change under streching,twisting and bending.  

In classical plane geometry two polygons are equal if all of their angles are equal and sides are of equal length this equivalence is called congruence. In topology two shapes are considered equal if one can be twisted,streched or bent into another this kind of equivalence is called homeomorphism.This definition is clearly more loose than the previous definition of congruence and hence the collection of shapes topologically equivalent to each other is much larger than the collection of shapes geometrically (congruence) equal to each other. For example a circle and a square are topologically equal as a square shaped string can be transformed into a circle shape string even if they are not geometrically equal. Due to a large collection of shapes being topologically equal to each other it becomes difficult to prove if two shapes are equal to each other or not. For example it's obvious that the 2d plane is not equal to the 3d space topologically cuz one can't be streched/twisted into another but to prove this rigorously takes some effort. Or for example is the sphere topologically equal to the doughnut 🍩?. These questions are not so straightforward to prove rigorously and hence we have the subfield of topology called algebraic topology. It turns out that algebraic objects like the integers, rationals,etc are easier to study than shapes themselves so in order to make topology easier we assign an algebraic object to each shape, there are a lot of ways to do this and once we do this it becomes much easier to tell if a shape is different from another as we just need to show that the algebraic objects attached to the respective shapes are different. So that's the primary idea of Algebraic Topology to reduce topological questions to algebraic ones.There are many ways to assign algebraic objects to topological objects the most easiest to describe way is homotopy groups.

(Things are going to be a bit more complicated from this point)

The idea of homotopy theory is to extend topology one step further, in topology two shapes are considered equal if they can be continuously deformed into each other similarly in homotopy theory two functions between shapes f,g:X→Y are equal if they can be transformed continuously into each other. Two functions are equal in this sense they are said to be homotopic. Using this idea of homotopy we can form algebraic objects from topological objects called homotopy groups. Even though these homotopy groups are the easiest to define computing them or finding them for a particular shape is comparitively harder. We have a homotopy group of a shape for any integers n≥1. So we have 1-homotopy group,2-homotopy group,3-homotopy group and so on... . It's a massive open problem to find the general n-homotopy group of a m dimensional sphere. Since homotopy groups are hard to compute , mathematicians have constructed more easy to compute and stable analogues of the homotopy groups called stable homotopy groups and hence have established stable homotopy theory. The ideas of homotopy theory can be applied to a lot of cases which are not topological like purely algebraic cases so we have a much more general theory called abstract homotopy theory to be able to apply the ideas of homotopy theory to a lot of areas in maths.

An interesting result in homotopy theory is that if we restrict our attention to very specific algebraic objects called groupoids and restrict our attention to very specific topological objects called spaces of homotopy 1-type, then the theory of Topology becomes literally equal to the theory of Algebra ! more specifically the category of groupoids and the category of topological spaces of homotopy 1-type are quillen equivalent model categories.

Anyways there are other ways to assign algebraic objects to topological objects like simplical/cellular (co) homology groups , these are slightly more complicated to define than homotopy groups but more easy to compute. There are a lot of beautiful classical applications of topological homology theory I won't list out all of them but one is a result proved in 2020 by ATH Fung that every simple closed curve inscribes infinitely(uncountable infinity) many rhombuses , here inscribes means that the vertices of the rhombi lie on the curve. Also similar to the case of homotopy the ideas of homology can be applied to a lot of areas, this generalized study of homology is called homological algebra. The primary idea behind homological algebra is to study by how much a function f:X→Y fails to be surjective. We measure the failure of surjectivity qualitatively through algebraic objects called homology groups. Two examples of applications of homological algebra will be de Rham cohomology which helps us to do calculus on higher dimensional shapes called manifolds and in some sense measures the failure of the fundamental theorem of calculus in these higher dimensional shapes and the second example will be etale cohomology using which Alexander Grothendieck solved the second weil conjecture an important conjecture in number theory and algebraic geometry.

To end this I would like to describe a very recent development. With enough experience it becomes more and more evident that homological algebra is of central importance in a lot of areas of maths especially algebraic areas. But suppose we are dealing with objects which are both algebraic and topological it's observed that it's difficult to do homological algebra if we want to respect both algebraic and topological properties of these objects. So a lot of results of homological algebra fail for algebraic topological(objects which are both shapes and have an algebraic structure) objects. To solve this issue Peter Scholze and Dustin Clausen in the late 2010s created a new kind of mathematics called condensed mathematics. They use new kind of objects called condensed sets to deal with this issue.

There are several other areas of topology as well like differential topology , topological data analysis,topological quantum field theory, etc but it will take too long to describe them and my knowledge is also limited so I will end it here.

I hope this motivates you to explore topology in more depth and detail :)

I am writing this to give the general math interested audience a brief idea about topology.

Warning: I am trying to make this accessible to everyone so if you haven't done topology formally it's highly likely that you might get a wrong idea of some concept so don't take everything I say literally and look into these things deeper and more rigorously if you want clarity.

So let's begin with what even is topology? To answer this let's try to relate topology to something we probably have some idea about which is geometry. Topology is both related and independent of geometry in some way. If you consider geometry to be the study of shapes then topology becomes a subfield of geometry because we also study shapes in topology. But if you take a more rigid definition that geometry studies properties of shapes like length,angle,area, volume,etc then topology becomes independent of geometry because in topology we study properties of shapes which remain same even if we twist ,strech and bend the shape and angles,area, volume obviously don't stay the same under these transformations so they are not topological properties. Modern mathematicians usually consider geometry to be simply study of shapes and not put too many rigid conditions on the kind of properties we study cuz most modern geometric studies like differential geometry, algebraic geometry study more qualitative properties like the dimension of a shape,if it can be embedded in some other shape or not ,etc rather than more quantitative properties like length, angles,etc even though these concepts are still of importance but the focus shifts from these specific quantities to more general stuff. So according to the more modern loose formulation of geometry, topology is a subfield of it which studies shapes and properties of shapes which don't change under streching,twisting and bending.  

In classical plane geometry two polygons are equal if all of their angles are equal and sides are of equal length this equivalence is called congruence. In topology two shapes are considered equal if one can be twisted,streched or bent into another this kind of equivalence is called homeomorphism.This definition is clearly more loose than the previous definition of congruence and hence the collection of shapes topologically equivalent to each other is much larger than the collection of shapes geometrically (congruence) equal to each other. For example a circle and a square are topologically equal as a square shaped string can be transformed into a circle shape string even if they are not geometrically equal. Due to a large collection of shapes being topologically equal to each other it becomes difficult to prove if two shapes are equal to each other or not. For example it's obvious that the 2d plane is not equal to the 3d space topologically cuz one can't be streched/twisted into another but to prove this rigorously takes some effort. Or for example is the sphere topologically equal to the doughnut 🍩?. These questions are not so straightforward to prove rigorously and hence we have the subfield of topology called algebraic topology. It turns out that algebraic objects like the integers, rationals,etc are easier to study than shapes themselves so in order to make topology easier we assign an algebraic object to each shape, there are a lot of ways to do this and once we do this it becomes much easier to tell if a shape is different from another as we just need to show that the algebraic objects attached to the respective shapes are different. So that's the primary idea of Algebraic Topology to reduce topological questions to algebraic ones.There are many ways to assign algebraic objects to topological objects the most easiest to describe way is homotopy groups.

(Things are going to be a bit more complicated from this point)

The idea of homotopy theory is to extend topology one step further, in topology two shapes are considered equal if they can be continuously deformed into each other similarly in homotopy theory two functions between shapes f,g:X→Y are equal if they can be transformed continuously into each other. Two functions are equal in this sense they are said to be homotopic. Using this idea of homotopy we can form algebraic objects from topological objects called homotopy groups. Even though these homotopy groups are the easiest to define computing them or finding them for a particular shape is comparitively harder. We have a homotopy group of a shape for any integers n≥1. So we have 1-homotopy group,2-homotopy group,3-homotopy group and so on... . It's a massive open problem to find the general n-homotopy group of a m dimensional sphere. Since homotopy groups are hard to compute , mathematicians have constructed more easy to compute and stable analogues of the homotopy groups called stable homotopy groups and hence have established stable homotopy theory. The ideas of homotopy theory can be applied to a lot of cases which are not topological like purely algebraic cases so we have a much more general theory called abstract homotopy theory to be able to apply the ideas of homotopy theory to a lot of areas in maths.

An interesting result in homotopy theory is that if we restrict our attention to very specific algebraic objects called groupoids and restrict our attention to very specific topological objects called spaces of homotopy 1-type, then the theory of Topology becomes literally equal to the theory of Algebra ! more specifically the category of groupoids and the category of topological spaces of homotopy 1-type are quillen equivalent model categories.

Anyways there are other ways to assign algebraic objects to topological objects like simplical/cellular (co) homology groups , these are slightly more complicated to define than homotopy groups but more easy to compute. There are a lot of beautiful classical applications of topological homology theory I won't list out all of them but one is a result proved in 2020 by ATH Fung that every simple closed curve inscribes infinitely many rhombuses , here inscribes means that the vertices of the rhombi lie on the curve. Also similar to the case of homotopy the ideas of homology can be applied to a lot of areas, this generalized study of homology is called homological algebra. The primary idea behind homological algebra is to study by how much a function f:X→Y fails to be surjective. We measure the failure of surjectivity qualitatively through algebraic objects called homology groups. Two examples of applications of homological algebra will be de Rham cohomology which helps us to do calculus on higher dimensional shapes called manifolds and in some sense measures the failure of the fundamental theorem of calculus in these higher dimensional shapes and the second example will be etale cohomology using which Alexander Grothendieck solved the second weil conjecture an important conjecture in number theory and algebraic geometry.

To end this I would like to describe a very recent development. With enough experience it becomes more and more evident that homological algebra is of central importance in a lot of areas of maths especially algebraic areas. But suppose we are dealing with objects which are both algebraic and topological it's observed that it's difficult to do homological algebra if we want to respect both algebraic and topological properties of these objects. So a lot of results of homological algebra fail for algebraic topological(objects which are both shapes and have an algebraic structure) objects. To solve this issue Peter Scholze and Dustin Clausen in the late 2010s created a new kind of mathematics called condensed mathematics. They use new kind of objects called condensed sets to deal with this issue.

There are several other areas of topology as well like differential topology , topological data analysis,topological quantum field theory, etc but it will take too long to describe them and my knowledge is also limited so I will end it here.

I hope this motivates you to explore topology in more depth and detail :)

I had this question about ISI, like in the first year or the first few months do students from an olympiad background get more attention/popularity or better treatment? Like I have seen this in maths camps that someone with an olympiad background is given much more attention generally. A similar question will be," is there elitism in ISI?".