[Math] Landmarks of subjects of mathematics

big-picture

In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).

For example in natural number theory it is good to know quadratic reciprocity and in linear algebra it's good to know the Cayley-Hamilton theorem (to give two examples).

So, what is one (per post) deep and representative theorem of each subject that one can spend a couple of months or so to learn about? (In Combinatorics, Graph theory, Real Analysis, Logic, Differential Geometry, etc.)

Best Answer

Differential Geometry: the Gauss-Bonnet theorem.

I took a one-semester intro course on differential geometry class and we got to this towards the end of the semester, so I feel that a couple of months is an appropriate time frame for this theorem.

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Big List – Categories of Mathematics

I first refer you here, to the math subject classification system of the American Mathematical Society. I also refer you Arxiv's Math subject classification system. These are the two major systems that I use and that I refer to when classifying or looking for mathematics. As for the categories - these are often made the way they are due to historical events or interpretations.

In reference to the distinctions between 'first' and 'second' level math, and so on: I think that Arturo was basing these on necessary prerequisites. For example, one can take a first class on Algebra, Geometry, Elementary Number Theory, Real Analysis, or Topology without having taken any of the others. Of course, one might argue that there are many interconnections and that one would benefit from knowing algebra before learning number theory, or topology before real analysis, etc. I think this is true, but that it misses the point: it's not necessary at first.

On the other hand, algebraic number theory, algebraic topology, analytic geometry, etc (to directly quote your quote of Arturo) all require multiple previous topics, i.e. some mixture of topology, number theory, algebra, geometry, analysis, etc.

Math History – Understanding Bourbaki’s Style in Mathematics

From Leo Corry, Writing the Ultimate Mathematical Textbook : Nicolas Bourbaki’s Éléments de mathématique :

In the decades following the founding of the group, Bourbaki’s books became classic in many areas of pure mathematics in which the concepts and main problems, the nomenclature and the peculiar style introduced by Bourbaki were adopted as standard.

The branches upon which Bourbaki exerted the deepest influence were algebra, topology and functional analysis and they became the backbone of mathematical curricula and research activity in many places around the world.

Notations such as the symbol ∅ for the empty set, and terms like injective, surjective, and bijective owe their widespread use to their adoption in the Éléments de mathématique.

And :

Bourbaki’s extremely austere and idiosyncratic presentation of the topics discussed in each of the chapters – from which diagrams and external motivations were expressly excluded – became a hallmark of the group’s style and a main manifestation of its thorough influence.

Also the widespread adoption of approaches to specific question, concepts, and nomenclature promoted in the books of the series indicate the breadth of this influence.

Concepts and theories were presented in a thoroughly axiomatic way and systematically discussed always going from the more general to the particular, and never generalizing a particular result.

And :

In 1950 Dieudonné published, signing with the name of Bourbaki, an article that came to be identified as the group’s manifesto, “The Architecture of Mathematics”. Faced with the unprecedented growth and diversification of knowledge in the discipline over the preceding decades, Dieudonné raised once again the well-known question of the unity of mathematics.

Mathematics is a strongly unified branch of knowledge in spite of appearances, he claimed, and now it is clear that the basis of this unity is the use of the axiomatic method as the work of David Hilbert had clearly revealed starting from the beginning of the century.

Mathematics should be seen, Dieudonné added, as a hierarchy of structures at the heart of which lie the so called “mother structures”:

At the center of our universe are found the great types of structures, ... they might be called the mother structures ... Beyond this first nucleus, appear the structures which might be called multiple structures. They involve two or more of the great mother-structures not in simple juxtaposition (which would not produce anything new) but combined organically by one or more axioms which set up a connection between them... Farther along we come finally to the theories properly called particular.

In additon to Leo Corry's papers, cab be useful also his book :

Leo Corry, Modern Algebra and the Rise of Mathematical Structures (2nd ed - 2003).

Best Answer

Related Solutions

Big List – Categories of Mathematics

Math History – Understanding Bourbaki’s Style in Mathematics

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