10
votes
0answers
395 views

How come Cartan did not notice the close relationship between symmetric spaces and isoparametric hypersurfaces?

Elie Cartan made fundamental contributions to the theory of Lie groups and their geometrical applications. Among those, we can list the introduction of the remarkable family of Riemannian symmetric ...
-2
votes
2answers
622 views

Accidental, unplanned breakthroughs in Mathematics [closed]

In math/physics, or generally in science, there are many moments where the success and the triumph come from the accidental, unplanned attempts. Moreover, there are some cases that originally having ...
45
votes
18answers
7k views

Examples of major theorems with very hard proofs that have NOT dramatically improved over time

This question complement a previous MO question: Examples of theorems with proofs that have dramatically improved over time. I am looking for a list of major theorems in mathematics whose proofs are ...
4
votes
1answer
1k views

Algebraic number theory: building and simplifying

This is a somewhat subjective question, about the past, present and especially future of algebraic number theory. I'm not at all in this area, but I'd be interested in an answer. As we all know, ...
12
votes
6answers
809 views

Proof by `universal receiver'

Anyone following the news knows about the major breakthoughs that have taken place recently in $3$-manifold topology. These have come via a route whose big-picture I find to be conceptually ...
19
votes
0answers
1k views

more on “Transalgebraic Theories” (a 19th century yoga)?

Among the talks at occasion of the Galois Bicentennial, one is about "Transalgebraic Theories". Unfortunately I found only this article describing that fascinating idea as " an extremely powerful ...
15
votes
3answers
766 views

Thom's Principle: rich structures are more numerous in low dimension

Marcel Berger states Thom's Principle as: "rich structures are more numerous in low dimension, and poor structures are more numerous in high dimension." This is in Geometry II ...
6
votes
0answers
374 views

Does mathematical fecundity ever deviate from its applicability?

We are all familiar with Wigner's "unreasonable effectiveness of mathematics" thesis (1), and of Hardy's opinion that "the great bulk of higher mathematics is useless" (2). I am wondering if there are ...
48
votes
12answers
9k views

Logic in mathematics and philosophy

What are the relations between logic as an area of (modern) philosophy and mathematical logic. The world "modern" refers to 20th century and later, and I am curious mainly about the second half of ...
41
votes
14answers
5k views

Does any research mathematics involve solving functional equations?

This is a somewhat frivolous question, so I won't mind if it gets closed. One of the categories of Olympiad-style problems (e.g. at the IMO) is solving various functional equations, such as those ...
22
votes
3answers
2k views

Online math history lectures

This question is somewhat similar to this: Best online mathematics videos? I'm using the word "history" loosely here. What I'm looking for are those lectures that put various mathematical ...
64
votes
6answers
8k views

How to find ICM talks?

I am very interested in reading some and skimming through the list of invited talks at the International Congress of Mathematicians. Since the proceedings contain talks supposedly by top experts in ...
113
votes
130answers
26k views

Fundamental Examples

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please) I'd love to learn about ...