# Mathematical breakthroughs [closed]

When I read about mathematical history I hear of breakthroughs. For example, Cartesian geometry, Newton/Leibniz Calculus, and so on. My question is this: What are some recent epoch-making breakthroughs in maths? I'm more interested in theories/new branches of maths such as cartesian geometry and calculus, rather than theorems such as Fermat's Last Theorem and the Riemann Hypothesis.

Thanks.

EDIT: By recent breakthrough's I mean anything after the complete solution to "the riddle of the quintics" by Galois.

EDIT: Examples: Use of computers in mathematics (thanks to Kieren MacMillan), combinatorics, graph theory, etc. I would also appreciate it if you could use your intuition to guess at which new branches of mathematics might appear. For example, imagine you were placed in the mathematical world pre-Newton and knew about everything that was going on at the time, I'm sure it would be possible to anticipate Calculus. Is something like that possible today?

• You might have to say what you mean by "recent", bearing in mind that the influence of a theory (rather than a theorem) may take more time to become apparent. – mdp Oct 1 '14 at 18:47
• Theory of von Neumann and $C*$ algebras in the last 100 years; Non - commutative geometry (Connes & others), linear and non-linear progamming, stochastic ordinary and partial differential equations in the last 50 yrs, theory of 4 manifolds (Donaldson) in the last 30 years or so. – Paul Oct 1 '14 at 19:34
• Non-standard analysis – Paul Oct 1 '14 at 19:36
• This question is really too broad. A complete answer will include all major developments in all branches of mathematics during the last two centuries. – Per Manne Oct 1 '14 at 19:37
• @AnalysisIncarnate I don't see any objective way to determine what is a breakthrough, and what is of less importance. Even if you say that only the creation of a new field of mathematics qualifies, most fields people work in today did not exist two centuries ago. – Per Manne Oct 2 '14 at 0:35

Two things that happened after Galois and created new branches of mathematics:

1. Cantor's foundations of set theory.

2. The creation of functional analysis by the Polish school around Banach.

I suggest category theory, not because of the investigations of more or less exotic categories, but because of how different theories are coupled together through functors and become comparable.

Despite knowing little about it myself, I would say the use of computer applications to develop and/or verify mathematical proofs would count.

A lot of theory of computation (started with Turing, Von Neumann and others) brought areas of study for mathematics. An important one is Complexity Theory, which made possible to classify problems in terms of their computational difficulty.

This kind of thinking wasn't too much important before the computers arise, but today is very important and has a whole theory behind. One of the greatest open problems in Complexity theory is the famous P versus NP.

Far away from my expertise but I heard that Homotopy Type Theory may lay a new foundation to the way we analyze proofs.

(Google "The HoTT Book")

I suggest the rapid development in the early 20th century of game theory, largely through the work of von Neumann. Game theory now has important applications in various fields, especially biology and economics.

Linear algebra applications were theoretically doable, but often completely impractical before computers could be used to speed up computation. There are probably other topics that had a good theoretical base, but the "breakthrough" of fast computation made them finally a lot more actionable.

I will say convex optimization. Yes, convexity has been studied for a long time, but convex optimization as a separate branch is recent. there is a lot of development in last thirty years, especially in the methods related to numerical solutions.