the greatest mathematical discovery of the modern era is asymptoric methods of nonlinear mechanics ( N. M. Krylov, N. N. Bogoliubov and Yu. A. Mitropolsky).

In 1976, Wolfgang Haken and Kenneth Appel used a computer to prove the four color theorem. Andrew Wiles, building on the work of others, proved Fermat's Last Theorem in 1995. Paul Cohen and Kurt Gödel proved that the continuum hypothesis is independent of (could neither be proved nor disproved from) the standard axioms of set theory. In 1998 Thomas Callister Hales proved the Kepler conjecture.

the greatest mathematical discovery of the modern era is asymptoric methods of nonlinear mechanics ( N. M. Krylov, N. N. Bogoliubov and Yu. A. Mitropolsky).

In my understanding, the proof of the Poincare Conjecture by the Russian mathematician Grigory Perelman (2002-2003) is the greatest mathematical discovery in the modern era. This happened almost 100 years after Henri Poincaré formulated this hypothesis in 1904. Experts say that what matters here is not so much the result itself (although this result is in fact really very strong beyond any doubt, regardless of a million dollars as a prize) but as the method of proof itself, and this is especially very valuable. This result eclipsed other outstanding mathematical evidence as Fermat's Theorem, which was completely proved in 1994 by Andrew Wiles.

Griffith C Evans: Mathematical Theory of Competition, JPE 1928

He used the term competition correctly and preceded the so-called game theory by a long time but no one paid attention

Steven Smale on Structural Stability

Informative question.

I agree with Gennady Fedulov about the proof of the Poincare Conjecture by the Russian mathematician Grigory Perelman. This is undoubtedly the first place. Second place is the proof of Fermat's Theorem by Andrew Wiles.

The Prime Gap Theorem by Yitang Zhang

Thank you Melvin R Currie for your recommendation. I read with fascination about this important contribution. Amir

Yes, the Prime Gap Theorem is in the realm of pure mathematics and beautiful.

The question was what are *some* of the greatest mathematical discoveries. Gὂdel's undecidability theorems (1931) are among the greatest.

Thanks Joseph Horowitz and Melvin R Currie.

Mathematics is a backbone of all aspects of our life, means, all topics in mathmatics is extremelly important and powerful tool.

Calculus& Almighty formula

1. The proof of Fermat's last theorem by the English professor of Mathematics- Andrew Wiles

2. the proof of Poincare's conjecture by the Russian professor of Mathematics -Gregorio Perelman

I can add here a great scientific feat of Gikhman and Skorokhod:

the creation of the theory of stochastic differential equations and the theory of stochastic processes:

Gikhman I.I., Skorokhod A.V. Stochastic differential equations.

Springer, 1972.

Gikhman I.I., Skorokhod A.V. The theory of stochastic processes, vol I. Springer, 1974.

Gikhman I.I., Skorokhod A.V. The theory of stochastic processes, vol II. Springer, 1975.

Gikhman I.I., Skorokhod A.V. The theory of stochastic processes, vol III. Springer, 1979.

Madelbrot Set and fractal geometry gives quite explanation about natural phenomena.

Mandelbrot Set and fractal geometry gives quite explanation about natural phenomena.

Fractal Geometry, Julia and Mandelbrot sets

I agree Prof/Dr/Mr Sanjay Mali. With respect to your 'Mandelbrot Set and fractal geometry gives quite explanation about natural phenomena' and 'Fractal Geometry, Julia and Mandelbrot sets' I have been waiting some time for one (1) of our Colleagues to bring this up. I have a particular passion about Mandelbrot etc.

I agree Prof/Dr/Mr Sanjay Mali. With respect to your 'Mandelbrot Set and fractal geometry gives quite explanation about natural phenomena' and 'Fractal Geometry, Julia and Mandelbrot sets' I have been waiting some time for one (1) of our Colleagues to bring this up and/or remind us of it and its 'mathematical beauty'. I have a particular interest and passion about all to with Mandelbrot Sets, Fractal Geometry, Julia Sets and any related mathematical topics..

UHG Wildberger, New-look at multi-sets Wildberger, RatTrig Wildberger, the meta-mathematical(ness) of real numbers.

Simply begin reading any of Wildberger's work and learn more than you ever thought possible. Your brain will thank you.

Thanks to Peter Osudar for your suggestion. Amir

You are welcome! I do like to share and appreciate insights gleaned with respect to Wildberger's work and teaching mathematics to the young.

My current fascination is taking finite fields towards his tropical view on the foundations of mathematics to help navigate the combinatoric properties of a rubiks cube(and the chromoegemetric properties, see his work on Chromogeometry) that will give educators a mathematical manipulative to help guide students in a strongly tautological way towards exploring the already developed, discovered, and known landscape of mathematics.

Developing a maturity towards "speed" and "novel" and "comprehension" is key and these insights slip through the grasps of hard working educators all the time. With a time tested manipulative like a rubiks cube(starting with a 2x2) educators might finally have what they need to progress towards a classroom where students will feel like they have the "anchor necessary" to establish themselves and believe in the mathematics they are participating in.

Just my thoughts...

Big things are coming...

Dear Peter, your preventive and insights are very unique. Thank you for sharing. Amir

Excellent contributions on this discussion.

good question

For real math I find some difficulties and try to develop myself and I will follow this question