Apart from some of the obvious answers like Terrence Tao, Andrew Wiles and Grigori Perelman which other people have mentioned, I would like to add some less-obvious names:

Manjul Bhargava: He’s a field medalist, and has made fundamental contributions in number theory (Geometry of numbers), representation theory, p-adic analysis and theory of elliptic curves (analyzing the BSD conjecture). Check out his paper on Average rank of elliptic curves, the fact that he explains such a strenuous topic so easily and efficiently makes me admire his mathematical skills.

Ravi Vakil: He is an algebraic geometer and his research work spans on various topics like Gromov-Witten theory. He is one of the very few mathematicians, to have worked on topics like Schubert calculus. He has solved several old problems in Schubert calculus. Among other results, he proved that all Schubert problems are enumerative over the real numbers, a result that resolves an issue mathematicians have worked on for at least two decades. Ravi Vakil (with other mathematicians) has also provided solutions of previously asked Putnam Questions. You can find them here. Some of his solutions are so elegant and unique, ‘*mind-boggling’ *is the word which comes to my mind to describe it.

Anyhow, all these mathematicians (Tao, Wiles, Grigori, Manjul, Tate, Scholze etc.) are still popular among other mathematicians or aspiring mathematicians. These mathematicians are outstanding in their respective fields (or many other fields).

However, I would like to mention two other unheard names here. These two other people are not the best number theorists or algebraic geometers or topologists. I don’t even know if they’ve studied these topics extensively or not. I don’t even know if they’re officially mathematicians or not. I just know that they’re one of the best ‘integrators’ in the world.

**First:** The mysterious User Cleo (MSE guys will know). People refer to her as ‘the second goddess of Namagiri’ as her style is just like Ramanujan’s. Loads of results in integrals, series and astounding relations to the zeta function. Polylogarithms, Hypergeometric functions, Trigamma, Huwitz Zeta, Dirichlet eta functions, you name it and she has mastered them all. She is ‘mysterious’ as she has provided closed forms of such integrals which neither *Mathemamica, Wolfram* nor *Maple* could find. One of my personal favorites of her results is:

[math]\displaystyle \int_{-1}^{1} \dfrac{1}{x}\sqrt{\dfrac{1+x}{1-x}}\ln \left(\dfrac{2x^2+2x+1}{2x^2-2x+1}\right) = 4\pi\cot^{-1}\sqrt{\phi}[/math]

where [math]\phi[/math] is the golden ratio [math]\phi = \dfrac{1+\sqrt{5}}{2}[/math] (Prove this, I dare you, I double dare you!)

She also conjectured a few strange integrals with Airy Functions and their relations with the zeta function which have now surprisingly been proved true. I believe she is outstanding with special integral transforms (like Jacobi and Gegenbauer transforms) as well.

**Second:** Cornel loan Valean. Have you ever solved an American Mathematical Monthly problem? Most of those integrals/series/analysis/functional equation problems are proposed by this guy. One of the most brilliant applied mathematicians I’ve ever seen! Check out his recently published paper of cubic harmonic series: A master theorem of series and an evaluation of a cubic harmonic series, it’s worth appreciating.

His problems might look scary, but are extremely delightful and fruitful to solve, one of them being this:

I recently proved one of his famous results on Quora which was also asked on AMM recently, and I’m deeply honored to have received appreciation from him 🙂 ! Here is that problem:

Prove that [math]\displaystyle \int_{0}^{\infty} \int_{0}^{\infty} \dfrac{\sin x \sin y \sin (x+y)}{xy(x+y)} \mathrm{d}x\,\mathrm{d}y = \zeta(2)[/math]

Pure gold.