Comradeship // Freechat

I am a PhD student in mathematics at an elite school in the US. First, some questions you need to answer for yourself: What are you looking to get out of your PhD? What do you want to do after your PhD? How strong will your application be? What kind of schools are you targeting? Unfortunately, most students in the US go directly from their undergraduate studies into a PhD program in the US. I'm not aware of any highly-regarded Masters programs in the US in pure mathematics.

Unfortunately, saying that you are interested in algebra is a bit vague. Are you interested in commutative algebra, noncommutative algebra, representation theory, algebraic number theory, algebraic geometry, homotopy theory, or something else entirely?

I will be very direct and say that outside of western countries, most countries only have a small number of top universities with a significant number of mathematicians who are not analysts. The fact is that there are very clearly "prestige" areas of mathematics (differential geometry, some kinds of nonlinear PDE, algebraic geometry, number theory, some parts of probability theory, geometric representation theory, etc) and they are all concentrated mostly in western countries. This is especially true after the fall of the USSR caused many mathematicians to leave countries like Russia and Ukraine for the US or western Europe.

The communist nation with the most developed mathematical research is China, but outside of Tsinghua, Peking University, Fudan, and a few other top departments, the vast majority of Chinese mathematicians are analysts or completely applied mathematicians. In the last decade or two, Fields Medalist Shing-Tung Yau (丘成桐, Tsinghua) has spent most of his time trying to improve the quality of mathematics research in China while top mathematicians Gang Tian (田刚, Peking University), Jun Li (李骏, Fudan), and Yongbin Ruan (阮勇斌, Zhejiang University) have also returned from the US to lead research centers in China. They still have a long way to go, as seen in this lecture by Yau, but Zhejiang University did manage to hire complex geometer Song Sun (孙菘) away from UC Berkeley.

Since you will have a Masters degree, you can directly apply to PhD programs outside of the US. China would certainly fit your criteria of greater sense of community, less insane politics, and more communists, but I'm not sure if you speak Chinese. In any case, you would have to write your PhD thesis in Chinese if you do your PhD anywhere in China that is not Hong Kong, Macau (which is not strong in the subject at all), or Taiwan. You would be able to write your thesis in English at any European university, but on the other hand Europeans are quite reactionary. Japan is also quite strong in algebraic areas of mathematics. What is most important to you?