How do top graduate programs in mathematics verify the validity of an enrolee's undergraduate degree from abroad? E.g. IUM in Russia Even if the title might seem self-explanatory, I would ask you to read the entire post to understand what exactly I'm trying to understand here.

I'm from Russian Federation. We don't have a lot of departments in pure mathematics. Most of those undergraduate programs named "mathematics" are actually aimed at people who wish to teach mathematics to engineers/IT/finance people or those who themselves wish to get into engineering, IT or finance. Actually, when I say "we don't have a lot" I mean we have two and a half. No exaggeration here. My explanation is actually a bit oversimplified, but I hope you understand what I meant to say.

The funny thing is that the most renown for the quality of pure mathematics students russian school is called Independent University of Moscow. They offer free education with entrance based on how you perform during the process of education itself. It has no entrance exams, everyone can attend lectures at first, but to get the status of a "student" you must do well in a first three mandatory courses. Why did I called the situtation the "funny thing"? The reason it's degree is not recongized as an official undergraduate degree in Russian Federation. So, being called Independent University of Moscow, it doesn't an official university status in Russia. 

What I wish to understand is how top graduate schools in mathematics would treat a candidate with such a diploma. From what I understand, it has a good reputation in mathematics. Many people from there were accepted to the best graduate schools in mathematics all over the world. For example, Harvard, MIT. Some of those even had their IUM "degree" as the their only degree. That is, they didn't have any official russian undergraduate degree in any field.

But IUM doesn't have a lot of alumni. So, those might look like some special cases, even expections. Besides, all recent IUM graduates had an other, an official degree from other russian university. And the latest case I know of with a man being accepted to MIT with only IUM diploma is dated 2006. Even if such a practice was approprite in US before, it might have changed now. I'm not sure.

That said, I understand there were exceptional cases of enrolee's being admitted to top graduate school (in US) without any degree at all, based on their exceptional recommendations or their research. I know at least three of such people (all from my country, they applied in nineteen-eighties/nineteen-nineties to Harvard and MIT). What I'm trying to understand is how do graduate schools treat the lack of the official status of a foreign school with it being a strong place with strong students and strong professors. Another nuance I'm thinking is that it is a school from abroad. I'm not even sure how to US departments verify the validity of one's undergraduate/masters degree if it's not from US or EU. 

Of course, what I'm asking is about pure bureaucratic procedures. I absolutely understand that to have a chance in getting into top graduate school in mathematics you must have exceptional credentials: recommendations of renown professors or even your own research articles. But, say, one has it (for example, you can get those things during a successful process of getting a "degree" at IUM, that's the reason it has so little of alumni). But how would graduate schools treat such a potential strong candidate if his degree is not official in the country where it was obtained?
IUM (quite purposefully) distants itself from the "officially recognised" education and relies on informal recognition in a sub-group of well-established mathematicians. As an IUM graduate, you most likely will be recognised as an outstanding candidate for your ability to do maths. You will also have recommendations of internationally renowned mathematicians. Many academics will feel themselves very reassured hiring someone with IUM credentials. 

However, universities do not consist of academic only, and HR departments may have their views on the process, too. It is fair to say that even the best professors sometimes have only limited influence on HR when it comes to formalities. And of course, the list of recognised qualifications is one of such things. 

Unfortunately, there are no universal rules which HR departments apply. It is likely that for a particular unusual situation like the one you described, the decision will be made ad hoc by someone in HR department; once it is made, it is unlikely they will go back on it.

That is why many of IUM graduates prefer also to get themselves a degree from officially recognised university. Even if it does not give them the level of education compared with IUM, the official degree always works well to pacify the HR departments, while academics will always look at IUM degrees on your CV as something really outstanding.