Some remarkable statistics about the major of (pure) mathematics at several universities in the world

1. The number of PhD/Graduate students at some US universities

   Many US universities indicated all of their graduate students on their websites. For example, the following image shows the list of graduate students in mathematics at Harvard University. 

   Based on these lists, I just counted the number of graduate students at some universities and got the following result(s): MIT has 122 students, Cornell University has 80 students, Texas A&M University has 105 students, Stanford University has 65 students, Harvard University has 64 students, Yale University has 46 students, University of Utah has 90 students, The University of Chicago has 81 students, City University of New York has 76 students and University of California, Berkeley (UCB) has 175 students. 

2. The number of PhD/Graduate students from several universities who later became speakers at International Congress of Mathematicians

   Being invited at International Congress of Mathematicians (ICM) was a big honor. Many mathematicians mentioned this in their CVs as awards instead of talks or presentations as usual. Cornell University (Ithaca, USA) even posted the list of all of its faculty members and former students who were invited to this mathematical event. Therefore, this statistics might give certain pictures of the "best students" at each university. The list of speakers at ICM can be accessed via wikipedia or IMU's website. Meanwhile, Mathematics Genealogy stored information about thousands of people who obtained a PhD in mathematics or related areas (e.g. statistics, theoretical computer science, ...). 

The information shown in Mathematics Genealogy Project. The flag
indicates the country in which the corresponding PhD degree was obtained.

   By combining these two tools, the number of PhD/Graduate students from each university who spoke at every ICM can be surveyed, which gives the following overall picture. First, in recent years, a large proportion of speakers obtained a PhD degree from US universities, especially from top ones such as Princeton University, UCB, ... However, things were quite different 30 - 40 years ago. At that time, there might be more than 10, or even more than 15, speakers at each ICM who obtained a PhD from Lomonosov State University (Russia), which were larger than that of Princeton or UCB. 

   For US universities, it can be seen that universities with higher rank (according to QS, for example) tend to have more PhD students who became ICM speakers at later stages. However, this might be wrong (in general) for those in other countries. For example, Hebrew University of Jerusalem (Isarel) had many such former PhD students, such as Noga Alon, Elon Lindenstrauss (Fields 2010), ..., but its rank in mathematics (rank 201 - 250, according to QS in 2025) is much lower than that of Technion - Isarel Institute of Technology (rank 111, according to QS in 2025), which is perhaps less famous for its alumni in mathematics. Moreover, there might be also other factors that affect such numbers, such as the number of students each year (which can be estimated via the numbers in section 1) and non-mathematical factors (funding, everyday life). 

   Since the information about a few mathematicians are not available, only the overall picture, which does not depend too much on a few changes in the data, will be indicated. The overall picture might be a good way to indicate the best countries/institutions to study mathematics in the world, depending on the career goal of each individual . For example, from the numbers in the 1980s and in the 1990s, it can be seen that Russia has/had an extremely strong community in mathematics, which was perhaps true about 30 - 40 years ago. 

3. Different numbers that contribute to the rank indicated by QS Ranking

   The rank indicated by QS Ranking depends on a certain list of factors, including Employability, Academic Reputation, a few indexes and Global Engagement, as shown in the image below. The focus of this section is the difference that can be seen by restricting ourselves to a single factor alone. 

   
   First, we restrict ourselves to "Academic Reputation". A notable thing is that University of Bonn (Germany) ranks 41st in general, but it seems to be a part of the top 10 or top 15 in terms of "Academic Reputation". In 2024, its "Academic Reputation" point was higher than that of many universities with higher rank, while it has low score regarding some other factor(s). 

4. Application fees when applying to graduate program in mathematics at different universities: does it depend on the universities' reputation? 

 

   In general, the answer should be "No". Here are some examples that I found when I applied for graduate programs at different universities: 
   + Carnegie Mellon University, which ranked 12th in mathematics in 2025 according to QS Ranking, doesn't require any application fee 
   + Some good universities, such as Rutgers University, University of Utah, Ohio State University, etc, only require $65 - $70, which are relatively cheap in comparison to other universities

   + Many good universities in Germany, such as University of Bonn, University of Hamburg, etc, doesn't require any application fee

   + Many universities in California require international applicants to pay an application fee of $155, which is extremely high. These universities range from high-ranking ones such as University of California, Berkeley and University of California, Los Angeles to lower-ranking ones (University of California, Santa Cruz, etc.)