The Abdus Salam ICTP offers a one year Diploma in HEP, CMP, ESP and Math. This one year programme is unique in the sense that it prepares students from developing nations to pursue a PhD in US or Europe or other parts of the world. I am an alumni of this programme. I studied at ICTP in 2014-15 batch of the Math Diploma. This will be the first time that I will recount my experiences. Please note that I speak only for the Mathematics Diploma section.
The programme is a year long intense programme with lectures for roughly 8 months divided into two semester, followed by one semester of a short project work. The lecturers are mostly scientists at ICTP, while for some courses there are other lecturers depending on who is available at the institute at that time. There are tutorial sessions for subjects where students are not very well prepared and they are mostly done by postdocs at ICTP. The courses covers all the basic courses that one might expect to study at the graduate level and sometimes they go beyond the normal curriculum. Needless to say the lecturers are experts in their own fields and this is reflected in the lectures. ICTP has NO bad lecturers. If you want to sample some lectures then you can go here where all the lectures are videotaped and made available online as soon as the lecture has finished. Some of the courses, like Differential Geometry, Complex Analysis and PDEs are gems and should be given a try.
After two semesters of courses with exams, the students move on to the final phase, that is of a short project work under some scientist at ICTP (not necessarily a permanent scientific staff of the institute). In mathematics, mostly this means trying to understand a particular topic or paper and then writing it out yourself and presenting it at the end of the stipulated time (around 3 months or so). In other subjects (most notably ESP and CMP), students also do original research and the works are normally published in reputed journals (an alumni couple of years back even published in Physics Letters B).
This covers the educational aspect of the course. But ICTP is much more than lectures. The one year that I spent there was memorable due to numerous factors. I will just list a few of them below:
The Marie Curie Library has access to any book on physics and mathematics that has ever been published (this is a slight exaggeration, but it is mostly true). The library staff are very friendly and even being a student I could order a book for the library which I wanted to read and the library did not have it (it took roughly two weeks for me to get the book in my hand). If you want to study then there can be no better place.
The city of Trieste is fantastic, very close to the Adriatic sea (in fact, one building of ICTP lies next to the sea), it provides awesome atmosphere for study and research. The Italian cuisine is pretty good, needless to say and the city is not as expensive as Milan or Rome. ICTP provides a stipend of 800 Euros which is more than enough to get by. Also the fact that everyone knows and respects ICTP in Trieste is an added bonus.
Everyone at ICTP is friendly. And I literally mean everyone. In my one year I did not find one single person who did not exchange a smile or stop of talk or discuss something. It is a huge family under one umbrella. I am sure that even if I go to ICTP after decades and mention that I am an alumni, there will be no dearth of people to talk and reminiscence.
All facilities provided at ICTP are topnotch. Students get almost the same benefits as any member of staff. And if there is a problem, it gets fixed soon enough so that it does not bother you.
Everyone speaks English. This can be difficult in the city where people do not speak English, but it was a good opportunity for me to pick up basic Italian.
If there is something in mathematics or theoretical physics that you want to learn or understand, chances are very high that there will be someone in ICTP who knows a lot about that particular topic and would be able to lend you a helping hand. It is not uncommon to pass by blackboards filled with esoteric symbols, even in the bar or lobby.
The scientific who’s who visits ICTP on a regular basis. You may randomly bump into someone who is very famous in his or her field and you would not even realize it. During my stay there, at least 4-5 Nobel laureates and 4-5 Fields Medalists visited the institute at some point or the other.
There is no lack of scientific activity in ICTP. At a given time, there will be 2-3 conferences or workshops going on and if something catches your fancy then you can just attend it without anyone questioning you (provided you do not miss any classes).
Italian coffee is awesome!!
The friends you will make in that one year will give you a new perspective of different cultures, places and what not.
These are just some of my personal feelings. There is a joint ICTP-SISSA PhD programme of which I do not know very much. But it should be awesome as well.
tl;dr: ICTP is a magical place
[This was an answer to a question in Quora that I was asked to answer. The original question can be viewed here.]
P.S. There are so many things to write about my experiences at ICTP that it would be too large a post, so I will write some more at a later time.