Congrats for being in the position to choose among several great universities. The majors that you are interested are very strong at Princeton. All college rankings can be somewhat helpful in understanding the strengths of each university. Here are the grad department rankings by USN&WR.
Math: Princeton #1, Brown #14, Yale #9
Physics: Princeton #3, Brown #35, Yale #12
Computer Science: Princeton #8, Brown #49, Yale #20
Engineering Princeton #17, Brown #47, Yale #38
In your fields of interest Princeton has the higher ranked department. Seventeen Princeton alumni and faculty have won Nobel Prizes in Physics. The two major international awards in mathematics are the Field Medal and the Abel Prize. Most of the Field Medal winners have been Princeton students, Princeton faculty, or have lived in the Princeton area. Filed Medal winners since 2000:
• 2018 Akshay Venkatesh *PhD 2002 at age 20; thesis Limiting forms of the trace formula;
Institute for Advanced Study 2005-06 & 2017-; at age 11 he competed at the 24th International Physics Olympiad in Williamsburg, Va., winning a bronze medal. His PhD adviser, Peter Sarnak is now at IAS.
• 2014 Manjul Bhargava *2001, Professor since 2003
• 2014 Maryam Mirzakhani, Princeton mathematics professor from 2004 to 2010; 1st woman Field Medal
• 2010 Cédric Villani, Institute for Advanced Study, 2009
• 2010 Stanislav Smirnov, Institute for Advanced Study
• 2010 Ngô Bảo Châu, Member of the Institute for Advanced Study in Princeton, 2 IMO gold medals
• 2010 Elon Lindenstrauss, Professor from 2004 to 2010; also at IAS
• 2006 Andrei Okounkov Professor from 2002-2010 Princeton University
• 2006 Terence Tao *96, a PhD at age 20; scored 760 on his math SAT at age 8; International Mathematical Olympiad, first competing at the age of 10, winning a bronze, silver, and gold medal
• 2002 Vladimir Voevodsky (Institute for Advanced Study, Princeton, 2002-2017)
Professor Manjul Bhargava has been teaching a freshman course explaining mathematics using card tricks. This years Abel Prize winner is Karen Uhlenbeck who was a visiting senior research scholar in mathematics at Princeton. She is the seventh Abel Prize recipient associated with Princeton. Robert Langlands received the 2018 prize for a theory he developed while an associate professor at Princeton; the 2016 prize went to Andrew Wiles, the James S. McDonnell Distinguished University Professor of Mathematics, Emeritus; late University mathematician John Nash shared the 2015 prize with Louis Nirenberg of New York University; and the 2014 prize went to Professor of Mathematics Yakov Sinai. University alumnus and former mathematics faculty member John W. Milnor ’51 GS ’54, the 2011 prize recipient; and John Tate GS ’50, who received the 2010 prize.
Alan Turing, called the “father of computer science” received his PhD from Princeton in 1938. John von Neumann joined the faculty in 1930. All computers today are designed with a single memory for programs and data first described by von Neumann in a paper in 1945; computer scientists call this the von Neumann architecture. The first stored program computer, the MANIAC (mathematical and numerical integrator and computer) was built in Princeton starting in 1947. The first commercially available IBM computer, the 701 was built based on the MANIC design. Princeton engineers have had an important role in the development of networking and the Internet. David Boggs BSE ’ 72 was a co-inventor of Ethernet. Robert Kahn MS *62, PhD 64 co-invented the Internet protocol TCP/IP. The largest cloud service provider with over 2 million servers was started by Jeff Bezos EE & CS, ’86. Stanford, Princeton, and MIT are frequently cited as having the best computer science PhD programs. Google has established an AI lab across the street from Princeton University. The lab builds on several years of close collaboration between Google and Professors of CS Elad Hazan *2006 and Yoram Singer, who will split their time working for Google and Princeton. Eric Schmidt BSEE ’76 is the former Executive Chairman of Google. The engineering school founded in 1889 was one of the first engineering schools in the US.
Good Luck in your decision.