Dating at least from the days of "God of Manga" Osamu
Tezuka's interest in the genre, science fiction has long been one of
the major streams of anime. Given the fictional science focus of many anime titles, it should
come as no surprise that MIT is occasionally mentioned in anime. Of
course, MIT usually is used to fill the role of "generic
non-Japanese source of advanced technology", but it is rare for any other American
college to be mentioned in anime, while MIT's appearances are
frequent and widespread.
Mostly, this prominence is probably just a matter of MIT being a
well-established, globally-recognized brand when it comes to advanced
technology and science, or educating people who are good at that sort
of thing. But MIT shows up in other places as well.
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MIT vice-president and dean Vannevar Bush plays a major role in
Serial Experiments Lain, particularly in episode 9.

As pointed out in Thought
Experiments Lain, Serial Experiments Lain's "Knights
of the Eastern Calculus" may be inspired by the Knights
of the Lambda Calculus, membership in the outer circles of which has long been one of the privileges granted to survivors of
6.001.
We have to admit that this is a bit of a stretch and is
certainly utterly obscure outside the confines of 6.001. What
probably happened is two independent sets of people were inspired
by a cool phrase from hacker lore (given LISP's long-time
popularity at MIT, it's reasonable to suppose that the phrase
may have originated here).
Really this entry exists because the web needed a place
for the following screenshot of Lain using LISP on her portable
Navi:

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Uri-P, the lunatic chief engineer of the Nadesico says
in episode 12 that he tried to get into MIT. He failed seven
times, no doubt much to the relief of the people who would have
had to compete against him in the 2.70 contest!
He says he modelled the virtual reality representation of Omoikane, the
Nadesico's computer, on MIT's library (and it does look a
bit like the view from the mezzanine in Hayden Library):

One of the five sets of Magi supercomputers attacking the Magi at
NERV in The end of Evangelion is located in Massachusetts.
We think it's fair to count this. Of course, it's clear that
MIT no longer exists in these days after the Second Impact (at least
not on the present banks of the Charles).

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In the first Patlabor movie E. Hoba, the author of the
Labor virus, graduated from MIT, and one or two mentions were
made of efforts by people at MIT to eradicate the virus.
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The character Alexander Howell from Yuu Watase's Ayashi no
Ceres is a graduate of MIT.

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The person Vision was hunting early in Bubblegum Crisis
episode 7 was an MIT graduate (he got an MD! presumably from the
medical school that will be established sometime in the next
thirty years). Mention of this shows up on a data screen when
Sylvie agrees to have the Knight Sabers act as his body guard.
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In the Magical Girl Pretty Sammy Tenchi Muyo manga story-line,
Washu graduated from MIT at age 11.
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11 seems to be a good age to graduate from MIT.
In Paniponi Dash, 11-year-old teacher and robotics genius,
Rebecca Miyamoto (recently arrived from America), says that she attended MIT
(though she has a great deal of trouble pronouncing
"Massachusetts").

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In Hanaukyo Maid Tai, episode 13, the character Shinshei goes to a
leading, albeit unnamed, university:

| Anyone who has seen MIT's Great Court | can have no doubt what university it was: |
 |  |
Tech is hell.
In episode 8 there's more conclusive evidence. A few minutes
into the episode we see a shot of her desk, on which sits a
letter designating her as an MIT fellow.
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In Amazing Nurse Nanako episode 4 "Firecrackers'', we
meet "Dr. Kyoji.... the youngest foreign student in MIT
history'' (younger even than Washu and Becky?) "from
Japan.... earned a doctorate degree''.
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One of the members of Heroes (evidently a Pillows cover-band),
on-stage in the last episode of Beck:
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A! My Goddess's Keichi and his sempais go to Nekomi Institute of Technology, but
they cosplay as some other institution's cheerleaders.
(Though that looks like a "neko" (cat) design behind the
initials, so maybe it's just a misprint. But the image is too
good to pass up.)
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In Boku no Sexual Harassment Episode 2, one of the
characters went to MIT, and the action is set in Boston. There
is absolutely no way we will post a picture from this
one.
This one is another stretch, since she never actually
says which college in Boston she attended, nor which
anime club she was a member of while here, but we like to think that
Genshiken's Kanako Ohno's Boston-area anime-club membership card
looked something like this:
And we miss her presence at our cosplay workshops.
In episode 2 of Kamen no Maid Guy, the masked male maid Kogarashi
offers to help his "master", high-school student Naeka Fujiwara, with her math
homework. When she disdains his abilities, his fellow maid Fubuki alerts Naeka that:
When Naeka professes ignorance of this "MIT", Fubuki gives a brief explanation
of MIT's name, location, and prestige as a parade of Nobelists passes over
a shot of Killian Court.
Fubuki finishes by calling it "A gathering of the elites around the world! You can also call it the peak of mathematics."
In episode 10 of Real Drive, which takes place in 2061, an AI named
Eliza is causing trouble in cyberspace. And where do AIs come from?

In Toradora!, student council president Kanou Sumire goes to an
American university, intending to become an astronaut. We see her get a
postcard from Japan:
(Apparently she's living in building 5.)
Gundam Seed's George Glenn, a prodigy aerospace engineer for the Atlantic Federation's Federal Aeronautics and Space Administration, also obtained his doctorate degree from MIT at the tender age of 17. He eventually reveals himself to be humankind's first Coordinator, or genetically-enhanced human being, which may better explain this achievement (which comes on top of being an Olympic silver medalist, an American football player, and an Air Force pilot).
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The manga Q.E.D, which has yet to be adapted into an anime (but has a live-action version!), features yet another MIT alumni, Sou Touma, a boy who graduated from the Institute at 15 and returns to Japan to attend high school and solve random mysteries.