Math E-320: Spring 2015
Teaching Math with a Historical Perspective
Mathematics E-320:
Instructor: Oliver Knill
Office: SciCtr 432

Computing lecture

For the P-NP problem, here is the Clay math entry.
Here is a Turing machine emulator. This was a Javascript implementation from 2000. Unlike the above implementation, it only implements one machine. Here are some "Busy beavers", Turing machines which produce a long string of 1, before halting.
Here is the trailer of the "imitation game" from which we have seen some scenes.

We have seen some old computers in class. Here are some links

Dynamics lecture

On pages like this are some javascript animated systems. This was done 15 years ago. In the mean time there is new technology like Webgl. An example in fluid dynanmics. Webgl is quite amazing. here is an other example.
A schlarpedia entry about the Sitnikov problem.

Crypto lecture

May 20: We talked about Diffie-Hellman encryption. A related vulnerability is called log jam". From that article

To exploit vulnerable connections, attackers must use the number field sieve algorithm to precompute data. Once they have completed that task, they can use it to perform man-in-the-middle attacks against vulnerable connections in real time. Using academic-level hardware, the researchers required just two weeks to generate data needed to attack the two most commonly called prime numbers 512-bit Diffie-Hellman uses to negotiate ephemeral keys. Those two data sets allow the attackers to compromise about 92 percent of sites supporting the export cipher. It wouldn't require much additional work to generate data needed to attack the remaining sites. The work required to precompute data needed to attack 768- and 1024-bit primes is orders of magnitude harder, but the researchers said the load is nonetheless within the means of state-sponsored eavesdroppers. In a research paper titled Imperfect Forward Secrecy: How Diffie-Hellman Fails in Practice, the researchers speculate the technique may be the means the National Security Agency reportedly uses to routinely break millions of encrypted connections. Documents leaked by former NSA subcontractor Edward Snowden revealed the mass crypto attacks but didn't say how they're carried out. Besides attacking HTTPS-protected Web and e-mail sessions, the researchers said, the same technique may be used to break SSH and VPN connections, too.


Analysis lecture

Topology lecture

Probability lecture

Please send questions and comments to knill@math.harvard.edu
Math E320| Oliver Knill | Spring 2015 | Extension School | Harvard University