Beauty in ML, Langlands problem, Dark Matter and Kerberos
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My weekend exploration…
Elegance in ML
Some random posts on twitter rekindled my fascination with ML. Chris Olah has an amazing explanation on the beauty in ML. Got to learn that even group theory is applied to ML. Link
Langlands Problem
The narrator began by explaining the origins of a problem that involves uniting two branches of mathematics: algebra and harmonic analysis. The first person to consider this connection was the mathematician Srinivasa Ramanujan, who made a conjecture that an infinite series in algebraic terms could be expressed in modular form, which is a symmetrical structure in harmonic analysis.
Many years later, Andrew Wiles used this principle to prove Fermat’s Last Theorem by contradiction. He discovered that all elliptic curves could be described as infinite series, and that these curves also have a modular form. This led to the popular conjecture that all infinite series have a modular form, which would mean that an elliptic curve must also be modular. However, this conjecture was later disproved by the existence of the Frey elliptic curve, which is not modular. This result directly implies that there is no solution to Fermat’s Last Theorem.
Kerberos
Kerberos is an auth method where trust is put on a Key distribution center(KDC). Here are the steps:
Server A wants to securely talk to Server B on the same network but it doesn’t know how. Server A tells the KDC’s Server to let it talk to the ticket granting system(TGS) which is part of KDC.
A sends some info about itself, a random number to KDC. KDC finds if it’s an authenticated user and sends {Kat, Time to Live, same random number} all encrypted with Kas(public key of Server A and Server S. Also to notify TGS of this it sends other packet with {Kat, some info on A, TTL} encrypted with Kst. Thus TGS knows that this has arrived directly from the Server of the key.
Server A sends some info on B, TTL to which TGS send {Kab, TTL, some random number} encrypted with Kat alonwith { Kab, TTL, some info on A} encrypted with Kbt so server B knows that server is legit and have authenticated with KDC first. Thus
Dark Matter
Dark Matter is not so fascinating, I have known about it for quite some time now but I was unaware of the working of experiments. So here is my exploration of it with Derek:
DAMA-LIBRA found a strange oscillating curve for the amount of concentration of DM around our galaxy. The reason that they hypothesized is that DM is not spread uniformly around our galaxy but rather it is divided rather in unequal proportion thus causing it to be dense for some time while sparse for some other.
To prove this hypothesis DAMA-LIBRA teamed up with an observatory in Australia because seasons are polar opposites so their curve would be slightly different/opposite. The rest of the video describes the experiment in Australia.
They used 7 cylinders each containing 1 Kg of pure NaI which when hit by WIMPs(candidate particle for DM) emits a photon which causes scintillation which are detected by photodetectors. Pure NaI contains radioactive Potassium which decays into electron and gamma rays which also produce scintillation so they are submerged in a liquid that just detected gamma particles. Thus if some detectors fire simultaneously then it was decay of radioactive Potassium. Cosmic rays which consist of muons might also disturb the experiment thus a Cosmic ray detector is placed at the top of the container. As the experiment is run underground there might be traces of radiation thus walls are painted to absorb it. Finally the whole chamber is surrounded by 120 tons of steel and plastic. It would take about a decade before the results show up.