The whole thing is unsubstantiated FUD. I base my judgment on the slides at
The whole argument boils down to:
a) there has recently been huge progress [*] in solving the Discrete Log Problem over fields of small characteristic;
b) progress in solving the DLP have historically implied progress in factorization, and vice versa;
c) factorization breaks RSA, and solving the DLP breaks DSA;
d) thus RSA and DSA are dead, move to ECDSA.
The fallacy of it is that in b) and c), the DLP is exclusively over fields of huge characteristics (thousands of bits), making the algorithms in a) powerless. The slides do not hint at the faintest research lead towards moving to huge characteristics. Best argument is that "renewed interest could result in further improvements".
One the positive side, the author is honest: "I’m not a mathematician, I just play one on stage".
[*] See e.g. this recent paper and its references
Razvan Barbulescu, Pierrick Gaudry, Antoine Joux, Emmanuel Thomé: A quasi-polynomial algorithm for discrete logarithm in finite fields of small characteristic