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Great talk

Very Interesting Peter ! I've understood almost everything

how does mass have length unit ? are we redefining mass? Not sure how it makes sense to talk about mass just related to manifold curvature, without any stress tensors or (energy) density ? Interesting talk thank you

talks too slow, why is it necessary to write rather than have prepared?

Bookmarked to go over in detail later.

Fermions don't quite behave as infinitesimal, they behave as finite range. The difference is that infinitesimals don't anti-commute, it's a huge difference. The anti-commutation make all the integrals pfaffians, which are bounded combinatorial computations on finite volume, unlike integrals.

I wish someone would attempt to put these RG proofs into a rigorous computational axiomatic system, I don't believe the construction of infinite products of Gaussian measures is rigorously done. This is not to cast doubt on the results, they are certainly correct, and the proofs are certainly the right proofs conceptually, but they don't fit well into standard rigorous set-theoretic systems. An example of a construction which looks absolutely innocent is the "Haar measure on the product of a lattice of circle groups". Another is the product measure for the Ising model. If there is a reference constructing these measures set theoretically carefully, I would be happy to see it, the references I have seen (e.g. Glimm and Jaffe) are cavalier about rigor (even though it doesn't look like it superfically). An example of where you can go wrong in rigorous proofs, the measure is defined over certain sets of spin-configurations. To say 'the correlation decays with such and so correlation length' is an EXTREMELY subtle statement about sets of configurations where the measure is concentrated, you can't use samples from the distribution in rigorous set-theoretic proofs, even though they are used everywhere in informal arguments. This makes it next to impossible to actually define the properties of RG limits in rigorous set theoretic mathematics, they are made easier when you can speak about probability naturally, without the old measure paradoxes that came from the non-measurable sets.

How can one get admission to IHES in Mathematics?

In japanese please.

It was a very descriptive lecture. Thank you very much.