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КОМЕНТАРІ
@shubhamjat6926 7 місяців тому
Great talk
@jeannedescombes9561 8 місяців тому
Very Interesting Peter ! I've understood almost everything
@Unidentifying Рік тому
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
@lenmargolin4872 Рік тому
talks too slow, why is it necessary to write rather than have prepared?
@moshecallen Рік тому
Bookmarked to go over in detail later.
@annaclarafenyo8185 2 роки тому
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.
@annaclarafenyo8185 2 роки тому
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.
@truebaran Рік тому
Hello Anna, for a careful construction of Gaussians measures you can check for example Berezansky&Kondratiev Book ,,Spectral methods in infinite dimensional analysis''. Regarding the Haar measure please remember that Haar measure always exists on a (locally) compact group: and the product of compact topological group is again compact so for a Haar measure there is no problem. As it comes to the Lebesgue measure in infinite dimensions there is none. You may be also interested in Connes&Marcolli view on the renormalization via universal Cosmic Galois Group and Hopf algebras. Finally I recommend a paper by R. Langlands ,,Renormalization fixed point as a mathematical object''. In case of any problems with finding the literature you can contact me truebaran(at)o2.pl
@tomhutchcroft2855 3 місяці тому
There's no problem constructing infinite-volume limits of lattice models; this is completely standard. See e.g. the book of Friedli and Velenik for a mathematical treatment. For the specific thing about the Haar measure on the product of countably many circles, this is no problem because e.g. the countably product is a locally compact group. In general there is no problem taking countably many independent copies of a random variable; this is something that will be done in any book on measure-theoretic probability. This is not at all where the technical problems with doing rigorous treatments of RG lie, to say the least!
@annaclarafenyo8185 3 місяці тому
@@tomhutchcroft2855 The problems are subtle, and will become apparent when you enter proofs into a proof-assistant, so that you try for actual full rigor. The issue is that there is no lifting theorem for results about deterministic objects to results about random variables, which means that every theorem you prove for a deterministic object of some sort, no matter how trivial, has to be reproved for random versions of the same thing. This nobody ever does, leading to lapses in rigor which are not in any way essential, but make it impossible to determine if the theorems are rigorously proved. This is not obvious from standard presentations, you have to see examples of these paradoxes to understand where they come from, they come from bad foundations.
@audaceamenvioutoudossou-ol8541 5 років тому
How can one get admission to IHES in Mathematics?
@IhesFr 5 років тому
Hi, to come to IHES for a research visit, you have to apply through our website: www.ihes.fr/en/applications/
@takahirokobayashi1385 7 років тому
In japanese please.
@takahirokobayashi1385 7 років тому
It was a very descriptive lecture. Thank you very much.