So I bumped into this article Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces and Essential self-adjointness of the wave operator and the limiting absorption principle on Lorentzian scattering spaces.
If you care about index theorems or quantum field theory on curved manifolds these results are a big deal. Math phys is finally making some head way into physically meaningful functional analytic results in gravity.
If you know about Connes standard model then development of the spectral action principle for (even a small class) of Lorentzian manifolds is particularly interesting.
There was a talk about this, in a simple case, this morning by Elmar Schrohe. If you'd like to get a feel for what this all means.
Title: Index Theory for Fourier Integral Operators and the Connes-Moscovici Local Index Formulae
Abstract: The index theory for operator algebras generated by pseudodifferential operators and Fourier integral operators, more specifically Lie groups of quantized canonical transformations, has attracted a lot of attention over the past years. It can be seen as a universal receptacle for a wide range of index problems such as the classical Atiyah-Singer index theorem, the Atiyah-Weinstein problem, or the B"ar-Strohmaier index theory for Dirac operators on Lorentzian spacetimes. It also includes work by Connes-Moscovici, Gorokhovsky-de Kleijn-Nest, or Perrot.
In my talk, I will focus on the particularly transparent situation, where the pseudodifferential operators are Shubin type operators on euclidean space. We first study the case, where the Fourier integral operators are given by metaplectic operators, then we add a Heisenberg type group of translations, so that we obtain the quantizations of isometric affine canonical transformations.
We find a cohomological index formula in the first case. In the second, our algebra encompasses noncommutative tori and toric orbifolds. We introduce a spectral triple $(\mathcal A, \mathcal H, D)$ with simple dimension spectrum. Here $\mathcal H=L^2(\mathbb R^n, \Lambda(\mathbb R^n))$ and $D$ is the Euler operator. a first order differential operator of index $1$. We obtain explicit algebraic expressions for the Connes-Moscovici cyclic cocycle and local index formulae for noncommutative tori and toric orbifolds.
There is a youtube channel with these talks on it (as well as other stuff): https://... keep reading on reddit ➡
Seeing Geometry / The Matrix After 15 Minute Meditation
Upon meditating for roughly 10 minutes, a geometrical pattern begins to emerge. It’s not my imagination or thoughts. It’s like I’m tuning into the right side of my brain, and if I tune into the left, that’s when the thoughts arise and my imagination runs wild.
But after sitting for 10 minutes and starting to concentrate on the shapes, i see it grow and expand, turning into various things. When I first became aware of it, it looked like a squirming bug, a parasite even, but further inspection and concentration reveals to me that it’s something much more. I was taken aback as I watched it turn into a car, weird looking language writings, a spider looking being that probably wasn’t a spider, lines, and so much more.
It looks just like the digital rain from the matrix. If I would meditate for longer than 10 minutes it would gain more color. Green - then Blue - then Red - then Orange - then Yellow - and finally purple. The purple one is the hardest to get to, although that’s just due to my impatience. A good 30-40 minute sit would do the trick but lately I haven’t felt the need to meditate any longer than 15 minutes at a time. When this thing started revealing itself to me, I would gain immense amounts of energy, and expanded awareness, balanced emotions, and a flow state.
I used to always meditate for 1 hour at a time, but now I don’t need to do that for peak results. My 5 senses are heightened to a tee after 15 min, I feel the energy flowing through my body but it feels different than jing or chi. It’s like rainbow colors? Upon typing that, I realize it could have something to do with chakras, but it seems like something much more divine. It’s really not my imagination doing this, it’s as if I’m becoming aware of something that was always there. When I see the geometry, I feel the energy swirling from my left brain going into the right, like the right side of my brain is taking over and it feels like one of those weather forecast screens when a region is on hurricane alert.
It almost feels like a permanent acid trip, and the symptoms match a kundalini awakening. When I imagine things with my left brain, there’s normally a snake that refuses to leave my mind, always a snake. I used to be afraid of it, but I’ve come to realize it’s a feminine energy and that made accepting it a lot easier. I can smell my aura at this point, it smells like a freshly opened flower that takes the frontline for my nat... keep reading on reddit ➡
whats your best achievement? my best one was darkness keeper: 181 att and worst fail is 89% and you
My friend and I found an Out of Bounds spot in the Reservoir. The photos attached are some of our findings. All geometry found is located through the middle taken door in the Reservoir Boss Room which can be seen in the 6th photo. The middle door is still blocked off, which may indicate that any secret related mission is still time-gated. This is an obligatory first post notice.
[https://medi... keep reading on reddit ➡
Hey, I made a visual guide to help my guild with Inerva Darkvein prog, and they seemed to like it, so I figured I'd post it here as well. Here is the link.
I hope someone can get some use out of it as well!
There's been a flurry of activity recently about using algebraic geometry and category theory to replace topology as the foundations for functional analysis. In particular, Peter Scholze and Dustin Clausen have developed a program of condensed mathematics to do this (and there was a post last week about formalizing one of the main results in Lean).
As someone who is not a functional analyst, but somewhat adjacent to it, I've been reading about this and wrote a blog post with some thoughts. In short, I'm excited to see what new ideas this has to offer but also skeptical that it can efficiently handle questions that are needed for PDE analysis, so I think it is probably better suited as a companion theory rather than a full replacement.
Please let me know if you have any thoughts/ comments.
Edit: Peter Scholze responded to this post. I've posted his comment as an addendum at the end of the post. It's very much worth reading.