Available online today (if your institution is paying…) from Cambridge University Press are two volumes well-worth spending some time with: New Spaces in Mathematics and New Spaces in Physics. These contain write-ups based on a workshop organized back in 2015 by Mathieu Anel and Gabriel Catren, the videos of which are available here.

It would be hard to write in any detail about the wealth of material in these volumes, so I’ll mainly just link to the essays that seemed especially interesting to me:

- Microlocal analysis and beyond by Pierre Schapira.
- Spaces as infinity groupoids by Timothy Porter.
- Sheaves and functors of points by Michel Vaquié.
- Stacks by Nicole Mestano and Carlos Simpson.
- The geometry of ambiguity: an introduction to the ideas of derived geometry by Mathieu Anel.
- Geometry in dg-categories by Maxim Kontsevich.
- Noncommutative geometry, the spectral standpoint by Alain Connes.
- Supergeometry in mathematics and physics by Mikhail Kapranov.
- Derived stacks in symplectic geometry by Damien Calaque.
- Twistor theory: a geometric perspective for describing the physical world by Roger Penrose.

For several decades now one often hears from prominent theoretical physicists that “Space-time is doomed”, to be imminently replaced by something new coming out of the latest ideas about fundamental physics. For a long time the claims of this sort getting the most attention were from string theorists, and in these volumes Marcos Mariño explains these in his Stringy geometry and emergent space. More recently, Nima Arkani-Hamed has been making well-publicized claims along these lines, with space-time to be replaced with volumes of objects in Grassmanians such as the amplitudehedron.

A large fraction of the theory community is now working on things like “it from qubit”, which propose to somehow get space-time emergent out of things like qubits or quantum information theory. For most of this kind of thing, I’ve found it hard to figure out exactly what the proposal is for the more fundamental objects from which space-time is supposed to emerge. One recent extreme proposal, by Sean Carroll, has the virtue of specifying what the object is (a self-adjoint matrix acting on a complex vector space), but I don’t think there’s a plausible route from that to our observed physics.

As many of the articles linked to above should make clear, mathematicians have over the past centuries developed a range of deep and surprising ideas about new sorts of ways to think about space and geometry. This activity continues: Peter Scholze’s perfectoid spaces and condensed mathematics are examples of new directions of this kind, too new to make it into these volumes.

Of all of these ideas, the ones that at the moment I find most compelling are the twistor geometry ideas of Roger Penrose, and I’ll have much more to say about those in another blog post soon.

Dear Peter,

great news 🙂 I have been really waiting to read these two volumes. I personally think that there is a great need to look into such mathematical structures (mostly from (higher) category theory and (algebraic) non-commutative geometry) in order to define an appropriate notion of “geometry”, “localization” and “holonomy” adapted to quantum physics. I feel that currently there is really a big and unjustified hype about *emergence* of space-time (a “fashion” even stronger than certain propaganda on strings and related ideas, since it is shared by essentially all of the competing lines of research on quantum gravity): everything geometrical is considered to be a macroscopic feature *emergent* from non-geometrical or pre-geometrical phases. There is something that I find quite weird and wrong about such ideology of emergence: four-dimensional Lorentzian manifolds might well turn out to be some “large-scale limit”, but this does not mean that a geometrical description at the quantum level is impossible and uninteresting (and these developments in mathematics seem exactly to suggest the contrary).

Best Regards.

Dear Peter,

Any book recommendation for twistor theory at physics postgraduate level? I’ve been quite mystified as to what’s going on there.

Thanks

A totally minor and irrational thing: the cover designs are not consistently laid out between the two volumes!

More constructively, there are pointers to relevant pages, talk videos and links to some arXiv/draft versions at https://ncatlab.org/nlab/show/New+Spaces+for+Mathematics+and+Physics

Wow, it’s out! I’ve got a chapter in here, and I’ve been waiting for it to come out for many years. It’s a good thing I read your blog.

Way off topic (no need to post) but there’s a new image out on the black hole you show above the blog.

https://www.space.com/first-black-hole-image-polarized-m87

ay,

Thanks. When I get a chance I may update the image that appears on the blog header.

Just to let you know strings are still making headlines:) https://www.theguardian.com/science/2021/apr/03/string-theory-michio-kaku-aliens-god-equation-large-hadron-collider