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.