theHigherGeometer

From July 2024, links to the following AMS white papers in the pdf at the link below:

• Committee on the Profession: “Questions artificial intelligence raises for the mathematics profession, (PDF)” a broad discussion of how AI might affect our profession.

• Committee on Education: “Artificial intelligence: Challenges and opportunities in postsecondary mathematics education, (PDF)” discussing what we should think about as educators – ethical concerns, new opportunities, and curriculum development.

• Committee on Publications: “Artificial intelligence: Publishing in mathematics, (PDF)” which considers implications of AI for peer review, research integrity, copyright, and publications.

• Committee on Equity, Diversity, and Inclusion: “Equity, diversity, inclusion, and artificial intelligence: Issues for mathematicians to consider, (PDF)” discusses research opportunities around fairness and equity, as well as ethical questions around training data and large language models.

• Committee on Science Policy: “Harnessing the power of science policy with mathematics” – how AI and mathematics can enrich one another, and how to enable interested mathematicians to get involved. These are intended as living documents to be updated regularly with resources and guidance

https://www.ams.org/about-us/AMSWhitePaperonAI.pdf

Lawrence Lessig ( @lessig@mastodon.world ) eat your heart out.

Well, one might argue about Farmer's Union, but patriotism demands I acknowledge it as the state cold drink.

(This was posted by a specific politician elsewhere, but I'm removing the attribution because I think this should not be partisan. Please note that there's a small caveat: new small parties and independents can still get donations, to offset the incumbent effect of major parties)

RE: @nbourbaki@mathstodon.xyz

No way! Here's a machine translation of the contents of the two new chapters (which, incidentally, are finally covering the material Grothendieck did in his PhD in the 50s!)

"This new fascicle of the Book on Topological Vector Spaces is devoted to tensor products of locally convex spaces, as well as their applications.

Chapter VI presents the theory of topological tensor products. It introduces the notion of a tensor construction, which equips the tensor product of locally convex spaces with a locally convex topology. The properties of two tensor constructions, called minimal and maximal, are detailed, along with their uses in describing certain common function spaces. The chapter continues with a study of dualities and envelopes of tensor constructions, followed by a proof of Grothendieck's fundamental inequality.

Chapter VII opens with a study of nuclear maps between locally convex spaces, whose properties are analogous to those of operators defined by continuous kernels. It examines notions of trace and determinant for nuclear maps between Banach spaces. Finally, the class of nuclear spaces is presented, which combines several of the themes of this volume."

RE: @albertcardona@mathstodon.xyz

Unsourced rumour, but one can I guess tell if it worked, and Elsevier-published textbooks just don't last like they used to.

Here's a question I hope to ask on MathOverflow soon, in more detail.

Consider some geometric objects in algebraic geometry, that form an algebraic stack (fibred in groupoids). I'm going to state things about these objects, as desiderata.

Now consider arbitrary morphisms between them, in such a way that the hom-sheaf is representable by something (algebraic space, scheme, I don't mind).

And now I want there to be some kind of "numerical" invariant(s) attached to each morphism, that live in something like a partially-ordered monoid. Composition should correspond to multiplication, more or less. Maybe only with an inequality or similar.

I would like the map from the hom-sheaf to the monoid to be algebraic, this one I have no idea about, except in the one example I currently have: elliptic curves and isogenies, with degree as the invariant (as the space of isogenies is a disjoint union of pieces of equal degree, and deg is Zariski-local constant I believe).

The more sophisticated thing I'm thinking of is something like taking a spectral space associated to the partially-ordered monoid (something something Alexandrov topology/specialisation order) and then try to do something like sending a linear map to its rank, which is really just capturing the stratification of the space of linear maps. Here one has some kind of tropical monoid, but also composition of linear maps only respects things up to an inequality, which feels like how one can encode metric spaces using enrichment, à la Lawvere. I feel like I can get this one to work just in a topological setting, or perhaps diffeological.

This is a bit of a vague idea, and it's been sitting lower down on my to-do list for a while. So I'm airing it now

A good essay analysing one of my favourite #Tolkien stories, much less well-known https://www.elfenomeno.com/en/info/ver/28796/the-notion-club-papers-the-great-experiment-of-jrr-tolkien

Another day, another bad news article about the merger.

Here's a random result which I suspect is true: take a proper, cocompact Lie groupoid. So all the automorphism groups of objects are compact Lie groups, and the (topological) space of isomorphism classes is compact.

Then there should be a presheaf on the groupoid (that is, on the manifold of objects, and equivariant for the groupoid action) that assigns the appropriate equivariant/groupoid vector bundle K-theory to an open (there's details about compactness here: perhaps one needs to be tricky with the site, and use local compactness etc, or perhaps used compact regular closed subsets or something).

And I think this should be related to a constructible sheaf either on the groupoid like this, or on the quotient space, using the nice stratification that comes from work of Weinstein, Crainic–Struchiner, Pflaum–Posthuma–Tang on local linearisation. The stalk at an object (or iso class) should be representation rings of the automorphism group. If the automorphism group is connected, I could imagine that the conjugation action inducing isomorphisms on the representation rings is such that you have something on the quotient space, without that I'm not so sure. It might be ok.

I had vague ideas that I wanted to use this for looking at a groupoid analogue of the result about the Brauer group compared to torsion classes in cohomology, but it's been a while, and I can't fully reconstruct the link. I have other ideas that are separate from this, and which feel more promising, but for my sins, I have to write up the proof for compact manifolds first.

Fun discussion involving @tao@mathstodon.xyz about Lean security vulnerabilities (more at link)

https://leanprover.zulipchat.com/#narrow/channel/423402-PrimeNumberTheorem.2B/topic/LeanCert.20for.20numerical.20log.20bounds.20.28re.3A.20PNT.23892.2C.20PNT.23914.29/near/572854830