Dustin Clausen and Peter Scholze are giving a course together this fall on Analytic Stacks, with Clausen lecturing at the IHES, Scholze from Bonn. Here’s the syllabus:

The purpose of this course is to propose new foundations for analytic geometry. The topics covered are as follows:

1. Light condensed abelian groups.

2. Analytic rings.

3. Analytic stacks.

4. Examples.

Yesterday Clausen gave the first lecture (video here), explaining that the goal was to provide new foundations, encompassing several distinct possibilities currently in use (complex analytic spaces, locally analytic manifolds, rigid analytic geometry/adic spaces, Berkovich spaces). These new foundations in particular should work equally well for archimedean and non-archimedean geometry and hopefully will be the right language for bringing together the Fargues-Scholze geometrization of local Langlands at non-archimedean places with a new geometrization at the archimedean place. He describes as “(very) speculative” the possibility of a geometrization of global Langlands (with Scholze more optimistic about this than he is).

Tomorrow Scholze will take over, giving the next six lectures. Perhaps this characterization is a bit over-the-top, but seeing lectures of this sort and of this ambition taking place at the IHES brings to mind the glory days of Grothendieck’s years lecturing at the IHES on new foundations for algebraic geometry. I fear that keeping up on the details of this as it happens will require the energy of someone much younger than I am…

**Update**: Scholze’s first lecture is here. He gives his version of the motivation for these new foundations.

**
Update**: This sort of thing didn’t happen back in the days of SGA.

For those who haven’t watched till the end, here is a couple of interesting things from

the Q&A:

1) At 1:45:45 Clausen reveals that he and Scholze are trying to write a book at the same time as they are giving these lectures. He says it’s not clear whether they’ll release things sequantially or all at once.

2) Around 1:46:00 a member of the audience asks whether he can describe in one or two sentences some major applications of this theory. After some hesitation, Clausen answers that it proves “the most general possible Rieman-Roch theorems in analytic geometry”.

Eventhough these topics fly way above my head, I find the progressive discovery of these massive new theories by Clausen and Scholze fascinating to watch. What I find particularly striking is that this is the 4th semester-long course that they give around Condensed Mathematics since 2019, accumulating hundreds of pages of lecture notes… yet Scholze and Clausen still haven’t published any paper together to date! It’s like witnessing the slow unravelling of something gigantic with likely spectacular applications, but we are still in the build up phase, with still no major results in sight, at least that they deem worthwhile of publication.

Equally striking is that Scholze hasn’t published anything new on the ArXiv for more than 2 years now. He seems to be completely all-in on this! Unless of course he has some other academic activities that I am not aware of.

Off-topic but talking about the IHES they’re hosting a programme March next year “Quantum and classical fields interacting with geometry” and I notice Witten is on the list of speakers. There’s been lots of work on the semiclassical approach recently in connection with black hole physics so it’s perhaps good for the field to see stringy people move onto other areas.

https://indico.math.cnrs.fr/event/9486/

Antoine Deleforge,

One “big” application or motivation that they may have is possibly the discovery of something called “analytic K-theory”. If I remember correctly, Clausen said on Mathoverflow that the main motivation for condensed mathematics is to define algebraic K-theory “with proper support” which guides them to something they call “analytic K-theory” that they then use to define “modified Hodge Conjecture” in terms of their “analytic K-theory”, and Clausen said that the usual Hodge conjecture is equivalent to the statement that “analytic K-theory class can be continuously deformed into an algebraic K-theory class”.

About Scholze’s “upcoming paper/academic activities”, you can see from a paper published in Arxiv last month by Kieu Hieu Nguyen here https://arxiv.org/pdf/2309.16505.pdf that apparently Scholze has an upcoming paper as you can see in Remark 1.7 which I will quote here:

“A construction of Hecke eigensheaves associated with “generous” L-parameters and a proof of Harris-Viehmann’s conjecture for an arbitrary reductive group as well as various foundational results would appeared in the forthcoming work of Hamann, Hansen and Scholze [HHS]”

Yonz,

Thanks a lot for these pointers! Really appreciate.

Also, Garoufalidis and Zagier have a reference to a paper in preparation with Scholze, called The Habiro ring of a number field. Reference [36] in https://arxiv.org/abs/2111.06645

While I have nothing useful to say about the math as this is not my field, I thought I’d make the following interesting observation: Tate, apart from being the grandfather of this young field, is also Dustin Clausen’s grandfather.

Abbot of Nalanda,

Yes, and another reason why watching this unfold evokes the 1960s and the transformation of algebraic geometry during those years, when Tate was one of the leaders.

At the end of the 4th Clausen-Scholze lecture at IHES, there’s a bit of a shocker, at least to me. Scholze discuses implications from, and to, the Continuum Hypothesis, due to high cardinalities.

kitchin,

I agree that I was a bit shocked that CH was suddenly brought up seeminlgy out of the blue! But then, my impression is that he’s saying more or less “This thing is convenient to have for some proofs, but the price to pay is not to have CH.” Meaning, it’s a convenient tool if you are ready to get rid of CH, but there are probably other ways around in most cases. Did you understand what he presented at some deeper evidence against CH, or something like this ?

Something that surprised/impressed me from yesterday’s lecture (#5) is that he presented the notion of “Solid” condensed Abelian group in a completely different way from their previous lectures, which seems much more elementary and easy to grasp. Namely, a condensed Abelian group is solid simply if its null sequences are summable (in an appropriate formal definition of this). It can be “solidified” if it is possible to assign a unique object to each sum of null sequence. Proofs of well-behaveness seem to be much easier from this definition, and he ends up showing the two definition are the same. It also explains quite intuitively why R can’t be meaningfully solidified.

Answering a question at the very end of the lecture, Scholze hints that Clausen and him think they have a way to present Liquid vector spaces in a similar fashion, and are currently working out the details on whether it works. “We’ll see in a few weeks”. Very exciting!

kitchin: Connections between Ext-vanishing hypotheses for abelian groups and set-theoretic size issues are a classical topic – read about the “Whitehead problem” for instance. From this point of view, the appearance of these matters in lecture 4 was interesting, but maybe not so shocking.

This may be off-topic, but a week ago, Clausen gave a talk titled “What is the K-theory of the complex numbers?” at IAS/Princeton Arithmetic Geometry Seminar here: https://www.math.princeton.edu/events/what-k-theory-complex-numbers-2023-11-06t213000

Here’s the abstract:

“I will explain what the question means and how to make it precise. Then I will give a conjectural answer.

This is based on joint work with Peter Scholze.”

You can find the video on IAS YouTube channel.

Now after I’ve watched the Clausen talk at IAS here: https://www.youtube.com/watch?v=5QE__xdYtA0 I see that he’s promoting the course on analytic stack with Scholze at 34:56 which makes contact to his IAS talk to some extent.

In this talk, Clausen describes his joint work with Scholze in creating something they called Analytic K-Theory, and Clausen also lists some conjectures regarding Analytic K-Theory and some evidence as to why they should be true (including the fact that they proved the conjectures for degree less than or equal to zero). I would guess that they have a paper together on this topic, but haven’t posted it to the arxiv yet because, as Clausen said in the IAS talk, they believed that degree one of the conjecture is within reach.

I debated writing this earlier today, but that last comment (“I would guess that they have a paper together on this topic, but haven’t posted it to the arxiv yet”) pushed me over the edge. The constant obsessive speculation and textual analysis of which papers Clausen / Scholze might be writing is really too much. Likewise, we don’t need a news flash every time one of them gives a lecture at, say, the IAS.

Condensed math is a beautiful idea, but – like scheme theory! – it is ultimately a tool which lets one handle certain technical situations (in this case, “mixing topology and homological algebra”) with greater facility and new insights. It is definitely not magic, and there is certainly no royal road through it to “geometric Langlands over Z” or any such thing.