theHigherGeometer

@highergeometer

rimcræftiga bespoke constructions in categorified geometry since 2010 dude [bridged from on the fediverse by ]

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theHigherGeometer@highergeometer· Jul 7

Well, it's soon... #ICM2026

Screenshot of YouTube video thumbnail from the Simons Foundation titled "International Congress of Mathematicians: Opening Ceremony", scheduled for 23 July 2026, 22:20 (my time zone; probably 9am local time in Philadelphia)

theHigherGeometer@highergeometer· Jun 28

I really want to understand this (from https://arxiv.org/abs/0902.0431), so I can look at its restriction to s(u(2)xu(3)) in f_4, where this Lie subalgebra corresponds to say even the "standard" inclusion due to Todorov–Dubois-Violette, surveyed in https://arxiv.org/abs/2606.15235

Theorem 2.5.3 from the first linked paper, arXiv:0902.0431, which describes the Killing form of the complex simple Lie algebra f_4 in terms of an inscrutable formula

theHigherGeometer@highergeometer· Jun 25

Is someone about to get a year-long holiday from the arXiv? cc @arXiv

“Large Causal Models from Large Language Models” https://arxiv.org/abs/2512.07796

1. Fabricated (no such paper)

« Citation: Xiaomei He et al. A survey of event causality identification: Taxonomy, resources, and challenges. ACM
Computing Surveys, 2025. Preprint

« No paper of this title by Xiaomei He exists; the only ACM Computing Surveys paper of this name is Qing Cheng,
Zefan Zeng, Xingchen Hu, Yuehang Si, Zhong Liu, A Survey of Event Causality Identification: Taxonomy, Challenges,
Assessment, and Prospects (2026) https://doi.org/10.1145/3756009.

2. Wrong arXiv ID (resolves to an unrelated paper)

« Citation: Zhiting Jin et al. Causal inference and natural language processing: A review. arXiv preprint
arXiv:2012.14366, 2021.

« Resolves to: https://arxiv.org/abs/2012.14366 is Jian Zhang, Cunjing Ge, Feifei Ma, Counting the Number of
Solutions to Constraints (2020).

3. Fabricated (arXiv ID, author, title do not exist)

« Citation: Vladislav Kosoy et al. Do large language models have causal knowledge? arXiv preprint arXiv:2305.00000,
2023.

« Resolution: https://arxiv.org/abs/2305.00000 is not a working arXiv link, the title does not resolve to any paper on a
web search, and an arXiv author search returns no “Vladislav Kosoy”, neither does a Google Scholar search. A
researcher named Eliza Kosoy works in the area.

[truncated]

“CSQL: Mapping Documents into Causal Databases” https://arxiv.org/abs/2601.08109

1. Wrong first author
« Citation: Abhimanyu Dubey et al. The llama 3 herd of models. arXiv preprint arXiv:2407.21783, 2024
« The first author of https://arxiv.org/abs/2407.21783 is Aaron Grattafiori, the second author is Abhimanyu Dubey (and then a giant list, hence the et al)

2. Wrong title
« Citation: Prashant Garg and Thiemo Fetzer. Testing causal claims in economics. arXiv preprint arXiv:2501.06873,
2025. URL https://arxiv.org/abs/2501.06873
« Correct title: https://arxiv.org/abs/2501.06873 is Causal Claims in Economics (Garg & Fetzer, 2025); authors and ID
correct, the leading “Testing” is not part of the real title in either v1 or v2.

3. Fabricated (arXiv ID, author, title do not exist)
« Citation: Vladislav Kosoy et al. Do large language models have causal knowledge? arXiv preprint arXiv:2305.00000, 2023
« Resolution: https://arxiv.org/abs/2305.00000 is not a working arXiv link, the title does not resolve to any paper on a
web search, and an arXiv author search returns no “Vladislav Kosoy”, neither does a Google Scholar search. A researcher named Eliza Kosoy works in the area.

4. Wrong arXiv ID (resolves to an unrelated paper)
« Citation: Tuan Le et al. Multi-agent causal discovery using large language models. arXiv preprint arXiv:2404.12345, 2024
« Resolves to: https://arxiv.org/abs/2404.12345 is Shuai Li, Ming Gong, Yu-Hang Li, et al., High spin axion insulator (2024).

[truncated]

“PROMETHEUS: Automating Deep Causal Research” https://arxiv.org/abs/2605.12835

1. Fabricated (nonexistent DOI, wrong authors/year/volume)
« Citation: Xiaomei He, Yi Guan, and Min Chen. A survey of event causality identification: Taxonomy, resources, and
techniques. ACM Computing Surveys, 55(14s):1-35, 2023. doi:10.1145/3582128.

« The cited DOI does not resolve: https://doi.org/10.1145/3582128 returns HTTP 404.
« Areal paper with a similar title: Qing Cheng, Zefan Zeng, Xingchen Hu, Yuehang Si, Zhong Liu, A Survey of Event
Causality Identification: Taxonomy, Challenges, Assessment, and Prospects, ACM Comput. Surv. 58(3), Art. 59, 2026
https://doi.org/10.1145/3756009 (different authors, year, volume, and subtitle).

2. Wrong arXiv ID (resolves to an unrelated paper)
« Citation: Zhijing Jin, Bernhard Schélkopf, Peter Spirtes, and Kun Zhang. Causal inference and natural language
processing: A survey. arXiv preprint arXiv:2012.14366, 2021.
« Resolves to: https://arxiv.org/abs/2012.14366 is Jian Zhang, Cunjing Ge, Feifei Ma, Counting the Number of
Solutions to Constraints (2020).

3. Wrong author list (real paper, correct ID)
« Citation: Hao Duong Le, Xin Xia, and Zhang Chen. Multi-agent causal discovery using large language models. arXiv
preprint arXiv:2407.15073, 2024.
« Correct author list: https://arxiv.org/abs/2407.15073 is by Hao Duong Le, Xin Xia, Haijie Xu, Chen Zhang, the citation
omits Haijie Xu and renders “Chen Zhang” as “Zhang Chen”.

[truncated]

theHigherGeometer@highergeometer· Jun 19

I was really happy when I figured out how to define natural number division like this, the recipes I'd seen were very much more mathematical, in that they just gave a recursive formula, and it took a bit of trial and error to figure out I needed the […] [Original post on mathstodon.xyz]

$-- | recursion with parameters -- This defines a function \z,n -> J_{g,h}(z, n) as: -- J_{g,h}(p, 0) := g(p) -- J_{g,h}(p, Sn) := h(p, n, J_{g,h}(p,n)) iteratorWithState :: (a -> b) -> (a -> Nat -> b -> b) -> a -> Nat -> b iteratorWithState g _ p Zero = g p iteratorWithState g h p (Successor n) = h p n (iteratorWithState g h p n)$

-- | truncated division
-- divide m n = n % m, the largest integer q=n % m such that m * q ≤ n
-- note that the arguments are the wrong way around!
divide :: Nat -> Nat -> Nat
divide m n = iteratorWithState constZero h m n
where
h l' m' n' = nonZero l' * (n' + isZero (s m' `mod` l'))

-- | sensible infix operator for divide
(%) :: Nat -> Nat -> Nat
(%) = flip divide

theHigherGeometer@highergeometer· Jun 11

Who can find a copy of this? ;-)

RC 7574 (#32794) 3/23/79
Computer Science 41 pages
RATIONALIZING MANY-SORTED ALGEBRAIC THEORIES
Sam Kamin †

Computer Science Department
State University of New York, Stony Brook
Stony Brook, New York 11794

Typed by Marilyn McCrae and Martha Cooper on CMS (sk.2141)

ABSTRACT
An informal exposition of the work of the ADJ group on algebraic semantics is given. The intent is to introduce the concepts and terminology in an intuitive way with examples drawn from computer science. The reader is referred to ADJ’s own technical papers for a precise account and many additional examples.

------
† This work was carried out while the author was visiting the Mathematical
Sciences Department of the Thomas J. Watson Research Center in the summer of 1977.

theHigherGeometer@highergeometer· Jun 8

Even just from this snippet preview, I *knew* who the author was, before I even realised the snippet of a url gave a big clue....

[A] Mount Allison University
https://mta.ca » ~cat-dist > catlist 2
https://mta.ca/~cat-dist/catlist/1999/zf-010499

Recall that ZERMELO set theory (1908), which is essentially equivalent to the categorists' notion of ELEMENTARY TOPOS with natural numbers ... /~pt/ ...

theHigherGeometer@highergeometer· Jun 7

I suspect whoever wrote this Wikipedia sentence didn't use the words in the way I think they should mean. What does is mean for a pair of distinct sequences to "be the same algebraically" here? Obviously they are not identical or this transform wouldn't help […] [Original post on mathstodon.xyz]

"Given a sequence X = (x_n) with n = 0,1,2,3,..., Aitken's delta-squared process associates to this sequence the new sequence
[mathematical details]
with [explanatory equations[, Both are the same sequence algebraically but
the latter has improved numerical stability in computational implementation."

the phrase "Both are the same sequence algebraically" is highlighted.

theHigherGeometer@highergeometer· May 22

This paper by my colleague, published last year in Applied Categorical Structures, has had its formatting mangled. Apart from repeatedly switching fonts from serif to sans-serif bold after a diagram in proof environments, the last proposition's statement is […] [Original post on mathstodon.xyz]

Screenshot of a mathematics article with Proposition 7.3 stating

"Let C be a quasi-category. The canonical map sd_2 C to C^Delta[1] C."

Text underneath talks about the canonical map "sd_2 C to C^Delta[1]" (no trailing "C"), indicating some text got mangled and the statement compressed.

theHigherGeometer@highergeometer· May 19

Sometimes I still can't believe I once pulled off this proof, hard analysis not being my forté

Page of mathematics with complex and delicate sequences of inequalities

theHigherGeometer@highergeometer· May 11

Even in 2003's Sets for Mathematics (Lawvere and Rosebrugh) we see the same treatment of the fixed point theorem without cartesian closedness. I don't have access to a copy of the original edition of Conceptual Mathematics from the 90s, to check if its like this in there also, but I suspect so.

Screenshot of Theorem 7.12 in Sets for Mathematics, the fixed point theorem stated just using finite products again.

theHigherGeometer@highergeometer· May 11

PSA Lawvere knew his eponymous fixed-point theorem was better stated *without assuming cartesian closedness*, in contrast to his paper in the 1960s (reprinted in 2006 in TAC). Here's part of pages 304 and 306 in Conceptual Mathematics, the second edition from 2009:

Screenshot of "Diagonal Theorem" noted to be valid "In any category with products", on page 304 of Conceptual Mathematics.

Screenshot of Exercise 1 on page 306 of Conceptual Mathematics, "Cantor's proof, if you read it carefully, really tells us a bit more" and goes on to say how "weak parametrisation" is a sufficient assumption, namely every map is extensionally equal to one in the given family (i.e. takes the same value on global points, not necessarily being literally equal). And further down the page the comment "Our proof of Cantor's Diagonal Theorem is clearly valid in any category with products"

theHigherGeometer@highergeometer· Apr 21

I think the definition here is wrong: https://en.wikipedia.org/wiki/Supermanifold#Concrete:_as_a_smooth_manifold it claims the "one-dimensional space of Grassmann numbers" are essentially given by the exterior algebra on a countably infinite-dimensional vector […] [Original post on mathstodon.xyz]

Screenshot of the section "Concrete: as a smooth manifold" of the Wikipedia page 'Supermanifold', highlighting the claim that the 1-dimensional Grassmann numbers are an infinite-dimensional alternating algebra

Screenshot of the section 'Finite vs. countable sets of generators' of the Wikipedia page 'Grassmann number', highlighting the claim that using a certain topology on an infinite-dimensional alternating algebra "effectively reduces" the dimension to be finite.

theHigherGeometer@highergeometer· Apr 20

Nice to be coming back to this question after a decade, when I claimed something similar was true in much more generality, based only on a feeling, in discussion with Dorette Pronk. Perhaps I was overambitious, but this case seems doable. And will give a […] [Original post on mathstodon.xyz]

Mathematical scribbles on a tablet note-taking app

theHigherGeometer@highergeometer· Apr 10

A work colleague told us in a meeting today about the concept of 'ghost Sets' in SET, which are triples of pairs of cards ((c1,c2),(d1,d2),(e1,e2)) such that the three cards giving extensions of each given pair into a Set (i.e. the third point on the line in […] [Original post on mathstodon.xyz]

A drawing in black pen of a straight line with three circles on it, and from each circle descending two lines: one red, one black, each of which has two crosses on it in the matching colour. The six red and six black crosses represent cards on the table in the game SET, and they both determine the same Set of absent cards, represented by the circles.

theHigherGeometer@highergeometer· Apr 8

Hilbert, by Constance Reid. A great book, showing David Hilbert's passage from a bold young and ambitious mathematician to an old man surrounded by the ruin of the mathematics department in Göttingen in the 1930s. This helped me place a lot of names of […] [Original post on mathstodon.xyz]

Photo of the book 'Hilbert', by Constance Reid'. The yellow cloth-bound hardcover has Hilbert's signature debossed into the boards.

theHigherGeometer@highergeometer· Apr 8

The Legend of Sigurd and Gudrún, by J.R.R. Tolkien (edited by Christopher Tolkien) This is Tolkien's adaptation/telling of the matter of the Volsungs and the aftermath with the Huns, to bridge a most unhappy lacuna in the main source text for the Icelandic […] [Original post on mathstodon.xyz]

Photo of the cover of the book The Legend of Sigurd and Gudrún by Tolkien, with illustration a photo of an old Scandinavian wood carving

theHigherGeometer@highergeometer· Apr 8

The Sage of King Heidrek the Wise, by Christopher Tolkien. This was the published version of Christopher's graduate thesis (he did a B.Litt. degree, roughly equivalent to a research masters in literature) that did a critical translation and edition of the […] [Original post on mathstodon.xyz]

Photo of the cover of The Saga of King Heidrek the Wise, by Christopher Tolkien, in a modern paperback issue

theHigherGeometer@highergeometer· Apr 8

J.R.R. Tolkien's The Silmarillion, by Rhona Beare. Beare was an academic, a lecturer in classics in Newcastle here in Australia, and when younger had a correspondence with Tolkien in the 1950s and 60s. One of her questions led to the correction to the 2nd […] [Original post on mathstodon.xyz]

Photo of the cover of Beare's book "J.R.R. Tolkien's The Silmarillion". Part of the series Babel Handbooks on Fantasy and SF Writers, volume 10, from Nimrod Publications. The cover is yellow and has a black stylized illustration that looks like something from a piece of Classical pottery.

theHigherGeometer@highergeometer· Mar 22

🤔 https://www.reddit.com/r/math/comments/1s0gm23/algebraic_topology_in_the_horror_movie_ring_1998/

Screenshot of reddit a comment and a reply:

"mjec - 7h ago
You're telling me they made a whole movie called Ring and didn't use the distributive property even once?!"

"No-Onion8029 - 7h ago
If someone distributes the movie to you, you die."

theHigherGeometer@highergeometer· Mar 16

Little ol' physics student not-yet-higher geometer making his first foray into constructing interesting examples (well, here it was a counterexample to a naive application of the formalism of Bouwknegt–Evslin–Mathai that applied to principal U(1)-bundles with […] [Original post on mathstodon.xyz]

Screenshot of 'Appendix B - Cohomology groups of T / zeta'

"The following are the integer-valued cohomology groups of the manifolds T / zeta and T-tilde ...."

what follows is the construction of a fibre bundle over a twisted torus with fibre the circle, but structure group Z mod 2 - it is not orientable as a bundle.

theHigherGeometer@highergeometer· Mar 13

For example:

"Verification pays off

- In the course of the verification, I had to prove Expr.hasBoundVar is correct
- This function works by accessing a 64 bit metadata field
- Thad to prove some bit tricks correct, but this worked out alright
- But the metadata calculates the depth of bound variables, which is a priori
unbounded, so this is an overflow situation
- The code was checking for overflow, but it used Lean’s panic function to do
it, and this function doesn’t actually crash the program, because it’s a pure
function with no exit path
- Instead, it returned the default value for the type (0), which is the worst
possible answer in this situation
- By pushing back the counterexample situation to the entry point I was able to
construct a concrete proof of false [#8554]."

Verification pays off

- The lean4lean repository contains bugs-found.md which tracks kernel bugs
caught so far
- divergences.md contains intentional divergences from the Lean kernel:
- Weird behaviors or bugs that haven't been fixed upstream yet
- Necessary limitations of lean4lean, such as native decide
- Some optimizations that the original kernel missed
- Primitives such as Nat .add are checked to ensure that kernel-accelerated
computation on literals is conservative over the naive equational specification

theHigherGeometer@highergeometer· Mar 12

Lawrence Lessig ( @lessig ) 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 […] [Original post on mathstodon.xyz]

Things that should be illegal in South Australia:

1) Calling a Parmi a Parma.
2) Farmer's Union not being the go to iced coffee.
3) Donations to political parties.

And now #3 is a reality.

theHigherGeometer@highergeometer· Mar 9

From bad AI maths papers on the arXiv to terrible AI Tolkien videos on YouTube... The title here is a mashup of three different books: the 1977 published Silmarillion, the 2007 standalone novel The Children of Húrin (massively expanding the story in chapter […] [Original post on mathstodon.xyz]

YouTube thumbnail image generated by AI, with nonsensical mashup of texts as the title, described in the post

theHigherGeometer@highergeometer· Mar 5

From @alexcorner I both love and respect this result ... but so much meme potential. "...and [Gurski]".

A diagram expressing a result in higher category theory, stating that there is a biadjoint biequivalence between ddBicat_s-Cat and BrMonCat. One of the functors, U, is explicit, the theorem is stated that U is a pointwise biequivalence. The corollary is that there exists the functor in the other direction, following "from U being a pointwise biequivalence and [Gurski]".

theHigherGeometer@highergeometer· Mar 5

Here's the abstract from the link

Peter Scholze — Geometric Langlands, after Gaitsgory, Raskin,...
At its core, the Langlands program seeks to give a description of the vector space of automorphic forms. This is a space of functions on a locally symmetric space X/Gamma, where Gamma = G(Z) subset G(R) is an arithmetic group, equipped with its Hecke symmetries. One seeks a “dual” description, in terms of L-parameters, which are roughly representations of the absolute Galois group with values in the Langlands dual group.

Similar conjectures exist in the function-field case; in this setting, the quotient X/Gamma can be interpreted as the F_p-points of the stack Bun_G of G-bundles on a curve C over F_p. Gaitsgory–Raskin, partly in large collaborations, proved a fine version of Langlands’ conjecture in this setting, in the everywhere unramified case. Notably, they define a vector space on the Galois side, as global sections of a sheaf on the moduli stack of L-parameters, and relate this to the space of automorphic forms, compatibly with the Hecke symmetries.

Their proof is a culmination of many decades of work in the geometric Langlands program, and consists of two key steps: 1) The space of automorphic forms is the trace of Frobenius on a category of l-adic sheaves on Bun_G. 2) This category of l-adic sheaves on Bun_G is equivalent to a category of coherent sheaves on the stack of L-parameters. The starting point for 2) is the observation that this conjecture makes sense for curves over any field, and any[...]

theHigherGeometer@highergeometer· Mar 2

@pschwahn sorry, too good not to immortalise

Image of Mathstodon user Paul Schwahn (@pschwahn) looking shocked, with both his hands clasped to his forehead as if in disbelief, with text from a post underneath saying flatly "They made the isospectral drums"

theHigherGeometer@highergeometer· Feb 24

This paper was covered 1] in Scientific American [https://arxiv.org/abs/2602.01820 - a quantitative version of Faltings' Theorem/Mordell Conjecture. Obviously take it with the caution needed for a preprint, but this is huge if true […] [Original post on mathstodon.xyz]

Theorem 1.3 of the linked paper, stating a uniform upper bound for the number of K-points on an algebraic curve of genus at least 2 over a number field L, in terms of the genus and the rank of its Jacobian.

theHigherGeometer@highergeometer· Feb 18

Pigs have flown.... (Paper was on the arXiv in September 2022)

Download citation

Received
07 February 2024

Accepted
19 August 2025

Published
18 February 2026

theHigherGeometer@highergeometer· Feb 15

First time I've seen a mathematics paper acknowledge the ONI. I know in the US you get/got a lot of cool pure maths funded by AFOSR, but here it's basically just the ARC in my experience so far. I cautiously say that I'm glad if there's another potential […] [Original post on mathstodon.xyz]

"1.2. Acknowledgements. This research was funded by the Australian Government through the Office of National Intelligence grant # NI240100145. We thank Tomas Lasic Latimer for fruitful discussions regarding trace identities in finite characteristic that led to the proof of Lemma 3.4. We also like to thank Claude Viallet and John Roberts for interesting discussions related to the topic of the paper."

theHigherGeometer@highergeometer· Feb 10

Fun discussion involving @tao 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