PCA-like method for filtering known variances

Principal Component Analysis is used to reduced the dimensions of atmospheric pressure grids (lat X long X time) into their most important modes of behaviour (e.g, the North Atlantic Oscillation is the PC1 of the North Atlantic pressure field).

I would like to know if there is a way to find the modes/splits of variance within a dataset (such as pressure grids) that are most explained by a seperate dataset. I suppose the analogy would be the cross-correlation compared to the autocorrelation.

Thanks!

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Every non-abelian $p$-group has a non-inner automorphism of order $p$

It is a well-known conjecture in group theory stated that every non-abelian $p$-group has a non-inner automorphism of order $p$.

Last December, someone uploaded a paper Some results of cohomological properties of $p$-group and non-inner automorphism with order $p$ on non-abelian finite $p$-group to arXiv claiming to have proved this conjecture. However, the paper contains numerous English grammatical errors and is poorly readable. Have anyone read this paper, and is its proof correct?

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What is the current status of Lee and Li's $10/8$-conjecture in $4$-manifold topology?

Lee and Li conjectured the following.

$10/8$-conjecture. If $X$ is a closed, oriented, smooth $4$-manifold with even intersection lattice $aU \oplus bE_8$, where $U$ is the hyperbolic lattice, then $a \geq |b|$.

This is equivalent to $b_2(X) \geq \frac{10}{8}|\sigma(X)|$, where $\sigma$ is the signature, and hence the name. (Note that if $X$ is spin, Matsumoto's $11/8$-conjecture proposes an improvement by replacing $10/8$ with $11/8$.)

There is a claimed proof by Kim, but there seem to be some doubts (see Remark 5.4 in Hamilton). However, other people seem to be using this result for their main theorems (see this and this).

I lack the background to actually verify Kim's proof. So, my question is, have the doubts been cleared, or are there any other updates?

(I have a similar question here on the $3/2$-conjecture, in case anyone is able to help with that.)

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Optimising the known bounds for Kalton's constant

Let $\mathcal A$ be an algebra of subsets of a set $\Omega$. For $\Delta>0$, call a function $f\colon \mathcal A\to \mathbb R$ $\Delta$-additive if $f(\varnothing)=0$ and $$ |f(A)+f(B)-f(A\cup B)|\leqslant \Delta $$ whenever $A,B\in\mathcal A$ are disjoint.

Kalton and Roberts proved that there is a universal constant $K$ such that, for every $\Delta$-additive $f\colon\mathcal A\to\mathbb R$, there is a finitely additive signed measure $\mu\colon\mathcal A\to\mathbb R$ satisfying $$ \sup_{A\in\mathcal A}|f(A)-\mu(A)|\leqslant K\Delta. $$ Let $K_{\mathrm{KR}}$ denote the best possible such constant.

The currently known bounds seem to be $ 3\leqslant K_{\mathrm{KR}}\leqslant 38.5. $ The lower bound $3$ is in https://arxiv.org/abs/2003.01193. My question is the following.

Question. What are the best currently known methods, analytic or computational, for improving either side of the estimate $$ 3\leqslant K_{\mathrm{KR}}\leqslant 38.5? $$

More concretely:

1. Is there a finite-dimensional optimisation problem, linear programme, semidefinite programme, or mixed-integer formulation whose value gives a certified lower bound for $K_{\mathrm{KR}}$?

2. Can the Kalton-Roberts proof, or later refinements of it, be sharpened in a systematic way to reduce the upper bound?

3. Are there known extremal or near-extremal examples beyond the Pawlik-type constructions and the examples used to obtain the lower bound $3$?

4. Is there a useful finite-algebra version of the problem? For instance, if $$ K(n) $$ denotes the best constant for set algebras on an $n$-point set, is anything known about the growth, monotonicity, or computability of $K(n)$, and about the relation $$ K_{\mathrm{KR}}=\sup_n K(n)? $$

The motivation is partly that the gap between $3$ and $38.5$ is very large, and the problem appears to have a strongly finite/combinatorial flavour once one restricts to finite set algebras. I would be particularly interested in approaches that turn the problem into a search for extremal approximately additive set functions.

Computational and AI-assisted approaches are very welcome. Recent progress on problems such as the Erdős unit distance problem suggests that even modest automated optimisation or guided search may uncover better finite configurations, which could then be certified rigorously.

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When do (or don't) residue fields generate the derived category of a ring?

Let $R$ be a commutative ring, and $D(R)$ its derived category of unbounded chain complexes. I'm interested in when the residue fields $k(\mathfrak{p}) = \mathrm{Frac}(R/\frak p)$ for $\mathfrak p \in \mathrm{Spec}(R)$ generate (or don't generate!) $D(R)$ as a localizing subcategory. In other words, when do/don't we have $$ D(R) = \mathrm{Loc}(k(\mathfrak p)\mid \mathfrak p \in \mathrm{Spec}(R)) $$ This is equivalent to the claim that $$ \mathrm{Hom}_{D(R)}(k(\mathfrak p),E) = 0 \text{ for all } \mathfrak p \in \mathrm{Spec}(R) \implies E = 0. $$

• If $R$ is noetherian, the residue fields always generate, for example by Neeman's work.
• Neeman has also given an example of a non-noetherian ring $R$ where $\mathrm{Spec}(R)$ is a point, and for which the residue fields don't generate. Thus, there doesn't seem to be a good topological condition on $\mathrm{Spec}(R)$ which implies that the residue fields generate.
• There are other known examples where residue fields do not generate. For example, if $R$ is an absolutely-flat ring that is not semi-artinian, then this fails by Theorem 4.7 of Stevenson's work.

Here is an example where I do not know the answer either way: take $R = k[x_1,x_2,\ldots]$. I suspect that the residue fields do not generate in this case. I even have a candidate that might not be built from residue fields: let $I = (x_1^2,x_2^3,\ldots)$ and consider the injective hull $E(R/I)$. By Example 9.1 here this is an indecomposable injective which is not of the form $E(R/\mathfrak p)$ for any prime ideal $\mathfrak p$. If one could show that any non-zero morphism $f \colon k(\mathfrak p) \to E(R/I)$ must be injective, then there are in fact no non-zero morphisms. But I do not know if this injectivity must hold.

---

The ideal example would be to find a ring $R$ generated by residue fields, whose spectrum does not satisfy a topological property we call weakly noetherian (or to show that finding such an $R$ is not possible). This is described a little more in this answer, but this means that the Hochster dual of $\mathrm{Spec}(R)$ satisfies the $T_D$-separation axiom (each of its points is locally closed). For example, the infinite polynoimal ring considered above is an example of a ring whose spectrum is not weakly noetherian. Every noetherian ring has weakly notherian spectrum, but there are also other examples (for example, $\mathrm{Spec}(R)$ is weakly noetherian if $\mathrm{Spec}(R)$ is Hochster scattered).

---

To summarise:

1. Are there any interesting classes of rings for which we can say generation by residue fields holds, or doesn't hold?
2. Can we say anything in the case that $\mathrm{Spec}(R)$ is not weakly noetherian, for example, for the infinite polynomial ring mentioned in the question.

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What happened to the fourth paper in the series "On the classification of primitive ideals for complex classical Lie algebras" by Garfinkle?

In a series of papers in Compositio Math. entitled On the classification of primitive ideals for complex classical Lie algebras I, II and III, Garfinkle describes an algorithm that allows one to determine the fibers of the Duflo map, and hence the left (or right) Kazhdan-Lusztig cells in the Weyl gorup for a Lie algebra of type $B$ or $C$.

Several places, both in reviews and in the papers themselves, mention is made of a fourth part which will deal with the additional things needed to treat type $D$, but I have not been able to find this paper anywhere (it is certainly not listed on MathSciNet).

What happened to this paper? Was it ever written? And if not, is there some other paper that describes what happens in type $D$?

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Rank of a sequence of covariance matrices

Let $X_i$ ($i=1, \dots$) be an orthonormal basis for $L^2(\Omega, \mathbb P)$. In particular, it holds that $$\mathbb E[X_iX_j] = \delta_{ij}.$$ Now take $Z\in L^2(\Omega, \mathbb P)$ and define $\tilde X_i:=ZX_i$. Let $\Sigma_m$ be the covariance matrix of $\tilde X_1, \dots, \tilde X_m$. Are there conditions on $Z$ ensuring that the rank of $\Sigma_m$ is bounded in $m$?

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What is the current status of the $3/2$-conjecture in $4$-manifold topology?

In Kirby's "Problems in low-dimensional topology" from 1997, Problem 4.93 is the following.

$3/2$-conjecture. An irreducible, simply connected, closed, smooth, spin $4$-manifold $X$ satisfies $$\chi(X) \geq \frac{3}{2}|\sigma(X)|,$$ where $\chi$ is the Euler characteristic and $\sigma$ the signature.

By simply connectedness, $\chi(X) = b_2(X) + 2$, and by spin and Rokhlin's theorem, the intersection lattice of $X$ is $aU \oplus 2kE_8$, where $U$ is the hyperbolic lattice. So, the conjecture is equivalent to $a \geq 4|k| - 1$.

(Caution: The conjecture is also stated on p. 247 of Scorpan's "Wild World of 4-manifolds", but he seems to be implicitly assuming simply connectedness in that section.)

However, in Kirby's new "K3: A New Problem List in Low-Dimensional Topology", this problem does not seem to appear, even though other related problems, such as Matsumoto's $11/8$-conjecture, do.

So my question is, is the $3/2$-conjecture resolved?

(I have a similar question here on Lee and Li's $10/8$-conjecture, in case anyone is able to help with that.)

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Smallest counterexample to Stein's conjecture?

An equi-$n$-square is an $n$ by $n$ array of cells filled with the symbols $1,2,\dots,n$ so that each symbol occurs exactly $n$ times. (Every Latin square of order $n$ is an equi-$n$-square, but the converse does not hold.) For $k \le n$, a partial transversal of size $k$ in an equi-$n$-square is a collection of $k$ cells in which the cells all have different rows, columns, and symbols. Stein conjectured that for each $n$, every equi-$n$-square has a partial transversal of size $n-1$. It is also known that an equi-$n$-square always has a partial transversal of size $2n/3$.

For each positive integer $n$, let $s_n$ denote the largest integer such that every equi-$n$-square has a partial transversal of size $s_n$. Pokrovskiy and Sudakov showed that Stein's conjecture is false, by constructing for large $n$ a family of counterexamples. From this construction we therefore know that $s_n \le n - (\ln n)/42$ for all sufficiently large $n$.

My question is quite concrete:

Is anything known about the smallest $n$ such that $s_n \le n-2$?

References:

• S. K. Stein, Transversals of Latin squares and their generalizations, Pacific J. Math. 59(2) 567–575, 1975.
• Alexey Pokrovskiy and Benny Sudakov, A counterexample to Stein's equi-$n$-square conjecture, Proc. AMS 147(6) 2281–2287, 2019. https://doi.org/10.1090/proc/14220

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Finite element method inverse estimate

$\DeclareMathOperator\diam{diam}$Looking for a proof in the literature of the following lemma:

Let $K\subset\mathbb{R}^d$ be a bounded domain. Let $P_X$ be a finite dimensional subspace of $\mathcal{H}^k(K)$ then

$$\left\|v\right\|_{\mathcal{H}^k(K)} \leq C \diam(K)^{-k} \left\|v\right\|_{L_2(K)}$$

for all $v\in P_X$ and where the constant $C$ does not depend on $\diam(K)$.

There is a proof in The mathematical theory of finite element method by Susanne C. Brenner, L. Ridgway Scott (p111), but they do not check if the constant $C$ is independent of $\diam(K)$.

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What should one do with small numerical improvements to great results?

This is a question from someone who is not a professional mathematician.

I was reading the preprint https://arxiv.org/pdf/2605.20579v1 and found I could improve the exponent in the lower bound slightly from 1.014114 to about 1.032. This is through different types of computational search rather than any particular mathematical insight.

The main improvement comes from taking $|T|$ much larger, since the Golod--Shafarevich constraint gives roughly quadratic capacity in $|T|$, allowing many more useful primes in $S_{\mathbb Q}$ and hence a larger value of $\delta$ in Sawin's exponent formula.

Will Sawin states in his paper that the existing exponent was not carefully optimized.

They options as I see them are:

• Keep it private
• Put it on a blog or personal website (I don't have either)
• Make a MO post about it
• Something else?

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Can an asymptotic formula imply eventual submultiplicativity?

Suppose a function $Q(N)$ satisfies the asymptotic formula [ Q(N)=\exp(\sqrt{N})\left(1+O\left(N^{-1/2}\right)\right) ] as $N\to\infty$.

Is it possible to deduce from this asymptotic formula that [ Q(M)Q(N)\geq Q(M+N) ] for all sufficiently large integers $M$ and $N$?

More generally, are there standard criteria under which an asymptotic formula of the shape [ Q(N)\sim f(N)e^{g(N)} ] implies eventual submultiplicativity?

I am also interested in whether one can expect the stronger inequality [ Q(M)Q(N)\geq Q(M+N) ] to hold for all positive integers $M$ and $N$, or whether asymptotic information alone is usually insufficient for proving such finite inequalities.

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For which n can we solve this problem to find a reverse magic square?

Recently, I saw a question on Puzzling Stack Exchange:

Is it possible to fill the $121$ entries in an $11×11$ square with the values $0,+1,−1$, so that the row sums and column sums are $22$ distinct numbers?

This problem is interesting for me because its definition is the reverse of the magic square. I think to solve this problem we must use something like design theory. Also, for $n=6$ this problem has an answer.

For which integer number $n$ does this problem has a solution? Is this problem well known in math?

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Is $\mathbf{FreeAb}^\mathrm{op}$ accessible?

Consider the category $\mathbf{FreeAb}$ of free abelian groups. It is not $\aleph_1$-accessible (MO/511426), but it cannot be decided if it is accessible (MSE/720885). My question here is concerning the dual category.

Question. Is $\mathbf{FreeAb}$ is coaccessible?

Of course, we should ask first if this is decidable or not, but I think it should be. Actually, I think that the answer is No, and that $\mathbf{FreeAb}$ does not even contain a small limit-dense subcategory. But I don't know how to prove that. Clearly, it has a cogenerator, $\mathbb{Z}$.

What's good to know: the forgetful functor to $\mathbf{Ab}$ preserves all limits; use the generator $\mathbb{Z}$.

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What can mathematicians do to mitigate the deleterious impacts of AI?

The following question is the sort of question that I would normally vote to close, so I will not be offended if it is closed, but I will let the community decide whether to make an exception. (The community wiki button seems to have disappeared, but this question should surely be marked community wiki if it is not closed.)

Except for a few extreme optimists, almost everyone agrees that AI can have some deleterious impacts. Are there concrete steps that the mathematical community can take, corporately, to try to mitigate those deleterious impacts?

There are, of course, many good things that can be done that have nothing to do with mathematics or the mathematical community specifically. But here on MO, I would like the focus to be on mathematics-specific suggestions.

I'm asking this question in part because of a comment by David Roberts:

I'd love to see even more discussion from senior figures who being given free access to the most powerful tools (eg in Tao's upcoming ICM lecture....) that sets out a firm stance around AI companies potentially poisoning the well for people not yet tenured.

I think this is an intriguing suggestion but I think more details need to be spelled out for it to be actionable.

My personal preference is for suggestions that as many mathematicians as possible can rally behind. For example, as we can see right here on MO, there is currently no consensus among mathematicians about just how capable AI models are (or are likely to be in the near future) at doing mathematical research, so a recommended course of action that assumes a particular view about how capable the AI models are is likely to receive support from only a limited segment of the community. But it's my feeling that there are some things that are less controversial and that could receive support from a broader swath of the community.

That said, I don't want to dissuade people from giving answers below that might seem to be "radical." Who knows, maybe a seemingly "radical" proposal will actually receive a lot of support. More important, I think, is that the proposal be actionable, and potentially able to achieve real, positive change.

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Integrality certification for product of two matrices $A B^{-1}$

Let's consider two non-singular integer matrices $A,B \in\mathbb{Z}^{n\times n}$. I want a test to check if $A\times B^{-1}$ is integral (or no denominators). I am referring to Colton Pauderis and Arne Storjohann's "Deterministic unimodularity certification" in van der Hoeven, Joris (ed.) et al., Proceedings of the 37th international symposium on symbolic and algebraic computation, ISSAC 2012, Grenoble, France, July 22–25, 2012. New York, NY: Association for Computing Machinery (ACM) (ISBN 978-1-4503-1269-1). 281-288 (2012), MR3206315, Zbl 1308.65067. The inverse of a matrix $M$ can be written as a $x$-adic expansion: $$ M^{-1}=c_0+c_1x+ c_2x^2+\cdots $$ If the matrix $M$ is unimodular, the $x$-adic expansion is finite and Storjohann provides a fast algorithm to check this. (Here $x\in \mathbb{Z}_{>2}$ is relatively prime to determinant of $M$). My approach was: I computed few $x$-adic terms of $B^{-1}$. Say: $B^{-1}=b_0+b_1x+\cdots$. I can write matrix $A$ as a finite $x$-adic expansion. Let $A=a_0+a_1x+\cdots +a_m+x^m$. Similarly, if $A\times B^{-1}$ is integral, I should be able to write it as a finite $x$-adic expansion. Hence, I checked if $a_0 \times b_i$ (or $a_j\times b_i$) becomes zero at some point. I test for some examples (for matrices over Number fields), but it didn't work. Is there any problem with the theory? or Can someone suggest me a way to certify integrality of the product $A\times B^{-1}$.

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Relation between compact vertical cohomology and local cohomology groups

I'm reading the books by Bott & Tu and Milnor & Stasheef simultaneously. The following is my doubt:

The Thom isomorphism in Bott & Tu is obtained as $H_{cv}^{*+n}(E)\rightarrow H^*(M)$, where $\pi\colon E\to M$ is $n$ plane bundle over the manifold of dimension $m$ manifold $M$ and the isomorphism is given by the integration along fiber map $\pi_*$. By the projection formula, the Thom isomorphism $\mathscr{T}$, inverse to $\pi_*$, is then explicitly given by $\omega\mapsto \pi^*(\omega)\wedge\Phi$, where $\Phi=\mathscr{T}(1)$ is the Thom class of $E$.

In Milnor & Stasheef, the isomorphism is $\mathscr{T}\colon H^*(M)\to H^{*+n}(E,E_0)$, where $E_0$ is the complement of zero section and the map factors through $H^*(E)$ i.e. $H^*(M)\xrightarrow{\pi^*} H^*(E)\xrightarrow{\smile\text{fundamental class}} H^{*+n}(E,E_0)$.

I know that $H_{c}^{*+n}(E)=\varinjlim H_{c}^{*+n}(E, E-K)$ over the directed set of compact subsets of $E$. My question is, how can I relate $H^{*+n}(E,E_0)$ and $H_{cv}^{*+n}(E)$.

Any hint would be helpful. Thanks.

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Iterated free extensions of categories by adjunction residues

Context & Motivation

Let $F : \mathcal{C} \to \mathcal{D}$ be a functor between small categories. Passing to presheaf completions gives an adjoint triple $$\mathrm{Lan}_F \dashv F^* \dashv \mathrm{Ran}F : \hat{\mathcal{C}} \rightleftharpoons \hat{\mathcal{D}}$$ where $F^* = ({-} \circ F)$ is precomposition. The unit of the left adjunction, $$\eta : \mathrm{Id}{\hat{\mathcal{C}}} \Rightarrow F^* \circ \mathrm{Lan}_F,$$ measures the failure of the round-trip to recover the original presheaf. Motivated by Spivak's database-migration framework, we think of the loci where $\eta_X$ fails to be an isomorphism as the residue of the translation. We want to internalize this residue by extending $\mathcal{C}$ to a category $\mathcal{C}'$ in which the failure witnesses become primitive objects.

The Construction

Fix $F_0 : \mathcal{C}_0 \to \mathcal{D}$; the target $\mathcal{D}$ is held fixed throughout.

Step 1 — Residue carrier. For each $v \in \mathcal{C}n$, the unit component $$\eta{y(v)} : y(v) \to F_n^*,\mathrm{Lan}_{F_n}(y(v))$$ is a natural transformation in $\hat{\mathcal{C}}_n$. Since $\hat{\mathcal{C}}_n$ is complete and cocomplete, form:

the kernel pair $K_v = y(v) \times_{F_n^*\mathrm{Lan}{F_n}(y(v))} y(v)$ (pullback of $\eta{y(v)}$ against itself — trivial iff $\eta_{y(v)}$ is already monic), and the image $Q_v = \mathrm{Im}(\eta_{y(v)})$ (via the epi–mono factorization — equal to $F_n^*\mathrm{Lan}{F_n}(y(v))$ iff $\eta{y(v)}$ is already epic). Define $\mathrm{Carr}(\eta_n)$ to be the full subcategory of $\hat{\mathcal{C}}_n$ on ${K_v, Q_v \mid v \in \mathcal{C}n,; \eta{y(v)} \text{ not an iso}}$. These are objects that exist in the presheaf completion but are not representable in $\mathcal{C}_n$; the next step makes them primitive.

Step 2 — Free extension. Define $$\mathcal{C}_{n+1} ;=; \mathcal{C}_n \sqcup^{\mathrm{free}} \mathrm{Carr}(\eta_n)$$ as the finitely-presented category whose generators are $\mathrm{ob}(\mathcal{C}_n) \sqcup \mathrm{ob}(\mathrm{Carr}(\eta_n))$, whose morphisms are generated by those of $\mathcal{C}_n$, those of $\mathrm{Carr}(\eta_n)$, and new anchor arrows $K_v \to v$ and $v \to Q_v$ for each failure locus $v$, quotiented only by the relations already present in $\mathcal{C}_n$ and $\mathrm{Carr}(\eta_n)$ — no new relations are added.

This yields a canonical promotion functor $P_n : \mathcal{C}n \to \mathcal{C}{n+1}$, which is faithful and reflects isomorphisms but is not full. The functor $F_{n+1} : \mathcal{C}_{n+1} \to \hat{\mathcal{D}}$ is defined by the universal property of the free extension: it agrees with $F_n$ on $\mathcal{C}_n$ (via $P_n$) and sends each carrier object $K_v, Q_v$ to its image under the canonical comparison map $\hat{\mathcal{C}}_n \to \hat{\mathcal{D}}$ induced by $F_n$.

The key structural property — the edge-locality invariant — is that the free extension creates no new morphism between two $\mathcal{C}_n$-objects that previously had none. Formally: for all $X, Y \in \mathcal{C}_n$, if $X \not\to Y$ in $\mathcal{C}n$, then $P_n(X) \not\to P_n(Y)$ in $\mathcal{C}{n+1}$. This holds by construction because all anchor arrows are of the form $K_v \to v$ or $v \to Q_v$: they leave from or arrive at a single $\mathcal{C}_n$-vertex and cannot compose into new $\mathcal{C}_n$-to-$\mathcal{C}_n$ paths.

Step 3 — The tower. $$\mathcal{C}_0 \xrightarrow{P_0} \mathcal{C}_1 \xrightarrow{P_1} \mathcal{C}_2 \xrightarrow{P_2} \cdots$$

Questions

Q1 (Prior art). Has this construction — iteratively adjoining the limit/colimit witnesses of an adjunction unit as free generators, then iterating — appeared in categorical logic, type theory, or model theory? It feels analogous to Turing–Feferman reflection progressions (adjoining consistency statements as axioms and iterating through ordinals), but cast over the free-category monad rather than a deduction relation. Is there a known categorical treatment?

Q2 (Stabilization). Under what conditions on $F_0$ does the residue carrier $\mathrm{Carr}(\eta_n)$ become empty at some finite stage — i.e., when does the tower stabilize? A natural conjecture: stabilization at stage 1 is equivalent to $F_0$ being fully faithful. Does stabilization at stage $n$ have a characterization in terms of some "$n$th-order" faithfulness condition?

Q3 (Lan-faithfulness frontier). A necessary condition for $\mathrm{Carr}(\eta_n)$ to be trivial is that the unit $\eta^{(n)}$ be componentwise monic — equivalently, that $\mathrm{Lan}_{F_n}$ be faithful. The boundary here is non-trivial:

$F$ fully faithful $\Rightarrow$ $\eta$ iso (classical); $\eta$ componentwise monic $\Rightarrow$ $F$ faithful (evaluate at coyoneda); $F$ faithful $\not\Rightarrow$ $\eta$ componentwise monic: the bicyclic monoid ($\langle a, b \mid ab = 1\rangle$ as a one-object category) gives a faithful functor whose Lan-unit collapses distinct elements of a torsion left-act. Does faithful + every test presheaf torsion-free imply $\eta_X$ monic?

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Hausdorff dimension of Apollonian circle packing, 1.305686729, 1.305688 or yet something else?

I am interested in the Hausdorff dimension of the Apollonian circle packing. There seem to be two numerical calculations of the value:

1.305686729(10)

from P.B Thomas and D.Dhar, The Hausdorf[sic!] dimension of the Apollonian packing of circles Journal of Physics A: Mathematical and General, Volume 27, Number 7, April 1994

and later from Curt McMullen (he does not cite Thomas and Dhar):

1.305688

McMullen, Curtis T. Hausdorff dimension and conformal dynamics. III. Computation of dimension. Amer. J. Math. 120, no. 4, 691–721. 1998

His algorithm is also explained in Section 5.5 on page 156 in "Indra's Pearls" by Mumford and Series (beware of some typos on that page!).

It is somewhat unsatisfying to have two contradicting numerical approximations. The OEIS gives 1.3056867 and does not cite McMullen, see A052483 (as of today).

With modern computers it might be worthwhile trying to settle this question. Curt McMullen has a C-program on his website that can calculate the Hausdorff dimension, that I am interested in: hdim.tar on http://abel.math.harvard.edu/~ctm/programs/index.html

Running

./hdim -a -e .00005

gives as output

Apollonian gasket


Epsilon   Dimension   Cover  Matrix Steps
5.00e-05 1.305687542911558287346746 76

So I guess after 76 steps (and about a week of computation time) we get 1.30568754291 as an numerical approximation.

• Does this mean that actually the first few digits are 1.305687?
• What are the correct first few decimal digits of this number?
• What are the best proven exact bounds and what are the most promising numerical experiments to get a good approximation?

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Incentivizing mathematics in an era of AI-accelerated proof abundance

Despite Timothy Gowers's relief that this recent AI solution of the Erdos Unit Distance Problem only involves a counterexample and not a proof in the positive (see the companion document in the link above), it seems appropriate for the community to weigh our commitment to the value of mathematics as a source of clarity and understanding, a point of view expressed by the late Bill Thurston in an MO answer here.

Despite the clickbait title, I should say that I'm not intending this post to be polemical, as my interactions with the Lean formalization community have convinced me that human beings have a lot of mathematics to do even with assistance from AI. (Despite my saying this, I am not encouraging AI contributions to Mathlib. In fact, I implore you not to try to PR AI generated code to Mathlib!) I'm writing this question because I believe it is important for the broader mathematical community to think about it.

Although Searle's Chinese Room Experiment can be critiqued in many ways, if we imagine ourselves as the person in the room with the Chinese lookup dictionary in the experiment, it's easy to imagine that a proof generated by a computer with minimal human input would function as the dictionary from the point of view of the user. Per capita, the amount of clarity and understanding has not increased by the generation of the proof by the computer.

Although it is probably overly simplistic, I believe the above paragraph points to the problem we face. Even if AI is able to answer all of our research problems, this does not guarantee an increase in per capita clarity and understanding, or even clarity and understanding within small communities of mathematicians.

The idea of mathematical understanding needs clarification. There is a nice essay, Mathematical Understanding by Jeremy Avigad (to Appear in the Blackwell Companion to the Philosophy of Mathematics, Alexander Paseau, editor.) in which it is posited that the mathematical understanding is completely characterized by abilities. If we aim to explain what we understand about a bit of math, we point to what we can do with it. I think it is reasonable to take the degree of mathematical understanding as a constellation of abilities as a working concept, at least.

My question is the following: Up until now, we have used the ability to be the first to solve an open question as one of the primary measures of mathematical value. This theorem economy has been the basis for the livelihood of mathematicians. What will we replace this by if and when AI can solve mathematical problems as effectively as we can?

(Note: this is already a problem...in that it is not clear to what extent papers today have been aided by ideas generated by LLMs. Comparing a mathematician of today to one of 10 years ago seems difficult.)

Oh, and it certainly won't be offensive to me if this question is closed, as it certainly is a soft question (please mark it community wiki, moderators, if you do not deem to close it!)

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