Introducing MathHelm: A Place to discuss AI-generated papers

13 November, 2026 in advertising, guest blog, math.GM | Tags: , , , , , , , , | by | 2 comments

[This is a guest post by the MathHelm scientific committee, whose membership is listed at the bottom of this post. -T.]

Mathematics depends on our regular conversations with colleagues and with the wider public. They help us share ideas, deepen our collective understanding, see where the field stands, and decide what to explore next. They also keep us connected. As technology changes rapidly, talking through uncertainty and supporting one another matter for our well-being and our shared effort to understand the universe.

On October 6, 2026, OpenAI released 722 mathematical manuscripts generated by an internal AI model, claiming significant results across many research areas. This release will undoubtedly bring significant changes to the mathematical community, even though most of these papers were not verified by us humans and among them, many results did not have a companion lean-verification.

Like many others, I was interested in several of these papers. I also found the whole collection intimidating. Their presentations often differ from the scholarly conventions I knew, and I could not quickly assess most of the arguments on my own. It was natural that I wanted to discuss with colleagues, but often felt hesitant to speak before I felt confident in my judgment. Sometimes I wasn’t even sure whom to discuss with. A few colleagues felt the same way. Together, we created MathHelm: a place for everyone to share comments, ask questions, and think together on massive AI-generated papers, including these 722 papers and any other similar paper.

Each of us only has the expertise to assess a small part of such a broad collection. Useful discussions were already appearing online: explanations of an idea, questions about a proof, or missed connections to earlier or ongoing work and related concerns. People’s comments are scattered and easy to miss. Bringing them together saves time for everyone and gives the community a clearer picture.

MathHelm is open to researchers worldwide, with discussions primarily in English. We invite mathematical researchers to register under their real names and discuss AI-generated papers and related mathematical questions. Comments can address correctness, readability, originality, main results, or key ideas. We hope everyone could be as objective and specific as possible, and include references where appropriate. You can also upload your own PDF notes.

The forum was founded by the scientific committee listed below and operates on a nonprofit basis to serve the mathematical community. It focuses on mathematical content and takes no predetermined position for or against AI. Our policy prohibits any organization from using forum content to train AI models.

Our guiding principle is simple:

Mathematicians have no obligation to check AI-generated papers. But if you happen to read part of one, your observations may be valuable to the whole community.

Even small comments such as a question about the proof method, an explanation of a single step, or a useful reference can help. You do not need a verdict on an entire paper to join the conversation.

Whatever tools we use, mathematics should remain a shared human pursuit: people should be able to understand its ideas, question its claims, and help shape its future. We hope MathHelm can offer a little steadiness in this age of rapid change, and a place to keep learning from one another.

MathHelm Scientific Committee

Xinyi Yuan (Peking University), Chair

Yi Ni (California Institute of Technology)

Xiaoming Sun (Institute of Computing Technology, Chinese Academy of Sciences)

Liang Xiao (Peking University)

Yuan Yuan (Westlake University)

Ruixiang Zhang (University of California, Berkeley)

Fan Zheng (ICMAT–CSIC, Spain)

247A, Notes 4: Weighted estimates and pseudodifferential operators

12 November, 2026 in 247A - Classical Fourier Analysis, math.CA | Tags: , | by | 1 comment

— 1. Weighted inequalities —

In the previous notes we have taken a linear or sublinear operator {T} on a measure space {X} and established {L^p} mapping properties, such as

\displaystyle \int_{Y} |Tf(y)|^p\ d\mu_Y(y) \lesssim \int_{X} |f(x)|^p\ d\mu_X(x).

These estimates reveal information about how the height and width of {Tf} is controlled by the height and width of {f}. However, these estimates do not indicate where the support of {Tf} is located, due to the rearrangement-invariant nature of the {L^p} norms. One way to establish this type of control is to start establishing weighted {L^p} estimates, such as

\displaystyle \int_{Y} |Tf(y)|^p w_2(y)\ d\mu(y) \lesssim \int_{X} |f(x)|^p w_1(x)\ d\mu_X(x). \ \ \ \ \ (1)

 for various pairs of weight functions {w_1: X \rightarrow {\bf R}^+} and {w_2: Y \rightarrow {\bf R}^+}. Roughly speaking, these weighted estimates are a variant of the usual {L^p} estimates which assert that if {f} avoids the regions where {w_1} is large, then {Tf} avoids the regions where {w_2} is large. If {X=Y} and one can establish the above type of estimate for many pairs {w_1, w_2}, with {w_1} “looking similar to” {w_2}, then these estimates begin to confirm an assertion that {T} does not significantly change the support of {f}.

One can view the estimate (1) as an assertion that {w_1} is at least as large as some sort of “nonlinear adjoint” of {T} applied to {w_2}. For instance:

Exercise 1 Let {T=T_K} be an integral operator with bounded non-negative kernel {K:X \times Y \rightarrow {\bf R}^+}, and let {w_1, w_2} be bounded weights. Show that the inequality

\displaystyle \int_{Y} |Tf(y)| w_2(y)\ d\mu_Y(y) \lesssim \int_{X} |f(x)| w_1(x)\ d\mu_X(x)

holds for all simple functions {f} with finite measure support if and only if one has the pointwise estimate {w_1 \geq T^* w_2}.

We know that boundedness properties of an adjoint can be used to establish boundedness properties of the original operator. There is a counterpart for weighted estimates:

Proposition 2 (Equivalence of weight bounds and vector-valued {L^p} estimates) Let {1 \leq p < q < \infty}, let {A > 0}, let {T: L^p(X) \rightarrow L^p(Y)} be an arbitrary operator (possibly nonlinear!), and let {r} be such that {r' = q/p}. Then the following are equivalent:
  • (i) For any sequence {(f_n)_{n \in {\bf Z}}} in {L^p(X)}, we have

    \displaystyle \| (\sum_n |Tf_n|^p)^{1/p} \|_{L^q(Y)} \leq A \| (\sum_n |f_n|^p)^{1/p} \|_{L^q(X)}.

    In particular (setting all but one of the {f_n} equal to zero),

    \displaystyle \| Tf \|_{L^q(Y)} \leq A \| f \|_{L^q(X)}.

  • (ii) For every non-negative {w_2 \in L^r(Y)} there exists a non-negative {w_1 \in L^r(X)} with {\|w_1\|_{L^r(X)} \leq \|w_2\|_{L^r(Y)}} such that

    \displaystyle \int_Y |Tf(y)|^p w_2(y)\ d\mu_Y(y) \leq A^p \int_X |f(x)|^p w_1(x)\ d\mu_X(x) \ \ \ \ \ (2)

     for all {f \in L^p(X)}.

Proof: By dividing {T} by {A} we may normalise {A=1}. We first show that (ii) implies (i). From the converse Hölder inequality we have

\displaystyle \| (\sum_n |Tf_n|^p)^{1/p} \|_{L^q(Y)}^p := \sup \{ \int_Y \sum_n |Tf_n(y)|^p w_2(y)\ d\mu_Y(y):

\displaystyle \| w_2 \|_{L^r(Y)} = 1, w_2 \geq 0 \}.

By monotone convergence we may interchange the sum and integral. From (2) and Hölder inequality we have

\displaystyle \sum_n \int_Y |Tf_n(y)|^p w_2(y)\ d\mu_Y(y) \leq \sum_n \int_Y |f_n(y)|^p w_1(y)\ d\mu_Y(y)

\displaystyle \leq \| (\sum_n |f_n|^p)^{1/p} \|_{L^q(X)}^p \|w_1\|_{L^r(X)}

where {\|w_1\|_{L^r(X)} \leq \|w_2\|_{L^r(Y)} = 1}. The claim follows.

Now we establish that (i) implies (ii). Fix {w_2}; we may normalise {\|w_2\|_{L^r(Y)} = 1}. We need to find a {w_1} obeying (2). From the above arguments we know that a necessary condition for {w_1} is that

\displaystyle \sum_n \int_Y |Tf_n(y)|^p w_2(y)\ d\mu_Y(y) \leq \int_Y |F(y)| w_1(y)\ d\mu_Y(y)

for all {f_n} and {F} such that {\sum_n |f_n|^p \leq |F|}. Motivated by this, given any {F \in L^{r'}(Y)}, we consider the “capacity”

\displaystyle \lambda(F) := \sup \{ \sum_n \int_Y |Tf_n(y)|^p w_2(y)\ d\mu_Y(y): \sum_n |f_n|^p \leq |F| \},

where {(f_n)_{n \in {\bf Z}}} ranges over all sequences {L^p(X)} with {\sum_n |f_n|^p} bounded pointwise by {F}. It will suffice to locate {w_1 \in L^r(X)} with {\|w_1\|_{L^r(X)} \leq 1} such that

\displaystyle \lambda(F) \leq \int_X |F| w_1\ d\mu_X

for all {F \in L^{r'}(X)}, since we may then substitute {F = |f|^p} and observe that {\lambda(|f|^p) \geq \int_Y |f|^p w_2\ d\mu_Y}.

Observe from Hölder that

\displaystyle \begin{array}{rl} \sum_n \int_Y |Tf_n(y)|^p w_2(y)\ d\mu_Y(y) &\leq \| (\sum_n |Tf_n|^p)^{1/p} \|_{L^q(Y)}^p \|w_2\|_{L^r(Y)} \\ &\leq \| (\sum_n |f_n|^p)^{1/p} \|_{L^q(X)}^p \\ &\leq \| F \|_{L^{r'}(X)} \end{array}

and so we have the boundedness

\displaystyle \lambda(F) \leq \| F \|_{L^{r'}(X)}.

We also easily verify the superlinearity properties

\displaystyle \lambda(cF) = |c| \lambda(F);\quad \lambda(F+G) \geq \lambda(F)+\lambda(G)

for all non-negative {F,G \in L^{r'}(X)}. Observe that the sets

\displaystyle A := \{ F \in L^{r'}(X): \| F\|_{L^{r'}(X)} < 1 \}

and

\displaystyle B := \{ F \in L^{r'}(X): F > 0; \lambda(F) > 1 \}

are algebraically open (see appendix), convex, and disjoint. Thus by the geometric Hahn-Banach theorem (see appendix), we can find a linear functional {\nu: L^{r'}(X) \rightarrow {\bf C}} such that {\nu < 1} on {A} and {\nu > 1} on {B}. (This is a typical use of the Hahn-Banach theorem: philosophically, this theorem asserts that in linear or convex programming, any possibility which is not obviously excluded by the constraints is in fact permissible. Here, it is the vector-valued inequality in (i) which shows that all the “obvious” obstructions to the estimate (ii) do not exist, and Hahn-Banach tells one that there are no non-obvious obstructions.) The former implies that {\nu} is a continuous linear functional with norm at most {1} and thus {\nu(F) = \int_X F w_1\ d\mu_X} for some {w_1 \in L^r(X)} with {\|w_1\|_{L^r(X)} \leq 1}. The latter implies that {\nu(F) \geq \lambda(F)} for all non-negative {F}, and in particular (setting {F := |f|^p}) that

\displaystyle \int_Y |Tf|^p w_2\ d\mu_Y \leq \lambda(|f|^p) \leq \nu(|f|^p) = \int_X |f|^p w_1\ d\mu_X

as desired. \Box

Remark 3 By combining this with Exercise 51 from last week’s notes, we obtain a remarkable fact: if {T: L^2(X) \rightarrow L^2(Y)} is a linear operator, then there is a unique continuous linear extension from {L^q(X)} to {L^q(Y)} for {q > 2} if and only if for every {w_2 \in L^r(Y)} there exists {w_1 \in L^r(X)} with {\|w_1\|_{L^r(X)} \lesssim_T \| w_2\|_{L^r(Y)}} for which one has the weighted estimate (1), where {r := (q/2)'}. Thus in the case of linear operators there is an equivalence between higher unweighted {L^p} estimates and weighted {L^2} estimates.

We thus see that weighted estimates can be quite powerful; they not only imply unweighted estimates, but also vector-valued versions of those estimates. Indeed, more is true: if we have a sequence of operators {T_n} which obey weighted estimates with the same set of weights, then we automatically obtain vector-valued estimates which involve all the {T_n} at once. For instance, if {p,q,r} are as in the above proposition, and for each {w_2 \in L^r(Y)} there was a {w_1 \in L^r(X)} (independent of {n}) with {\|w_1\|_{L^r(X)} \lesssim \|w_2\|_{L^r(Y)}} such that

\displaystyle \int_Y |T_nf(y)|^p w_2(y)\ d\mu_Y(y) \leq \int_X |f(x)|^p w_1(x)\ d\mu_X(x)

for all {n}, then we can conclude the vector-valued estimate

\displaystyle \| (\sum_n |T_n f_n|^p)^{1/p} \|_{L^q(Y)} \leq A \| (\sum_n |f_n|^p)^{1/p} \|_{L^q(X)}.

It is thus of interest to obtain weighted estimates, preferably those in which the final weight {w_2} is given explicitly in terms of {w_1}. A particularly pleasant situation occurs when {X=Y} and the weights {w_1, w_2} are in fact equal, although having estimates with asymmetric weights can still be very useful.

One particularly common (and useful) class of weights are power weights such as {w(x) = \langle x \rangle^\alpha} or {w(x) = |x|^\alpha} for Euclidean spaces {{\bf R}^d}, especially in PDE applications where such weights tend to appear quite naturally (for instance, via the introduction of Sobolev spaces). However, we will work here with more abstract weights. Here it turns out that one can often mimic the proof of an unweighted estimate to deduce the weighted counterpart.

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Lean Grimoire

11 November, 2026 in diversions, math.LO | Tags: , | by | 31 comments

Back in July, I started experimenting with using modern coding agents to build a number of mathematical visualization applets. With the extremely hectic events of the last few months, I had to set those projects aside; but I recently found a bit of time to work on one such project that I had started back then, which was to modernize a Javascript interactive textbook I had made back in 2018, entitled “QED”, and announced previously on this blog. The purpose of that textbook was to teach propositional and first-order logic in the form of a “crafting game”, where one applied various rules of logic to reach a desired conclusion from a given set of hypotheses.

With a coding agent, I rebuilt the game (which I have given the working title of “Lean Grimoire”) with a number of new features, for instance giving the option to show an in-progress Lean proof of one’s game state. Just for fun, I also added an option to play in a “fantasy mode” in which various game elements are depicted using fantasy-themed flavor text (for instance, mathematical statements become “potions”, mathematical expressions become “scrolls”, and so forth). Here is a typical screenshot, in which one has nearly completed the proof of the assertion A \implies A \wedge B given A \implies B:

I would be interested in having some playtesting of this app to make it more useful as a means to teach logical proofs and/or Lean.

“Six Math Essentials” now available

10 November, 2026 in advertising, book, math.GM | Tags: | by | 3 comments

My popular mathematics book, “Six Math Essentials” (Six Maths Essentials in the UK), which was previously announced on this blog in February, is now available (in print, digital, and audiobook formats).

Some related links:

Our Survey of Early Career Mathematicians

9 November, 2026 in guest blog | Tags: , , | by | 5 comments

[This is a guest post by Anna Brandenberger, Hanna Mularczyk, and Dora Woodruff (MIT). This blog post was initially written in a different file format and converted using AI. — T.]

In late September and early October, we distributed a survey of early career mathematicians. Our goals were to take a “temperature check” of the community, and to give early career mathematicians a centralized, targeted platform to share their views on AI (the survey was not a rigorous data analysis effort). The personal stakes are highest for grad students and postdocs, who face job insecurity and funding instability, so it’s important that our voices are taken into account. We worry that our concerns will be deprioritized, or even misrepresented, without collective resources like this survey.

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What should we teach our (undergraduate) students?

8 November, 2026 in guest blog, opinion | Tags: , , | by | 12 comments

[This is a guest post by Jana Gevertz and Steffen Marcus. This blog post was initially written in a different file format and converted using AI. — T.]

In the last few months AI has hit our discipline with a frightening force. Many of us, understandably, have turned our attention to our students. They are absorbing this change while still working out what kind of mathematicians, and what kind of professionals, they hope to become. On this blog, Álvaro Lozano-Robledo recently asked “what should we tell our students?” (What should we tell our students? | What’s new). In this post, we will focus on the natural follow-up question: “what should we teach them?”

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More posts

6 November, 2026 in opinion | by | 11 comments

As in previous posts, readers are invited to post links to additional articles in the comments.

Futures of Maths Launch

5 November, 2026 in advertising, guest blog | Tags: , , , | by | 7 comments

[This is a guest post by Francesco Fournier-Facio, Yara Kyrychenko, Jon Roozenbeek, and Julian Wykowski, cross-posted from the Proofs and Prompts blog. This blog post was initially written in a different file format and converted using AI. — T.]

Today we launch Futures of Maths, a new, living platform that asks active research mathematicians about the most pressing questions around maths and AI. A string of solutions to famous maths problems, found with the help of AI, has provoked a great deal of discussion among research mathematicians. We built Futures of Maths to understand how mathematicians are dealing with this change in their research and teaching, and where they see the field heading.

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More of my interviews and talks

4 November, 2026 in media | Tags: | by | Leave a comment

I am continuing in this current era to open up my blog to a broader range of voices on math and AI, with the majority of posts now being guest posts by other authors. My own views on these topics (which often overlap with, but are not identical to, those of the guest authors) are summarized on this page. I have also given a number of recent interviews and talks on these topics; I am listing some of these here for reference.

Disclosure: while many of these articles were solicited due to current events, some are also related to a long-planned launch of a book I wrote, to be released Nov 10.

Doctoral Advising Guidelines by the AHM

3 November, 2026 in guest blog, opinion | Tags: , | by | 34 comments

[This is a guest post by the AHM communications group. — T.]

We are happy to announce that the Teaching Working Group of the Association for Human Mathematics wrote a set of recommendations on doctoral advising aimed at departments, advisors, and students. The document contains five guidelines centered on prioritizing human-authored writing, avoiding overreliance on AI, confronting the ethical concerns of AI use, and improving the wellbeing of students, as well as concrete proposals on how to implement them. We encourage everyone to read and share the document, and we especially look forward to feedback and suggestions.

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