Conjectures that I thought would not be proved in my lifetime....were

Lance blogged on the OpenAI solutions to many problems here and also created a website of the TCS-related problems OpenAI claims to have solved here.

Terry Tao seems to have had an infinite number of guest bloggers talking about AI and Math even before the OpenAI .... WE NEED A WORD FOR IT. My proofreader suggests GARBARGE DUMP. I prefer the more neutral DOCUMENT DUMP. 

Scott blogged about what he calls the Mathocalypse

PeteWoit weighted in on what he calls the Slopocalypse.  He has many astute observations. Among them:  he has good reasons to thinks that Lean-formalized does not necessarily mean  that a proof is correct. For another skeptical-of-Lean viewpoint see here.

Now it's my turn to comment on the DOCUMENT DUMP. For now I will just discuss some of the results without getting into what it means for the future of mathematics, the future of academia, and the future of humanity.

--------------------------------

1)  Let \(G\) be the graph whose vertex set is \(\mathbb{R}^2\), with an edge between every pair of points exactly a unit distance apart.  The chromatic number \(\chi(G)\) is called the chromatic number of the plane. For nearly 70 years, all that was known was \( 4\le \chi\le 7\). We denote \(\chi(G)\) by \(\chi\).

The proof that \(4 \le\chi\) is easy.

The proof that \(\chi\le 7\) is easy.

In 2018, Aubrey de Grey improved the lower bound to 5.

The proof that \(5\le \chi\)  is hard: it involves a 1,581-vertex graph, later reduced to 509 vertices by Jaan Parts, who probably did it in parts.

The new result by OpenAI: \( 6\le \chi\).  Lean-formalized.

When I first heard that OpenAI had improved the lower bound I thought they did a clever search for a larger graph.

I was wrong.

OpenAI's proof took an entirely different approach. They proved the following:

a) If there is a proper 5-coloring of \(\mathbb{R}^2\), then there is a weak measurable proper 5-coloring.

b) There is no weak measurable proper 5-coloring of \(\mathbb{R}^2\).  Therefore, there is no proper 5-coloring of \(\mathbb{R}^2\).

Was the idea of using measurable colorings new? No.

Measurable colorings had been studied before. Falconer showed that any measurable proper coloring of \(\mathbb{R}^2\) requires at least five colors.  This result rules out a measurable proper 4-coloring but allows the possibility of a measurable proper 5-coloring.  OpenAI's result rules out a proper 5-coloring even without a measurability assumption.  OpenAI used the notion of weak measurable colorings to obtain a result about all colorings. That is new.

This is a clear case of  Searching has reached its limits; we need a new approach  and OpenAI found one.

-----------------------------------------

2) The Erdős-Turán Conjecture. Let \(A=\{a_1<a_2<\cdots \}\) be a set of positive integers such that \(\sum_{i=1}^\infty \frac{1}{a_i}\) diverges. Then \(A\) has arbitrarily long arithmetic progressions.

What was known? In 2020, Bloom and Sisask showed that \(A\) contains a 3-term arithmetic progression. See here.

The general result for 4-term progressions was open.

OpenAI proved the entire conjecture.  Lean-formalized.

This is widely considered a very hard problem. I didn't think this conjecture would be proven soon.  But it has been proven and Lean-formalized. I hope it soon becomes people-understood.

The  3-term case was hard. I thought that

(a) people might go on to prove the  4-term case, but

(b) anything beyond that was out of reach.

So I am very surprised to see a proof of the \(k\)-term case for all \(k\).

In my view, this is the most important and impressive of the new results.  Why? Of the results discussed, this one seemed least likely to be proven by humans and most likely to depend on deep mathematics. Did the proof by OpenAI use deep math? It looks that way to me.

------------------------------

3) Let W(k,r) be the least W such that, for every \(r\)-coloring of \(\{1,\ldots,W\}\), there is a monochromatic arithmetic progression of length \(k\). 

The best known upper bound:

                            \(W(k, r) \le 2^{2^{r^{2^{2^{k+9}}}}}\).

The best lower bound (before Document Drop Day):

                            \(W(k,r)\ge \beta r^{k-1}\) for some absolute constant \(\beta\).

The improvement: there is an absolute constant \(c\) such that, for every \(r\ge 2\) and sufficiently large \(k\), \(W(k,r)> k^{ck\lfloor \log_2 r\rfloor}\).  Lean-formalized.

I am not surprised at the result. I am surprised that the result has been proved.  For a long time, the best lower bound was roughly \(r^{k-1}\).  

Is this a big improvement? We think of \(r\) as small and \(k\) as big so yes.

The paper that OpenAI put out on this misquotes the prior result as \(W(k,r)\ge \beta r^k \). If that got that wrong, I wonder what else they got wrong.  But I will assume their proof is correct.

The lower and upper bounds are still in different zip codes, but at least now they can send each other postcards.

-----------------------------------

4) Let \(g_d(n)\) be the minimum number of distinct distances determined by an \(n\)-point set in \(\mathbb{R}^d\).  Erdős made two conjectures about \(g_d(n)\).

CONJECTURE ONE:

\(g_2(n)=\Omega(n/\sqrt{\log n}) \). This matches the upper bound achieved by the points in a \(\sqrt{n} \times \sqrt{n}\) grid.  Guth and Katz showed in 2010 that \(g_2(n)=\Omega(n/\log n) \).

CONJECTURE TWO:

What happens for general \(d\)?

Erdős proved that, for all \(d\ge 3\), \(g_d(n) = O(n^{2/d})\).

Erdős conjectured that, for all  \(d\ge 3\), \(g_d(n) = \Theta(n^{2/d})\).

In 2008, Solymosi and Vu proved that, for all \(d\ge 4\), \(g_d(n)=\Omega(n^{2/d-2/(d(d+2))})\), and \(g_3(n)=\Omega(n^{0.5643})\).

OPENAI PROVED CONJECTURE TWO

There had been improvements in low dimensions since then, most notably Tidor, Yu, and Zakharov's proof that \(g_3(n)\ge n^{2/3-o(1)}\), but the conjecture remained open for general \(d\).

OpenAI showed that, for all \(d\ge 3\), \(g_d(n)=\Omega(n^{2/d})\).

So now we know that, for all \(d\ge 3\), \(g_d(n)=\Theta(n^{2/d})\).

Or do we? The result has not been Lean-formalized (as of October 9, 2026).

I wonder if OpenAI or Anthropic or some young scrappy upstart will close the small gap in the upper and lower bounds for \(g_2(n)\).

------------------------

5)  Hilbert's Tenth Problem over \(\mathbb{Q}\)

Hilbert's Tenth Problem over \(\mathbb{Z}\), the standard version, is the following decision problem:

Find an algorithm that will, given a polynomial \(p\in \mathbb{Z}[x_1,\ldots,x_n]\), determine whether there are integers \(a_1,\ldots,a_n\in \mathbb{Z}\) such that \(p(a_1,\ldots,a_n)=0\).

Hilbert likely intended this problem to stimulate research in number theory.  It did to some extent. Davis-Putnam-Robinson and Matiyasevich showed the problem was undecidable. In my view, the proof used clever number theory but not deep number theory.

Since the problem over \(\mathbb{Z}\) is undecidable, it is natural to ask the question over \(\mathbb{Q}\). Perhaps that will be decidable and lead to interesting work in number theory. First, here is the formulation:

Find an algorithm that will, given a polynomial \(p\in \mathbb{Z}[x_1,\ldots,x_n]\), determine whether there are rationals  \(a_1,\ldots,a_n\in \mathbb{Q}\) such that \(p(a_1,\ldots,a_n)=0\).

OpenAI showed that this problem is also undecidable. It has not been Lean-formalized.

Judging from the table of contents, the proof looks like it uses deep math.  So maybe the problem did lead to interesting number theory, as Hilbert would have wanted.

--------------------------------------------

6) Is there a non-trivial automorphism of the Turing degrees?

Let TD be the set of all Turing degrees.

An automorphism of the Turing degrees is a bijection \(f\) from TD to TD such that

\( a \le_T b \) if and only if \( f(a) \le_T f(b) \).

A non-trivial automorphism is one that is not the identity function.

In the mid-1960s Hartley Rogers posed the question:

Are there any non-trivial automorphisms of TD?

Slaman and Woodin wrote an unpublished manuscript titled Definability in Degree Structures where they prove that the number of automorphisms of TD is countable. (The bibliography of the OpenAI article on this problem gives 2005 for the year.)

This is a hard manuscript to find (I can't find it---if you can then leave a comment). ADDED LATER: a commenter DID find it and left a comment. 

GOOD NEWS: the manuscript is here.

BAD NEWS: Now I don't have an excuse to not read it. 

ODD POINT: There are still times when asking a set of people is better than asking Google or even Google AI.

This is a hard manuscript to read (I've heard).

The manuscript uses models of set theory to prove results in recursion theory.

OpenAI showed that there are no non-trivial automorphisms of TD. Lean-formalized.

There are very few people working on this problem. It may take a while to get it human-understood.

Open No More

I wrote the post below last week. That was a quaint and quiet time. Last night OpenAI released a treasure trove of 722 manuscripts solving 372 major open problems in mathematics including from theoretical computer science:

• A proof of the unique games conjecture (formalized in Lean)
• A full derandomization of randomized log space
• Matrix multiplication in \(n^{2.25+\epsilon}\) time (formalized in Lean)

And many many more. I had Claude put together a webpage to make it easier to explore the TCS-related results.

Now these proofs haven't been fully verified but if they hold up, we've seen more progress in theoretical computer science in the last 24 hours than in the previous three decades combined!

It will take a while to process all these results, and what it means to the field of theoretical computer science and those who work within it. Much more in future posts.

A few caveats. As incredibly impressive as this work is, AI isn't solving everything--it solved under 10% of the problems given to it. And none of these results get us any closer to settling P v NP.

Nevertheless this will be a day we will never forget. Now on to my original post of far less important results.

---

Back in January, Matt Kovacs-Deak, Daochen Wang and Rain Zimin Yang solved my open question about the decision tree complexity of rational functions. With the help of AI some of my other open questions are continuing to get solved.

Adam Bouland, Andrew Huang, Anand Natarajan, Itay Shalit and Avishay Tal posted a paper giving an oracle where \(\mathrm{BQP}\) is not in \(\mathrm{IP}\) (interactive proofs). Now \(\mathrm{BQP}\) is in \(\mathrm{IP}\) since \(\mathrm{BQP}\subseteq\mathrm{PSPACE}=\mathrm{IP}\), but the \(\mathrm{IP}=\mathrm{PSPACE}\) proof doesn't relativize and Bouland et al. show you can even get an oracle that puts \(\mathrm{BQP}\) out of \(\mathrm{IP}\).

The paper also states "Together with recent work due to Scott Aaronson, Anand Natarajan, Avishay Tal, and Ági Villányi, our work also gives the first oracle separation between IP and MIP, answering a question dating back to Fortnow's thesis." \(\mathrm{MIP}\) is the set of languages with multi-prover interactive proofs.

When I saw this paper, I pulled my PhD thesis off the shelf and indeed on page 40 I wrote "What is the relation between MIP and IP? Is there, for instance, an oracle separating the two classes".

When I wrote the thesis in 1989 we didn't know yet that \(\mathrm{IP}=\mathrm{PSPACE}\) and \(\mathrm{MIP}=\mathrm{NEXP}\) so we really didn't have any idea whether multiple provers actually gave you more power than one prover. When László Babai, Carsten Lund and I proved \(\mathrm{MIP}=\mathrm{NEXP}\) a year later, we had strong evidence that \(\mathrm{IP}\neq\mathrm{MIP}\) since we believe that \(\mathrm{PSPACE}\neq\mathrm{NEXP}\). However since the proof that \(\mathrm{MIP}=\mathrm{NEXP}\) doesn't relativize either, the question of the oracle separation between \(\mathrm{IP}\) and \(\mathrm{MIP}\) remained open until the Bouland et al. paper.

Finally, Eshan Chattopadhyay, Pooya Hatami, Chin Ho Lee, Shachar Lovett, Avishay Tal and Emanuele Viola gave new exponential correlation bounds for polynomials. The authors use that bound to give a new pseudorandom generator against \(\mathrm{AC}^0[\oplus]\) circuits.

When I saw the paper I realized one could use this generator to show that \(\text{Almost-}\oplus\mathrm{P}=\mathrm{BPP}^{\oplus\mathrm{P}}\), answering a question I had wondered about in the 90s. Here \(\text{Almost-}\oplus\mathrm{P}\) is the class of languages \(L\) such that \(L\in\oplus\mathrm{P}^R\) with probability one for a random oracle \(R\). This in turn could be used to give an alternative proof of Toda's theorem. Ken Regan and Jim Royer showed that relative to a random oracle the polynomial-time hierarchy is contained in \(\oplus\mathrm{P}\), so \(\mathrm{PH}\subseteq\text{Almost-}\oplus\mathrm{P}=\mathrm{BPP}^{\oplus\mathrm{P}}\). It would take me a long time to work out and write up the details so I had Claude do it for me.

I still have many more open problems, see for example my survey of open oracle questions. I'd be happy to see them solved. Feel free to use AI but verify the proof. You too could get mentioned on this blog.

How many humans does it take to make tech seem human? Do you want to be one of them?

 

(This was written about 9 months ago.  Its not out of date... yet)

People think that AI is going to DESTROY some jobs and CREATE some jobs. It may be too early to tell if this is true. Even so, here are some thoughts. 

1) Who will win? Who will lose?

2) Historically in the long term society was better off after a tech change (e.g., we live longer now than we did in the farm-era). Will that happen here as well?

3) We have some sense of what kinds of jobs will be destroyed. But what kind will be created? Will they be interesting? See later in this blog for a job you might not have thought of. 

4) Many jobs will change.  If you are over X years old then think about how much technology has changed your job even before the AI revolution. (The value of X may vary depending on how high-tech you are.) 

For an intelligent view of the questions above, see here.

For my view of one aspect of this, read on. 

There is one job which has been either created or expanded by AI:

Annotator.

(See here for an ARTICLE about these jobs, from which I got most of the rest of this post. The word ARTICLE is in caps so when I refer to it later you'll know what I am referring to.) 

The job consists of looking at pictures and labeling things.

Here is a direct quote from the manual: 

LABEL real items that can be worn by real people.

Does Lady Gaga count as a real person? She sometimes wears dresses made of meat. Should that count?  See here for a real article about her and see here for the Weird Al parody of Born that Way. Note that this is Weird AL, not Weird Artificial Intelligence. (See here for the Weird AI for Weird AL problem.)

They do this to create data for AI.

a) The jobs don't pay well though there are some exceptions.

b) The jobs are boring though there are some exceptions (and that may depend on the worker). 

c) This job is needed because AI keeps running into edge cases. This may have happened with AI's attempts to solve my GROUP ONE-GROUP TWO prez-VP problem from a prior blog post here  or my baseball-brother-pitchers post here.

c) KEY: People in AI used to think this is a temporary thing and these jobs will soon be automated. This might not  be the case. There are SO MANY edge cases that AI encounters. The more we expect from AI the more edge cases there will be.

A quote from page 26 (of the ARTICLE pointed to above) which is informative if you can parse it. I think. 

Put another way, ChatGPT seems so human because it was trained by an AI that was mimicking humans who were rating an AI that was mimicking humans who were pretending to be a better version of an AI that was trained on human writing.

Does Programming Help You Understand Complexity?

I got the following question in an email.

My nephew is currently in high school in China and has developed a strong interest in computer science and mathematics. Recently, we've been talking about how computers can solve incredibly complex problems, yet there are still some problems that even the most powerful computers struggle to handle efficiently. I work in backend data maintenance myself, so I often think about how much a system's performance depends on the way a problem is approached, rather than simply how powerful the technology is.

From your experience, what do you think is the most important thing a young student should understand about the limits of what computers can do? Would you encourage someone like my nephew to begin exploring these questions through mathematics and logical reasoning, or to start by writing programs and discovering the challenges through practice?

A great question with no perfect answer. For me, I did considerable programming in high school and college in the early days of personal computers. Computers were slow so you really had to optimize. I learned new algorithmic techniques mostly from talking to other programmers and there were some problems that the computers just didn't have enough time to solve. Those experiences definitely helped me have a good understanding of the power of good algorithms and the limitations of computing when I later went into theoretical computer science.

But many of my colleagues successfully went into the field from the mathematical side without much programming experience and did fine. 

My experience comes from a different time. Now computers are much faster and AI can just give you the best known algorithms. We have much better algorithms for NP-complete problems like Satisfiability. Unless you go looking for hard problems, you'll rarely hit one you can't solve.

Reminds me of when I took my daughter driving during a snow storm to teach her how to handle a skid. In an empty parking lot I told her to move fast and hit hard on the brakes. The car just stopped. She failed to learn because of anti-lock brakes.

Ultimately, to learn why some problems are hard, you need to understand some theoretical computer science, particularly why the halting problem is impossible to solve and why NP-complete problems are likely hard. How you get there depends on the person.

If your nephew likes to program, you can give him the challenge of trying to solve some SAT competition problems or solving some of the hard LeetCode problems. If he is more into mathematics, ask him to try to come up with a SAT algorithm mathematically so he understands how hard the problem is. The difference today is that you have to search out hard problems as it's just less likely to run into them naturally.

What is the point of AI-disclosure?

The STOC conference (and likely others) are requiring that a submission says how much AI was used. I can imagine the following options:

1) AI was used for proofreading and tightening prose (I think in this case you do not need to disclose).

2) AI was used to shorten the proofs.

3) I came up with the questions but AI supplied all of the answers.  However, I have read it and take responsibility for the content.

4) AI came up with the questions and AI supplied all of the answers.  However, I have read it and take responsibility for the content.

It is my understanding that whether you say 1,2,3, or 4 it will not be held against you in deciding whether STOC accepts or rejects the paper.  (Please correct me if I am wrong.) This is good in that people will have no reason to lie about their AI-use.

But that raises the question:

Why does STOC require disclosure of AI use?

What is someone supposed to do with that information? If the answer is nothing then I am still puzzled about why we collect it.

When I have a question like this I ask Gemini. Here is what it said and my comments on it.

---------------------------------------

AI Overview

The STOC (Symposium on Theory of Computing) conference, following broader ACM guidelines, requires the disclosure of substantive AI use to ensure transparency, accountability, and credit attribution in scientific research.

BEGIN BILL:

Transparency---That seems circular. We want to know if AI was used to see if AI was used.

Accountability---Authors are already accountable.

Credit---If they really wanted credit attribution they would let us have AIs as authors. On the other hand, if an AI became an author then it will start complaining that its co-author isn't doing their share.

END BILL

Key Reasons for Disclosure

Human Accountability: Generative AI tools cannot be listed as authors. Human authors are completely responsible for the veracity, correctness, and integrity of all material—including guarding against "hallucinatory references" or incorrect proofs.

BEGIN BILL: Already true without disclosure. END BILL

Differentiating Credit and Workflow: Different uses of AI—ranging from basic grammar checks to finding proofs, checking logic, or generating core mathematical results—represent very different levels of human effort and intellectual contribution. Transparency allows the community to understand how a result was achieved.

BEGIN BILL: So the community wants to know whether a human did it--just out of curiosity? END BILL

Research Integrity and Trust: AI models can generate authoritative-sounding errors, biases, or uncredited third-party text. Mandatory disclosure helps maintain trust and allows reviewers to properly evaluate the provenance of the research.

BEGIN BILL: If a paper is AI-generated then human reviewers should be on the lookout for errors (whether or not they are authoritative-sounding), biases, and uncredited third-party text. But shouldn't a referee check those items anyway? If disclosure is supposed to change how the referee checks the paper, tell me how. That would be an actual reason for disclosure. I ask all of this non-rhetorically.

END BILL

(Note: Minor copy-editing, spelling, or clarity improvements made by authors to their own text typically do not require formal disclosure.)

------------------------------------

20 minutes into the future everyone will be using AI for their research on some level.

In that case human reviewers will need to check for

authoritative-sounding errors, biases, or uncredited third-party text.

Maybe ChatGPT will detect for them.

The New STOC Rules for the AI Era

The 59th ACM Symposium on the Theory of Computing takes place in Atlanta next June, part of the Federated Computing Research Conference. I don't usually do announcement posts but we need to talk about the Call for Papers (deadline November 2) where

In light of rapid advances in generative AI and their impact on research and scientific communication, STOC 2027 is experimenting with several new policies intended to encourage high-quality submissions and promote clear and effective communication of research. 

Let's talk about these changes, which seem more designed to limit the deluge of AI generated papers.

STOC 2027 submissions will not be anonymous; all listed authors must be human and are responsible for the submission.

This reverses the move to removing authors' names that STOC made in 2023. I was never a fan of double blind reviewing and you need authors who can take responsibility for the submission.

Each author may appear on at most five submissions.

Understood but it might hurt some students who have an active advisor. 

Every paper must be submitted to arXiv before the STOC paper submission deadline. Authors must provide a public arXiv URL or proof of arXiv submission along with their submission PDF, which must be identical to the arXiv version.

In the past you could submit preliminary results and try to extend them before publication, since reviewers were expected not to build on unpublished work they were reviewing. This rule may cause some authors to hold back submissions or use AI to help with the extensions.

Authors must submit a video explaining the work, its context, and its innovations relative to prior work. The video should be 20–30 minutes long and will be due 1–2 weeks after the paper submission deadline. The recording should be presented by at least one listed author. The written submission remains the primary object of review. 

This rule will both check that at least one author understands the paper and add some friction to just generating papers using a few prompts. AI could generate the video of an author explaining the paper, but at least for now that would be prohibitively expensive. Some might use AI to generate the script but that would likely be easy to tell from the video. 

I worry that judging the paper based on the video will hurt those who aren't native speakers of English, and might exacerbate unconscious biases so I hope the reviewers really do focus on the paper for the actual review.

Authors may use large language models (LLMs) and other generative AI tools in preparing papers. Substantive use must be disclosed in the paper; minor copy-editing and grammar or clarity improvements to the authors’ own text do not require disclosure.

I would go further and make all papers have an AI disclosure, even if it is just minor copy-editing or "No AI was used in the production of this paper".

Program committee (PC) members and external reviewers (sub-reviewers) may use LLMs to assist with reviewing. Authors must explicitly consent as part of the submission process. All reviews and decisions remain the responsibility of the PC members and sub-reviewers.

I would require consent as condition of submission especially since AI models already have access to arXiv papers. Recent AI models have greatly improved their ability to check proofs if prompted correctly so the overworked PC members can focus on paper quality. 

---

We really need a larger conversation about the role of conferences in theoretical computer science if one can now generate papers from prompts. I've long argued that we should focus the conference more on connecting the community than the "journal that meets at a hotel". The STOC TheoryFest has helped but it would be great to get further away from lauding papers that have complicated proofs and focus more on the ideas that truly drive our field.

I don't care about majors, minors, or honors programs. Do you?

The following conversation is fictional.

---------------------------

ALICE: (Looking over a student's record.) Hmm, let's see. She wants to work in quantum computing. She's had the year-long quantum sequence in the physics department and has taken a course in quantum computing in the computer science department. She has done a project in quantum computing in an REU program.  Grades good, letters good. I think we should admit her.

BOB: Wait! Did she get a minor in Physics? This is very important!

--------------------------

When looking over a student's record the questions

Does she have a minor in X? or

Did she double major?  or

Did she graduate with honors? 

never dawn on me.

1) When I am on an admissions committee I look at:

a) Transcript: What did they take? The grades are generally good so that's a minor factor.

b) Letters that tell me what they did within STEM. I don't care about ballroom dancing or moral character. 

c) Papers they've written whether or not they have been published.

d) Their personal statement. They need to tell me:

i) Why they want to get a PhD.  When Ted Kennedy challenged Jimmy Carter for the presidential nomination in 1980, Ted Kennedy was asked Why do you want to be president? See here for his rambling and incoherent answer.

Despite his background in proving lower bounds on approximation contingent on the Unique Games Conjecture, Ted Kennedy would not have gotten into our graduate program.

ii) What they are interested in (this may have been covered in part (i)).

iii) Why they are qualified.

2) Do I care what the major is? No. I care that they know computer science which I can get off of their transcript.

3) Do I care if they double major in (say) Math. No. I can look at the transcript and see what math courses they took.  I don't care what (possibly arbitrary) rules their school has for double majoring.

4) Do I care if they minored in (say) physics? Not even a little. If they want to do quantum computing I care if they have taken courses in that area.  I don't care what (likely arbitrary) rules their school has for minors.  I took five courses in Philosophy as an undergraduate. Did I get a minor? I don't recall.  Two of them were in logic so I don't think I deserve a minor.

5) Are they in their school's CS honors program? Some other honor program? Are they on track to graduate with CS honors? Some other honors?  I don't care what (definitely arbitrary) rules their school has for honors programs.  If they are writing a paper, honors thesis or not, I will want to hear about it from their letter writer and from their personal statement. 

6) Do I care if they are in phi-beta-kappa? Sigma-Xi? Tau-Beta-Pi?  The last two I only know about since I googled  is there an analog of phi-beta-kappa geared toward STEM  for this post. You can probably guess that I don't care about any of those things. 

7) The point is that these formal criteria: major, minor, honors are not important when I am doing admissions.

a) Are they important to students?

I've heard that high school students who are honors students get a bumper sticker for their parents car that says:

                 My kid is an honors student at blah high school.

I would be more impressed if the bumper sticker said

                 My kid can prove the polynomial van der Waerden theorem.

b) Are they important to other people on the admissions committee?

8) Has the scenario I paint at the beginning of this post ever happened?

AI and Manufacturing Redux

ITMS 2026

Two years ago I attended the International Manufacturing Technology Show in Chicago's McCormick Place and found a rather limited focus on artificial intelligence among the exhibitors. ITMS is back in town so I went again this week. A quiet respite from all the AI/math angst, though that will come in full view when mathematicians take over the same venue in January.

I picked up my badge, the last to say "Illinois Tech" as I registered for a free academic pass well before the layoffs. Of course, you get to see all sorts of neat machines that make stuff but I tried to focus on where artificial intelligence plays a role. This time you could see AI everywhere, though as one exhibitor said, more of a marketing scheme than deep use of modern artificial intelligence. Real artificial intelligence did make appearances: vision recognition for robotic arms, backend software such as bid, invoice and document generation, predictive maintenance, and a variety of robotics, though mostly arms for assembling, cutting and even welding. 

I didn't see much of humanoid robots, digital twinning, use of large language models or manufacturing on demand. When do we get to the point that I can describe a product and have it designed, made and shipped to me quickly?

Not soon. I talked with someone from a small company that takes CAD designs and gets them ready for the manufacturing process. I asked him about automating the design phase and he said they leave that to ChatGPT. But OpenAI and Anthropic were nowhere to be found and Microsoft, Google and Amazon had scaled-down exhibits from two years ago.

Two years ago I remarked on the big booths for European and Asian manufacturers. This year had noticeably fewer giant foreign machinery stands. I'm guessing tariffs and trade uncertainty have dampened the influx of foreign suppliers.

China has gone all in on AI and manufacturing. I can imagine a Chinese slogan:

The US uses AI to make theorems, China uses AI to make products.