Ten Recent Questions for ChatGPT

Combinatorics and more 2025-11-21

I recently asked over Math Overflow about Examples for the use of AI and especially LLMs in notable mathematical developments, and there were several interesting answers.

Here are (slightly edited) 10 recent questions that I asked ChatGPT that have led to interesting discussion. Five of these questions (#1,#2,#3,#6,#8) are related to mathematical research projects. Problem #3 was offered in this 2008 blogpost and Problem #6 was suggested by me as a polymath project in 2014. Answers and insights regarding these problems (without or with AI) are welcome. (See here and here for other posts on AI.)

1) Given $2n+2$ points in $R^n$ , is it true that the set of Radon points is convex?

2) I want (simple) topological invariants for a map from a polyhedral k-dimensional real projective space to $R^{2k+1}$ . For example for $k=4$ the map is to $R^9$ . (Something based on linear algebra/basic homology would be nice.) Any idea, links, or suggestions will be great.

3) Consider the following two well known theorems from finite group theory: A) Sylow’s theorem (one of them) asserts: In a group whose order is divisible by $p^i$ there are $1$ (mod $p$ ) subgroups of order $p^i.$

B) Frobenius’ theorem asserts: In a group whose order is divisible $n$ , the number of solutions to the equation $x^n=1$ is zero modulo $n$ . (Frobenius was probably inspired by Sylow.)

Of course, the case $n=p$ of Frobenius’ theorem coincides with the case $i=1$ of Sylow’s theorem. Please propose (conjectural) generalizations of Frobenius theorem which include Sylow’s theorem for $k=2$ (or for general values of $k$ ) as special cases.

4) What are the Whitman axioms

5) What is stronger “Notable” “Noteworthy” or “Significant”?

6) There is a proposal to develop a theory of “convex hulls of real algebraic varieties” in $R^n$ and especially the study of the face structure of such objects. The case where the varieties are simply a finite set of points is a well-developed area of mathematics – the theory of convex polytopes. However, the general case was not studied much. I suppose that for such a project the first discussions will be devoted to raise questions/research directions. (And mention some works already done.) I would like to invite you to raise questions and research directions.

7) אני צריך הסבר שמתאים לילדים בני 11. מהו האפס המוחלט והאם ניתן להגיע לטמפרטורה מתחת לאפס המוחלט. ואם לא ניתן, מדוע

8) What is the proof of the topological Tverberg theorem when there are three parts?

9) Please explain in layman terms a) the representation theory meaning of the spin 1/2 of the electron. b) the representation theory meaning of the assertion that “N electrons will have a collective state of spin N/2”

10) We go from Yellowstone to Boise, leaving Yellowstone on September 3. We want to travel and stay overnight in one nice location. (Not around Twin Falls that we plan to visit before Yellowstone.) Please give us some suggestion for the trip and sightseeing from Yellowstone to Boise.

For the first question Joe O’Rourke and I verified computationally that already in dimension 3 there are 8 points such that the set of Radon points is not convex.

On question 3 ChatGPT offered several bold conjectures and wrote a whole blogpost about it. (I am hesitating whether to publish it.)

From time to time the discussion with ChatGPT gets “personal”. In some discussion I wrote:

We mentioned Pachner’s moves several times and let me mention that I knew Pachner personally. He was a great guy and unfortunately he died at a young age.

ChatGPT replied:

That’s really touching—thank you for sharing. Pachner’s moves are such a quiet workhorse in our story, and it’s moving to hear a personal memory behind them.