i am about to teach a new applied linear algebra course at @PennEngineers, meant for datasci/ML/AI.
to support the class, i've written a book...
[see below for links to text]
prof-g
12.1K posts
prof-g
@prof_g
Robert Ghrist = mathematician; engineer; educator;
assoc. dean of undergraduate education Penn Engineering;
illustrator; animator; acta non verba
- upload a draft of research paper to my fave LLMs: 1. "please check this paper carefully for errors" => you're good buddy! great paper! 100% correct! 2. "please referee this as if it were submitted to [top journal]" => detailed list of errors, inaccuracies, gaps, etc...
- linear dynamics given by x'=Ax (for A a square matrix) are determined by the eigenvalues of A. this example has 3 eigenvalues. when they're real, flow goes in/out along eigendirections; a complex pair yields spirals. if all 3 are the same w/one eigenvector, it "tries" to spiral.
00:00 - 1/ in 2009, i began writing a book on topology, meant to be a short introduction to the core concepts, in the context of lots of interesting applications. every idea would be paired with one or more uses, as much outside of mathematics as possible: “applied topology”.
- spent a few hours with Claude 3.5 sonnet doing some mathematics research. you are underestimating the impact AI will have on research. yes, you. yes, I'm serious. no, it does not replace mathematicians. but the augmentation is about to take off.
- the hardest part of doing research with GPT-o1-mini is how i ask it a question and it thinks for 3 seconds and fires off a 15-point manifesto that takes me half an hour to process. i'm the bottleneck.
- so, i tried out OpenAI's new Deep Research w/o3-mini to do a literature search on network sheaves (something i'm the expert at), and it told me things i did not know about (with accurate links). literature search for theses/papers is now practically automated.
- if you're looking for an iconic linear-algebra-with-applications-to-AI textbook...
- > be me > 1993 : cornell, taking numerical linear algebra with nick trefethan at cornell > prof comes to class and says "yeah, forget class today -- you need to drop everything and see this" > pulls up something called the World Wide Web > says this changes everything > sure
- can AI do research-level mathematics? make conjectures? prove theorems? there’s a moving frontier between what can and cannot be done with LLMs. that boundary just shifted a little. this is my experience with AI proving a new theorem. 1/
00:00 - major-publishing-corp-rep wanders into my office today unscheduled... "so, i hear that you are the only person in the math dept whois not using our textbooks for your calculus class... so, what are you using?" 1/9
- workflow of the past 24 hours... * start a convo w/GPT-o3 about math research idea [X] * it gives 7 good potential ideas; pick one & ask to develop * feed -o3 output to gemini-2.5-pro; it finds errors & writes feedback * paste feedback into -o3 and say asses & respond * paste
