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]

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.

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/

topologist's breakfast

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

how i teach the kalman filter to 1st-year undergraduate engineering students @Penn : a thread 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 response into gemini; it finds more problems * iterate until convergence * feed the consensus idea w/detailed report to grok-3 * grok finds gaping error, fixes by taking things in different direction (!!!) * gemini agrees: big problems, now ameliorated * output final consensus report * paste into claude-3.7 and ask it to outline a paper * approve outline; request latex following my style/notation conventions * claude outputs 30 pages of dense latex, section by section, one-shot (!) ==== is this correct/watertight? (surely not) is this genuinely novel? (pretty sure yes) is this the future? (no, it's the present) ==== everybody underestimates not only what is coming but what can currently be done w/existing tools.

grad, curl, div

remember the double pendulum -- classic exemplar of chaos? well, it's time for a thread about double pendula, generalizing from 2-d to 3-d... 1/8

fellow profs: find a way to use or create low-cost texts or text-alternatives for your students. many, many students at the margin will be positively impacted. \end

Holy smokes... The new GPT-4o model is the first that has ever properly rendered a mobius strip. This is 1-shot, no editing.

gravitational 3-body problems are great for chaotic dynamics. one of my favorites is the sitnikov system: given two equal massive bodies rotating about one another on ellipses, add a third tiny mass perfectly poised between them. give it a kick & what happens?

taught my students how to analyze chaotic dynamics via the geometric lorenz attractor this week... such a beautiful branched surface...

always remember everything looks linear

the 1st video of my series w/ on Foundations of Topological Data Analysis is up on YouTube. it's about combinatorial simplices & simplicial complexes. (link below)

remember kids : don't do drugs -- do higher math

okay, i've had my first "holy 💩" moment with GPT-o1 for proving theorems. i've been working on a proof of [REDACTED] for weeks, cycling between claude/GPT/gemini, always with a subtle bug or misplaced inequality... always the same 3-4 proof ideas based on pattern-matching...

i got into a bit of an argument w/yan lecun last night at dinner about whether LLMs are useful in doing research... a few observations: * he was throughout polite & pleasant: we disagreed but it never got personal or even heated * he stated his views about the reasoning capabilities of LLMs (or lack thereof) * i stated my experience working on [ redacted latest research idea ] * i began to build an example of how LLMs are doing spatial reasoning, but as soon as i started with a simple case, he said "oh, that was discussed on X, and it's in the training data -- it's all just repeating what is in the training data" * that was basically the end of the argument from his view: biases confirmed, we're done here * as a followup, he was asked "how often do you use LLMs?" his response "not much at all... a little bit for coding sometimes" ===== one of the great things yan said to penn students was that "you might think there's nothing more to be done... but you can do so much more". that is 100% right on. ===== if you are keeping up with the latest models and working daily to push what can be discovered, you are so far ahead... maybe even ahead of the experts. you can do so much more.

1/ i am writing a book on applied dynamical systems. you can find a working draft here: math.upenn.edu/~ghrist/prepr…

this is the substance and essence of true love

current big gap in AI capabilities: knot theory. recognizing knot types, even in"standard presentation" (from knot tables) is very difficult, even for the best models right now.

a quick thread on what the @penn mathematics department is doing starting this fall to improve the student experience in calculus courses... 1/

what ties together machine learning, rational homotopy theory, rough paths, lie groups, topological data analysis, and how to teach undergraduate vector calculus? a thread… 1/

periodic phenomena (vibrations, tremors, mood swings?) often arise from a [supercritical] Hopf bifurcation, in which a spiral sink becomes a spiral source, spawning an attracting limit cycle as the parameter passes the critical value 1/4

we covered chaotic dynamical systems this week in class... it's *so* nice to see students' reactions to the existence of deterministic chaos after a semester's worth of rigorous prediction in systems...

this semester, i am teaching multivariable calculus for engineers @penn, using materials from the Calculus BLUE Project, a video-text on youtube. you can check out the (updated for 2020) trailer here... piped.video/watch?v=xZeqTptJ…

If I had a student who proved this optimization result, there's no way I would tell her "that's not real math, not by 10,000 miles" and berate her for not measuring up to Grothendieck. These LLMs are students, learning quickly how to do high quality math research.

1/ this past week in multivariable calculus @Penn the chain rule & its consequences...

teaching green's theorem, gauss's theorem, & stokes' theorem always prompts the question, "what is this good for?" although fluids & emag are important, it would be nice to have a more modern application... how about to data? 1/13

i find it difficult to teach "staircase diagrams" for 1-d maps on a chalkboard; even pictures in books are weak at illustrating stable vs unstable equilibria & dependence on initial conditions. animation is helpful in so many areas of mathematics, especially dynamical systems.

i did a simulation (in the beginning of 2020) of mackay's "anosov machine" exhibiting chaotic dynamics. did not get around to rendering it till today. it's pretty nice...

the branchline of the lorenz attractor a very strange attractor

like all things, this semester has come to an end... final lecture of my applied dynamical systems course @Penn

one of the classic nonlinear 2nd order ODEs is the Duffing equation, which exhibits a supercritical pitchfork: past a certain parameter value, a spiral sink turns to a saddle and sheds a pair of spiral sinks. what's so nice about that? next comes a novel simulation... 1/3

am teaching network dynamics this week; my fave example comes from neuroscience. TLNs = threshold-linear networks are simple system that model firing of neurons that can excite neighbors based on a directed graph. this leads to interesting periodic & chaotic dynamics. 1/3

prediction: vibe shift incoming. a subset of professors is going to finish classes/finals soon and check in with the latest AI models (after not playing with them for a year), testing out what AIs can do with their research ideas. these profs will then not sleep for a week.

what is this? it has something to do with a spinning tennis racket... 1/5

one of the things i did this semester in my multivariable calculus class was to give the option of doing an "extra credit" project. i've never done that before, and was inclined against it. but: it went so very well. let me tell you... 1/

the bogdanov-takens bifurcation is a great codimension-2 bifurcation of vector fields. within the 2-d parameter space (a,b) you have curves of: saddle-node bifurcations; hopf bifurcations; and homoclinic bifurcations. x' = y y' = a + by + x^2 + xy

tomorrow i am teaching coupled oscillators in my applied dynamical systems class. it all begins with a pair of independent identical pendula, just doing their thing... 1/9

visual representation of choosing a phd thesis topic

you're a math researcher. you understand what's coming. do you: 1) write up as much as possible now while you still can assist, putting all your time into working w/AIs to produce? 2) draft up as many crazy ideas into sketch/outlines as possible, wait two years, then say "go!"

shaking pendula can lead to interesting dynamics, but, of course, the natural next step is hang a pendulum from a pendulum. such a double pendulum is a classic example of chaotic dynamics, made possible by the "extra room" available in its 4-dimensional phase space

mathematical art is increasingly important in research & AI is about to revolutionize what can be done... 🧵

if you're taking/teaching applied dynamical systems this term & want to peek at a book-in-progress, i've updated the draft on my website... see below

I'm so proud of my daughter who just graduated with her degree in computer science... I knew a long time ago that she was destined to code...

the lorenz equations define a classic chaotic dynamical system. any orbit meanders about an attractor in a way that, though deterministic, appears impossible to predict. 1/4

when teaching bifurcation theory, it's really easy to find mechanical examples of some (zeeman's catastrophe machine being a classic!) for others, it's more of a challenge...

in my heart i know i'm a fool for paying $200/mo to openai... (and $20 to anthropic, $20 to google, $30 to midjourney, $?? to X, etc...) but i'm getting so many research leads & able to produce at such a pace... yeah, i'm gonna spend it.

1/ this week in multivariable calculus @Penn ... constrained optimization

august '24: got obsessed with using LLMs to prove a new theorem (that a conversation with claude had conjectured). worked on it for a month or more, mapping out the space of wrong proofs, till GPT-o1-mini nailed it... wrote it up & submitted to journal. here's what happened...

here's my new linear algebra text in multiple formats: physical edition : amazon.com/Linear-Algebra-Es… pdf : math.upenn.edu/~ghrist/prepr… text : math.upenn.edu/~ghrist/prepr… enjoy!

corollary of despair... mathematics departments that justify their existence on teaching calculus: time's up. you will not persist and may not deserve to. adapt now, using all tools at hand, while you can. not happy to bear this prophecy, but there it is...

for my applied dynamical systems course, i am preparing some materials on neuroscience for next month... my favorite part of this is the *threshold linear network* model... 1/

i am still checking it for correctness, but even if it's wrong, it's such a clever move, something like it is highly likely to work. i'm so very pleased: to have this degree of creativity and precision on tap is gonna make mathematicians deliriously happy...

quick summary: aug ’24 : claude-3.5/gpt-4o conjectured a new theorem. sept ’24 : generated many wrong proofs w/claude+gpt+gemini, mapping out the subspace of latent proof-space. sept 13 ’24 : gpt-o1-mini dropped & nailed a correct + elegant proof. oct 1 ’24 => arxiv preprint. 3/

oh, and, more trivially, teaching is a solved problem. i'm getting extraordinary performance improvement in my GPT tutors in the space of a year. soon, they will answer questions better than i do.

And to think just a month ago I had to double-mast...

one of the things you notice when you study chaotic dynamics in 2-d maps is the resemblance to mixing of fluids. the stretching & folding mechanisms are very similar.

> be me > late 2022 : chatGPT released > i'm the prof, teaching classes > i cancel lecture > "YO! -- drop everything and go long this rn fr" > students: "uh, whuh? do you really think this is gonna be bigger than the iphone? c'mon!" > me : let me tell you a story from 1993...

less than 9 weeks ago, i reported on how o1-mini was able to get a breakthrough in generating a proof of an AI-generated math conjecture. with the full o1 model out yesterday, i prompted it with the same initial idea to see what it would do...

You misspelled Briseis...

i've spent serious efforts working on developing math problems with an unambiguous (e.g., numerical) answer that gpt-5-pro cannot solve. it is *nontrivial* to do so. it was totally different even 4-6 months ago.

i have been teaching calculus to engineering students at penn for over a decade: they are getting better & better every year (modulo the covid blip).

one of the most beautiful facets of geometric topology is the study of contact structures -- these are nowhere-integrable plane fields (in 3-d). unlike vector fields, plane field are rarely illustrated/animated. (cause it's hard!)

would like to see a list of the many mathematicians who think that 0 is our greatest achievement

what is a good example of a saddle-node bifurcation in 1-d continuous-time dynamics? i'm so glad you asked... one standard example is a torqued pendulum. turn up the torque, and eventually the stable & unstable equilibria merge & annihilate: a saddle-node. 1/7

working on the mathematics of mechanical linkages... feels like i'm going back in time... (i was originally trained as a mechanical engineer)

3. homology

i pointed out the reversed inequality. gpt-o1-mini said “To address this, let's revisit and refine the proof...” >> 43 seconds of thought later << an entirely new, clever, correct proof. more elegant than the human proof. 17/

i think that every mathematician interested in visualization has to try drawing the hopf fibration of the 3-sphere in their own way at some point. welp, i guess it's my turn to try... 1/

the latest models (gemini-2.5-pro, grok-3, ...) are *so* good at vibing math research. tossing in old dead-end ideas from a decade ago & getting fresh insights & paths. things i wouldn't suggest to a grad student (too risky, not likely to work) are now game on...

i haven't been making many animations recently, but here's a new vid of an old standard: the lorenz attractor. in this, you can clearly see why it's often modeled as a branched surface.

this week in dynamical systems: examples of chaotic systems... i'm going to review one of my favorite -- the forced duffing oscillator. here is a physical simulation of it using springs to model the double-well potential.

instead of trying to get AI to prove the Big Conjecture, here's something you can do right now with high odds of success... the hidden 💎 search for math papers by profs at a top uni that are >5 years out and have <10 citations. pull up a stack of 'em. feed the stack to gemini (big context window) or grok (highly creative) and ask it to identify which (if any) has a "hidden gem" idea: something that would be useful in other parts of math or applications outside of math... then pick one (that interests you) and send it to gpt-5-pro with instructions to extract the hidden 💎 and start cutting & polishing... generalize, expand, look for connections, new conjectures, etc. then slowly build out the new results. write it up (claude suggested for this step). for a mathematician who can learn new things quickly, this is a 💎 mine... and it holds out hope that many "lost ideas" buried in obscure papers will eventually be unearthed and developed. i suspect [hope] that this process will be automated in the near-future. API calls & some rigging -- doable now in some form, though the human-in-the-loop seems critical atm.

my love-letter to the humanities: a book on mathematics & poetry. it starts with the question: "Where the Hell is Heaven?" thesis: in imagining the afterlife, poets from ancient times used the same imaginative skills that geometers and topologists use to imagine abstract realms... and the poets' imaginations presaged the geometers'... ------ a few hints (chapters from the book) 1. the most ancient accounts of the underworld are highly-structured and coordinate-based: go here, turn this way, walk this far. the land of the dead is euclidean. 2. as man looks up, from plato through dante, the heavens bend and spin in clockwork spheres; spherical geometry is born in the shape of the universe. dante even goes so far as, perhaps, to invoke the 3-sphere as a model of the universe, with earth at the bottom, and God atop. 3. milton is more abstract, making both heaven & hell spaces distinct from the physical universe; but satan's journey through Chaos from Hell to earth is through an entropic Void where one can neither fathom nor find a path -- until the precise direction is given, using which, the way is short. that is a perfect description of hyperbolic space, centuries before lobachevsky et al. 4. poets, both earlier and later, explored other geometries in speculative/spiritual works, including prototypes of fractal geometry (blake's "heaven in a wild flower"). 5. swedenborg, blake, and other contemporaries go so far as to describe abstract spaces of various sorts, including hints of non-hausdorff worlds, a superposition of heaven & hell. 6. finally, in the era of the modernists (eliot, joyce, beckett), the poetic imagining of heaven is a loss of faith -- everything is words and worlds built of words. this mirrors the contemporaneous rebirth of algebraic geometry, annihilating and rebirthing geometry as writ with variables. the entire book is an ode and nod to blake's marriage: of heaven & hell, image & verse, mathematics & poetry.

fun problem in matrix algebra from last week's multivariable calculus class @penn given two parallelograms based at the origin, find a linear transformation taking one to the other... students tried combining rotations, rescaling, shears, but there is a better way...

in 2-D continuous-time linear dynamics, the trace and determinant of the matrix suffice to tell you what type of equilibrium you have...

> a bunch of us grad students are like "whoa, check out this thing... address... wtf... > oh hey it's NASA > wtf! it's data! this is so cool! > yeah, okay, it was nice to not have class eh? > funny how profs get excited about stuff..." > > [ TIME PASSES ]

these are *1st-year* undergraduate students @Penn, working on their own in these weird times & learning really advanced stuff. i am so proud of their intellectual curiosity & verve. the future is going to be a better place because of students like these. //

the aizawa attractor a delightfully chaotic dynamical system

the secret reason why we can't compute stable homotopy groups of spheres 😭

> be me > early 2024 > /galadriel voice/ > i amar prestar aen… > han mathon ne nen… > han mathon ne chae… > a han noston ned gwilith…

just had my 1st ever experience of reading a new math preprint on the arxiv, noticing the choice of words, the over-use of bullet points, the super-short sections, the superfluous concluding paragraph in each section... yeah, this was probably written by a LLM.

The first two chapters of "Applied Dynamical Systems Vol 1" are now posted on YouTube... (this is a videotext on 1-d dynamics; prereqs = basic calculus) piped.video/playlist?list=PL…

calculus BLUE chapters 8-12 are now published, covering: differential forms, gauss's theorem, & stokes' theorem... or stokes's theorem... or the divergence theorem... or something ostrogromthing, whatever. just good, fundamental theorems.

next video in the new series with @viditnanda is up! Foundations of Topological Data Analysis section 1.2 on subcomplexes

i'll post full details once i know the ending to this story and can organize things... but i am now really impressed with what GPT-o1 can do for proofs...

henry wente: geometer & teacher. he left this world today: 20-jan-2020. he's the reason i'm a mathematician.

the pope studied math in philly? this is like the superbowl for me

multi-variable calculus student: "we don't need to worry about orientations in the real world, right?" professor, a topologist: <sobbing heavily>

surprising efficiency gains by sharing ideas between disparate domains is an underrated use-case for AIs. they pattern-match and are broadly-trained.