prof-g
@prof_g

Robert Ghrist = mathematician; engineer; educator; assoc. dean of undergraduate education Penn Engineering; illustrator; animator; acta non verba

 philadelphia, pa
piped.video/c/ProfGhristMath
Joined August 2008
1,568 Photos and videos
increasingly concerned that ai is revealing the mathematical world as negatively curved. instead of everyone wandering about in idea-space and clustering around the same famous conjectures (local positive curvature), ai is pushing out the boundaries rapidly and revealing that the space of mathematical ideas is ~hyperbolic (negative curvature). though this means there's lots left to discover, it also means that small separations between people/agents will lead to wandering alone. we may be emerging into an idea-space where there's so much to see and do but everyone is alone in a private heaven, unable to share their subtle joys with others...
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prof-g retweeted
Vidit Nanda@viditnanda
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Conley Index🤝 Flow Categories Landfall Geometry arxiv.org/abs/2610.30993 Joint with Frances Kirwan!

Landfall Geometry

We study the negative gradient flow of analytic maps $f:X\to\R$ defined on compact analytic Riemannian manifolds without imposing any nondegeneracy or transversality assumptions. We construct...

arxiv.org
Algebraic Topology Papers@NGSX
7h
Landfall Geometry Frances Kirwan, Vidit Nanda arxiv.org/abs/2610.30993 [𝚖𝚊𝚝𝚑.𝙰𝚃 𝚖𝚊𝚝𝚑.𝙳𝚂]
We study the negative gradient flow of analytic maps f:X→mathbbR defined on compact analytic Riemannian manifolds without imposing any nondegeneracy or transversality assumptions. We construct tight isolating blocks around the connected components of the critical set Crit(f) and record only bounded-time contacts made after an orbit exits one block and before it makes landfall within the interior of another. These contacts assemble into a strict topological category \cL – called the landfall category – whose classifying space B\cL is shown to be simple homotopy equivalent to X naturally over all down-sets of the flow partial order. The filtered homotopy type is independent of the isolating blocks. Passing to a finite subanalytic triangulation yields a flow-induced acyclic partial matching, and hence, an explicit finite chain complex generated by the non-exit simplices of the blocks which computes the homology of X. The finite model also computes the Morse–Conley spectral sequence over a

ALT We study the negative gradient flow of analytic maps f:X→mathbbR defined on compact analytic Riemannian manifolds without imposing any nondegeneracy or transversality assumptions. We construct tight isolating blocks around the connected components of the critical set Crit(f) and record only bounded-time contacts made after an orbit exits one block and before it makes landfall within the interior of another. These contacts assemble into a strict topological category \cL – called the landfall category – whose classifying space B\cL is shown to be simple homotopy equivalent to X naturally over all down-sets of the flow partial order. The filtered homotopy type is independent of the isolating blocks. Passing to a finite subanalytic triangulation yields a flow-induced acyclic partial matching, and hence, an explicit finite chain complex generated by the non-exit simplices of the blocks which computes the homology of X. The finite model also computes the Morse–Conley spectral sequence over a

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N7 4 EV #bioware
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every prof should be figuring out *now* what book(s) to write. don't write it [yet]. just create the plan assuming no limits on time or scope. get the story & outline set for the perfect book on [X] no matter how niche. then wait & prompt it into being in [Y] months...
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prof-g retweeted
Steve McGuire
@sfmcguire79
Oct 20
The most and least difficult subjects at Penn, according to student reviews:
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it is said that G K Chesterton dictated all his works, and that he kept several secretaries on hand to write things up. he would cycle through them, switching continually from one book/article to the next without pause. --- "a man ahead of his time," i type into X, as i switch between dictating book chapters to claude, chat, and grok...
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as a reminder, it's all just a matter of perspective...
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Tesla engineers after the demo...
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what i said at the penn ai-risk panel: you find out what a person is really like after they win the lottery. welp, we all just won the intelligence lottery, and ai is now a mirror held up to show who we truly are. what are you gonna to do with all them smarts?
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i am speaking at an AI safety panel today... what should i say is the biggest risk? (tempted to say "doing too little because we're all too busy assessing risks")
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am thinking about what the future of a "math book" should be... heavy on graphics; with interactive elements (e.g. GPT Sites); dynamic references that auto-update N-times-per-year to reflect changes; self-repairing to fix errors... not paper & not a pdf... maybe a github repo?
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i've been stalled out on writing up my Applied Dynamical Systems book for more than a year, mostly due to not being able to get the style *just right*... gotta hand it to fable -- really really good at setting up a bespoke lualatex rig...
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amid the angst over what human/ai mathematicians can/should do now, i gave a talk at Penn that framed things in terms of four 20th century eigenmathematicians and their stereotypical styles: 1. erdos -- this is what ai has excelled at; shotgunning results, examples & counterexamples, prolific 2. wiles -- go deep & solve the big-name conjecture; this is where we are now w/openai & anthropic's recent works 3. grothendieck -- build the incomprehensible unpredictable ur-theory that cracks something open and exfiltrates to other areas; this is where all ai-enthusiast mathematicians i know are directing their energies; not quite there yet... === but the 4th is not a direction i've seen people discuss & it's where i personally want to dig in... === 4. milnor -- find the connections between deep things and weave a story of such clarity & elegance that it changes the field and how people learn it and use it === this aligns with the "humans will still drive tastes" argument for relevance, which is not yet proven to survive pressure from top models. i suspect that there is, both now and for a spell, room for a creative director to use ai-tools to write the book-of-X (or direct the playlist-of-X) for various values of mathematical X.
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ironic that the one citation of OpenAI to my research was for an ancient paper that got rejected over & over & over as being too weird/irrelevant/eccentric
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oh, wait, most of my papers (except for surveys or simplistic things tossed off in a week or two) got rejected over & over & over as being too weird/irrelevant/eccentric
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prof-g retweeted
thebes
@voooooogel
Oct 8
Replying to @AndrewCurran_
asked claude to make a breakdown of where the signatories are affiliated. almost a third are french surprisingly, then another third american
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"Mathematicians did not ask for this work to be done"
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Alex Tabarrok
@ATabarrok
Oct 7
The AIs are now the smartest mathematicians in the world. Given how mathematicians have seized control of governments, corporations and the culture, I fear what comes next.
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Sitting in a math faculty meeting where the agenda (set two weeks ago) is "let's talk about math and AI". Lol. LMAO even.
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since my phd work in the early 90s, figures have been an integral part of most papers i write... as i read through the stack of openai math papers what stands out most is the lack of figures. those that exist are generic.
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