is there a finite-time singularity in fluids?
(euclidean 3-d euler/navier-stokes, smooth IC)
this is the data we collected at @CRMatematica today...
we voted on a scale from 0 to 10 where
0-nope! ==== 5-??? ==== 10-yeah!
here's the vote : euler in blue, navier-stokes in red...
but wait...there's more...
after voting -- in private! -- we then did a public vote-by-show-of-hands with yes/no/???
(all eyes were on terry tao, sitting in the back...)
then we did a second private vote...
can you guess how terry voted?
more data! the same exact question was asked at a meeting in 2007 at the 250th anniversary of the euler equations at a meeting of experts in Aussois, CH
(thanks to Kai Schneider for the data!)
here's what the data was in 2007
if you want to see the change in sentiment over time 2007 => 2024 1st vote => 2024 2nd vote, here's the euler equations...
and the same historical trends for navier stokes
so what have we learned?
* opinions have drifted more towards singularity
* opinions can change in real-time due to influence by neighbors -- very relevant to information flows...
* MS excel doesn't like indexing from 0 to 10
special thanks to kai cieliebak, @evamirandag , and kai schneider for suggesting and helping to get out the vote! (and for allowing me to conduct this little opinion dynamics experiment...)
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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]
this is a weird 300-page book, written in a weird manner.
i built a custom project on claude to assist with the creation.
i began on nov 2, 2024 & published the book on dec 28, 2024
claude captured my writing style (from other works), kept track of the outline, thematic elements, latex preferences. i would prompt it to write individual sections in latex; then i would assemble, edit, rewrite; claude also helped with editing through a separate process.
1/ new paper in PNAS (link at end)
led by @irishryoon -- @chadgiusti, greg henselman-petrusek, yiyi yu, spencer lavere smith & i tackle a key challenge in TDA:
how do you match topological features across populations or datasets?
🧠🔗
2/ the context:
neuroscience often involves comparing neural manifolds—geometric representations of neural activity—across subjects.
how can we align topological features (e.g., cycles) between different manifolds?
3/ feature matching in TDA is hard.
given two persistence diagrams, how do you reliably correlate significant features?
this involves balancing geometric structure with topological significance
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/
this is joint work with julian gould and miguel lopez, phd students in my lab @Penn .
this is also joint work with claude-3.5-sonnet, gemini-1.5-pro, gpt-4o, and gpt-o1-mini .
it’s a 25 page paper on network information flows and lattice theory.
(link to preprint at end) 2/
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/
1/ yesterday’s thread on the genesis of figures in “Elementary Applied Topology” ended on a cliffhanger: what do all the chapter heading illustrations mean? it’s a puzzle.
get ready for a myth/math mishmash.
2/ the title page & the back cover carry an iconic stylized pomegranate, with some motion induced on what looks like half-a-dozen seeds? hmmm… that reminds me of something…
3/ of course, this is a persephone myth – she who was seduced by hades and descended to the underworld. what does that have to do with topology?
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”.
2/ i wanted the book to have a lot of pictures – at least one per page on average – and i wanted the pictures to be the exercises. each picture is a puzzle, and if you understand the picture, then you have solved the exercise & understand something.