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@robertghrist
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@Penn mathematician; engineer; educator; assoc. dean of undergraduate education @PennEngineers ; illustrator; animator; e/acta non verba
philadelphia, pa
Joined August 2008
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Pinned Tweet
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...
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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.
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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”.
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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.
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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.
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> 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
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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/
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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
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how i teach the kalman filter to 1st-year undergraduate engineering students
@Penn
: a thread 1/9
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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
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
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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
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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?
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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)
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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...
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1/ i am writing a book on applied dynamical systems.
you can find a working draft here:
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a quick thread on what the
@penn
mathematics department is doing starting this fall to improve the student experience in calculus courses...
1/
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what ties together machine learning, rational homotopy theory, rough paths, lie groups, topological data analysis, and how to teach undergraduate vector calculus?
a thread…
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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
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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...
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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?
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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...
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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.
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the branchline of the lorenz attractor
a very strange attractor
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like all things, this semester has come to an end...
final lecture of my applied dynamical systems course
@Penn
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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
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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
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what is this?
it has something to do with a spinning tennis racket...
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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...
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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
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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
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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
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mathematical art is increasingly important in research & AI is about to revolutionize what can be done... 🧵
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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...
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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.
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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...
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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...
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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...
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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...
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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.
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> 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...
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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.
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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!)
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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
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working on the mathematics of mechanical linkages...
feels like i'm going back in time...
(i was originally trained as a mechanical engineer)
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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/
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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.
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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/
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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.
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in 2-D continuous-time linear dynamics, the trace and determinant of the matrix suffice to tell you what type of equilibrium you have...
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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...
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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.
//
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> 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 ]
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the secret reason why we can't compute stable homotopy groups of spheres 😭
Marc Andreessen and Ben Horowitz say that when they met White House officials to discuss AI, the officials said they could classify any area of math they think is leading in a bad direction to make it a state secret and "it will end"
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> be me
> early 2024
> /galadriel voice/
> i amar prestar aen…
> han mathon ne nen…
> han mathon ne chae…
> a han noston ned gwilith…
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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)
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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.
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next video in the new series with
@viditnanda
is up!
Foundations of Topological Data Analysis
section 1.2 on subcomplexes
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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.
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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...
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henry wente: geometer & teacher.
he left this world today: 20-jan-2020.
he's the reason i'm a mathematician.
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multi-variable calculus student: "we don't need to worry about orientations in the real world, right?"
professor, a topologist: <sobbing heavily>
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things that we're *not* doing in multivariable calculus (but we almost do...)
frenet frames for a curve
unit tangent, normal, & binormal vectors
very old-school; very fun; very sidequest
not on the main storyline for the course...
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i know that the theorem is novel [not in training] and true [we have a human-generated proof], but i wanted to get an AI-generated proof. GPT-o1-mini went down the usual path got stuck, then jumped to an extremely clever and elegant new approach!
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@chadtopaz
@amermathsoc
@ucdavis
Her statement was that we should not have political tests for employment in math. Including tests of noble values (diversity) as well as tests of values once considered ignoble (communism). She's right: once you admit the merit of political tests for employment, we all lose.
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1/ this week in multivariable calculus
@Penn
applications of saddle points...
everyone knows what you want to minimize (stress) and maximize (profit), but why would a saddle be useful?
let's play a game...
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the periodically-forced duffing oscillator induces a flow on the solid torus with some lovely chaotic dynamics. by following a set of initial conditions in the plane over time, you can see the stretching & folding lead to chaos.
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a man at my (small) church died of covid yesterday; he was my age & leaves a wife & teenage daughter.
my neighbor buried his 8yo daughter this weekend: cancer.
remember: thou art dust. do good in this sad world, if you can, while you have the chance.
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coming in spring 2020: Applied Dynamical Systems vol 1...
a video-text on YouTube
teaser trailer...
;-)
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1/ announcing a new paper w/
@_jakobhansen
read on to learn about:
> social networks
> opinion dynamics (e.g. political discourse)
> sheaf cohomology
> harmonic opinions
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this trend of iterating image creation by starting with a topic and repeating "make it more" is a good example of a dominant eigenvalue/eigenspace. note the convergence to something cosmic...
Obsessed with the new “make it more” trend on ChatGPT.
You generate an image of something, and then keep asking for it to be MORE.
For example - spicy ramen getting progressively spicier 🔥 (from u/dulipat)
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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.
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”.
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circles encircling circles...
(this took an entire day to render... 🥲)
(it's ya boi, the hopf fibration of the 3-sphere...)
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when i teach single-variable calculus, i like introducing special functions early: the bessel functions (such a J_0) are very easy to motivate...
but they are not all there is...
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one of the great examples of an interesting (discrete-time, 2-d) dynamical system comes from a bouncing ball on a sinusoidally-oscillating table, the analysis of which famously appears in the classic text of Guckenheimer-Holmes.
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taught 1-D bifurcations in applied dynamical systems...
it's a dictionary for how behaviors can change abruptly.
consider dx/dt = f(x, \mu) where \mu is a parameter.
you can explore the "multiverse" of systems at different \mu-values & get different behaviors...
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i'm teaching multivariable calculus this fall semester using the video-text from the Calculus Blue Project... and this semester, there is a physical book to go with it
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bessel functions are very groovy & appear in a number of physical settings: example = a radially-symmetric vibrating membrane has displacement given by the bessel function J_0 times a temporal sinusoid...
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i get a lot of questions about how to make math vids...
my workflow is so byzantine, it's not worth writing up.
but...for those making vids for fall semester in a hurry: a thread of advice...
/1
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since the "is math created or discovered" arguments are once again circling, it's time to share my view...
mathematics is incarnated...
🧵
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"in my role as co-director of the
@PennFirstPlus
program, one of the complaints we constantly hear is how students are struggling to afford these /homework systems/ that cost 90% of the book price, and are not covered by financial aid..." 4/9
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summary, first; details, following:
-> all lectures via short asynchronous vids
-> all "in class" time is active learning
-> no more midterms; weekly quizzes
-> no more $200 textbook/mymathlab
-> all homeworks online via canvas
-> no more 8am recitations
3/
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it's up! the next installment of FTDA = Foundations of Topological Data Analysis (with
@viditnanda
) is live...
it's all about (simplicial) maps
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last week in my applied dynamical systems class we talked about coupled oscillators. this elicited a lot of discussion about crickets, cicadas, moodswings, astrology, & much more...
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coming this summer on YT:
Foundations of Topological Data Analysis
a new video series joint w/
@viditnanda
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me, silently judging the entire publishing industry: "well, i need to go teach a room full of these students now, so if you'll excuse me, i'm going to work now..." 8/9
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all of these lectures & animations are available in volume 3 of Calculus BLUE available on youtube: see, e.g., and 8/9
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what well-known but infrequently-taught aspect of calculus is this illustrating?
& what is the deeper principle behind it?
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if you're teaching or learning multivariable calculus this semester, there's 25 hours of animated video lectures here, covering: matrix algebra & multivariate derivatives, Taylor series, optimization, integrals, probability, fields, & differential forms.
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coupling two zero-friction spinners with repelling magnets leads to a system with simple dynamics... or maybe not so simple. chaotic dynamics are often tame in the short-term.
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why learn differential forms & stokes' theorem?
besides the usual applications in physics, there are some lovely novel applications in data science...
it's time for the grand finale from the calculus BLUE project...
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