Shing-Tung Yau

“My proof, I told them [Andrew Strominger and Edward Witten], was motivated by physics, specifically the notion that even in a vacuum, a space with no matter, gravity could still exist. I felt certain that this must be important for physics, though I was not sure of the exact ramification.“

In “The Shape of a Life - One Mathematician's Search for the Universe's Hidden Geometry” by Shing-Tung Yau, Steve Nadis

“String theory further postulated that we inhabit a ten-dimensional universe consisting of the three familiar (and infinitely large) spatial dimensions, one dimension of time, and six additional miniature dimensions that are wound up into a tight coil and thereby hidden from view. The question that Candelas and Strominger, among others, were grappling with concerned the geometry of the six shrunken, or ‘compactified’ dimensions. What, exactly, is the shape into which these extra dimensions are confined? Strominger knew they needed a manifold, or space, with well-defined properties, including a special kind of symmetry called ‘supersymmetry,’ which turns out to be an intrinsic feature of the manifolds, of the variety called Kähler, whose existence I had proved. Supersymmetry is also a requisite feature of many versions of string theory, which is why it’s sometimes called ‘superstring theory’ instead.”

In “The Shape of a Life - One Mathematician's Search for the Universe's Hidden Geometry” by Shing-Tung Yau, Steve Nadis

I wear a giant panda suit outside a Panda Burger giving out promotional leaflets. As this job is a bit easy and I can do it without too much conscious effort... the only thing I have to watch out for is farting as it is unpleasant trapped in that panda suit... anyhow I digress ... this gives me a LOT of time to think about serious issues such as time and the merits of having a smart-watch. So I'm with you 100% about the conversation.

Mark Twain said that scientific facts give rise to speculations, which of course are tested if possible. For the most part, math is not about "numbers" but largely about properties of, and relationships among highly abstract objects. Indeed, mathematics as a profession is a risk and self-sacrifice. One has to devote time and effort to one's field before one gets to appreciate it and produce results worth of publication. But there is always a risk that, even if one gains an understanding - which in itself is rare and precious - it will not be followed by original results, stalling one's academic career. This stalling of career due to the lack of originality is normally a direct result of being risk averse and not pushing yourself hard enough. Mathematics is an essentially creative activity: you are bound to achieve something if you are genuinely interested...Tricky thing defining maths. Even if the definition is true, it never looks very interesting. Certainly not as interesting as mathematics itself. It's certainly made a wee bit of progress from counting. Over the last few thousand years... There was that Archimedes and that other Euclid guy. And that Al Khwarizmi dude. Some Newton bloke. Euler, Gauss, a whole truckload of Bernoullis, Fourier, Cauchy, Poincare, Riemann, Noether, Cantor, Goedel, Brouwer... feel like I've forgotten a few hundred really big names but I just can't put my fingers on them...Reducing maths to numbers is kind of like saying all cooking is really just a matter of making 2 minute noodles.

My querky moment while learning mathematics was during a moment of boredom when I took the differences of successive calculated polynomial values and continued taking differences of the results. It turns out this is the basis of the difference engine that Babbage designed, and how mathematical tables were created before the advent of electronic calculators and computers. Probably unsurprisingly I took up Engineering which makes use of a myriad of mathematical techniques and valid short cuts, many of which are never taught to scientists and mathematicians in my experience.

There's something sublime, mystical and ineffable about such problems. You'd think maths would be easy, just counting, but hidden within those ostensibly basic concepts are such convolutions and crenelations and complications. It's amazing that 1+1 can get to such things like Fermat's Last Theorem and imaginary numbers or that Calabi-Yau Manifolds can be applied to Physics, namely String Theory and General Relativity. Let alone whatever these things are on about.

I just wish Yau had written a more math-oriented biography. We don't really get math insights on how he got to prove some of the things important to Physics, namely the Calabi-Yau conjecture. It's all very vague... If you want that to dig deeper into the math part of some of these topics, you should read “The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions” by the same authors.

Coda: No-one uses Calabi-Yau in a sentence (apart from Woody Allen in a New Yorker piece). It inspired me...

I wish my house was a Calabi-Yau Space,
a place where I could tell fiction from fact
I'd invite politicians to sit in the middle
Then I'd focus the heat so it's hot as a griddle
I'd make then elucidate policies at length
And keeping them talking to sap all their strength
And right at the end I would shout and declare
"Your lies and deceit are now totally clear
My house has deciphered your thoughts and your words
And showed them as nothing but bright polished turds
I'm leaving you now and I'll never come back
This part of my house is now fading to black....

NB: It was kind of interesting to read about Yau’s take on the feud between Yau and Chern and also his attempt at explaining what happened with the Poincaré Conjecture (he was accused of “stealing” Perelman’s discovery by having some of his students develop a more rigorous proof of Perelman’s demonstration).… (més)

“The spaces Calabi envisioned not only were complex, but also had a special property called Kähler geometry. Riemann surfaces automatically qualify as Kähler, so the real meaning of the term only becomes apparent for complex manifolds of two (complex) dimensions or higher. In a Kähler manifold, space looks Euclidean at a single point and stays close to being Euclidean because when you move away from that point, while deviating in specific ways."

In “The Shape of Inner Space - String Theory and the Geometry of the Universe's Hidden Dimensions” by Shing-Tung Yau, Steve Nadis

“So what exactly had I [Yau] accomplished [proving the Calabi-Yau Conjecture]? In proving the conjecture, I had validated my conviction that important mathematical problems could be solved by combining nonlinear partial differential equation with geometry. More specifically, I had proved that a Ricci-flat metric can be found for compact Kähler spaces with a vanishing first Chern class, even though I could not produce a precise formula for the metric itself. [..] Although that might not sound like much, the metric I proved to be ‘there’ turned out to be pretty magical. For as a consequence of the proof, I had confirmed the existence of many fantastic, multidimensional shapes (now called Calabi-Yau spaces) that satisfy the Einstein equation in the case where matter is absent. I had produced not just a solution to the Einstein equation, but also the largest class fo solutions to that equation that we know of.”

In “The Shape of Inner Space - String Theory and the Geometry of the Universe's Hidden Dimensions” by Shing-Tung Yau, Steve Nadis

“And it is within this circle of tiny radius that the fifth dimension of Kaluza-Klein theory is hidden. String Theory takes that idea several steps further, arguing in effect that when you look at the cross-section of this slender cylinder with an even more powerful microscope, you’ll see six dimensions lurking inside instead of just one. No matter where you are in four-dimensional spacetime, or where you are on the surface of this infinitely long cylinder, attached to each point is a tiny, six-dimensional space. And not matter where you stand in this infinite space. The compact six-dimensional space that’s hiding ‘next door’ is exactly the same.”

In “The Shape of Inner Space - String Theory and the Geometry of the Universe's Hidden Dimensions” by Shing-Tung Yau, Steve Nadis

Disclaimer: I don’t believe in String Theory being the TOE. Despite this, Yau’s book gives us a very thorough state-of-the-art compendium on String Theory (and M-Theory) and its connection with Calabi-Yau Spaces.

The current cosmology standard theory (Lambda CDM) is like a table with four legs; cosmic background radiation, inflation, dark matter and dark energy. The problem is, all four "legs" don't stand up to scrutiny. He is also right about cosmologist can't explain the one universe we have, so invest multi-verses and branes etc, which does little more than shift the problem elsewhere. For 200 years everyone thought that Newtonian gravity was correct, until Einstein proved it wasn't. Einstein's theory is also just a good approximation, and needs to be replaced.

Penrose's fashion-faith-fantasy strictures on "string theory" apply also to "black-hole theory", but the latter has its origins in his own topology of a surface-of-separation around the singularity. To some of us, including Einstein, this meant this class of solution is unphysical. Penrose would do physics a great service if he recognised this basic flaw in his early work and looked properly into the solution class with ultra-high gravitational fields replacing the matter-singularity (gravastars and collapsars).

Two battery chickens are chatting. The first says that he believes that humans have the power of life and death over them, and apparently infinite control over their well-being. He also believes that humans not only create amazing music, but care about the well-being of chickens, and indeed of every single chicken in their care. The second chicken tells him not to be so stupid, how can that be when they live in such awful conditions, wings and beaks clipped, force fed, only to be slaughtered and then eviscerated by hideous metal contraptions. I posit that there are only two possible ways out of this paradox. One is that humans either don't exist or don't have much control over the lives of battery chickens. The other will, I am sure, occur to you over time.

It's a question of probability. The odds of a randomly generated universe being capable of generating the stability and complex chemistry necessary for any conceivable kind of life is mind-bogglingly staggeringly small. Therefore our universe is mind-bogglingly staggeringly improbable. It's not "arrogance" to notice this and ask why.

Hawking's original explanation was that there are a near-infinite number of other universe that we can't see or detect in any way, and we just happen to live in one of the exceptionally rare ones that support life. But this is a highly speculative and unsatisfactory explanation precisely because we have no evidence that a multiverse exists. It's hard to imagine a more extreme violation of the principle of Occam's Razor.

How would a young Sheldon Cooper look alike answer this? I thought he'd try asking how a 5D AdS=4 CFT theory could exist in a 4D=3+1 D space as our world appears to be. Rather than fob him off with some slightly patronising "we can also build a 4D AdS theory!" the answer he'd be seeking was related to compactification of the extra 6=10-(3+1) dimensions so a 5D AdS theory can indeed fit into our (theoretically plausible) 10D universe, that appears 3+1 D to us, with some dimensions left over..

Perhaps Lee Smolin's Fecund Universe explains the String landscape. In his model, black holes create new universes, with differing constants of nature, so there is a "selection" effect, where "successful" universes are the ones that make the most black holes. Perhaps the String landscape of different ways to do Calabi-Yau manifolds is reflected in the different universes produced by black holes? I read somewhere these C-Y manifolds can be a way to do Brahms-Dicke theory which as my limited understanding is, is kind of a "generalization" of general relativity, where Newton's constant G is actually a variable. I always enjoy generalizations wherever possible so I like Brahms-Dicke, and the fact that it can be incorporated into C-Y manifolds is even cooler. So if the landscape problem is only real problem, than maybe this can be solved by the Fecund Universe model. However if it is true that the Large Hadron Collider ought to have found super-symmetries (not my area for sure) and they have not, that is more concerning than the landscape issue. Maybe string theory will turn out to be a piece of an even bigger picture which would explain this problem with not seeing super-symmetries. Having a computer science background, I know a bit about Set Theory, so I am naturally intrigued by Causal Set Theory though in fairness I don't know too much about it. Maybe Causal Set Theory can one day replicate in some way the predictions of string theory (or could be "boiled down" so to speak to a string model) but also solve the shortcomings re. Supersymmetry. Would love a video to explain more about Causal Set Theory or other possible candidates out there to solve these issues. There will be no resolutions to any issues until the questions are answered: why are Haag's and Leutweyler's theorems true in Relativity (where c… (més)