Ravi Vakil: Algebraic geometry and the ongoing unification of mathematics

Abstract:
I will try to share a glimpse of this strange unification of many different ideas. This talk is aimed at a general audience, and no particular background will be assumed.
When we look carefully at nature, we can discover surprising coincidences, which suggest deeper underlying structure. The centrality of mathematics comes in part from the fact that seemingly unrelated ideas are often unified by some grand theory, which is far more powerful than the sum of its parts. Mathematics is most exciting when different ideas come together unexpectedly to give a new point of view. This is typified in algebraic geometry, and in the work of Deligne in particular, which brings together many themes in mathematics, including geometry, number, shape (topology), algebra, and more. This magic is the reason I became an algebraic geometer. For example, the theory of Pythagorean triples (such as ) connects geometry to the theory of numbers by way of algebra. This ancient example grows up to be the Weil conjectures, a wondrous prediction whose proof was finally completed by Deligne.
This lecture was given at The University of Oslo, May 22, 2013 and was part of the Abel Prize Lectures in connection with the Abel Prize Week celebrations.
Program for the Abel Lectures 2013
1. "Hidden symmetries of algebraic varieties" by Abel Laureate 2013, professor Pierre Deligne, Institute for Advanced Study, Princeton
2. "Life Over Finite Fields" by professor Nicholas Katz, Princeton University
3. "Mixed Hodge structures and the topology of algebraic varieties" by professor Claire Voisin, École Polytechnique and CNRS
4. "Algebraic geometry and the ongoing unification of mathematics", a science lecture by professor Ravi Vakil, Stanford University

[@ginomorales8989]

Acording to Fernando Zalamea, Grothendieck did not like the approach of Deligne. He considered it like a betrayal to his own vision on the subject. It was splendid, obviously, but Grothendieck considered it artificial, like a very ingenious and complex trick, not quite encompassed in the big scheme of fine topologies, topos, ramifications and all that. Like being Connes, spending decades developing noncommutative geometry, hearing that some of his students just solved the RH in two lines using undergraduate complex analysis.

[@McRingil]

Thank you fot the comment, where did you read this stuff?

[@ginomorales8989]

@@McRingil Hi, the book Grothendieck, by Zalamea (in spanish) says in p.259, fn.399: "The path adopted by Deligne (...) is completely distinct. Grothendieck never accepted that proof, to his viewpoint artificial, far from the natural road offered by the standard conjectures (...). That mathematical sadness is one of the diverse reasons (...) that made him go away from Deligne and the mathematical community." Zalamea is being polite describing this. You and anyone interested in the story can go to Recoltes et semailles, part two ("The burial"), by Grothendieck himself, where in section B ("Pierre and the motives") he explains in detail his "not quite positive" opinion about Deligne.

[@dronevlogsmath]

@@ginomorales8989 is the book available in English?

[@hambonesmithsonian8085]

What do you mean “solved the RH?”

[@Israel2.3.2]

Really interesting comments on Grothendieck towards the end. "Not quite human."

[@u.v.s.5583]

Most mathematicians are almost human almost everywhere, but Grothendieck was weird even for a mathematician, but in sporadic cases he was hyperweird.

[@ginomorales8989]

I heard once a quote describing him as "a student of a primary school from an advanced galactic civilization visiting us". Somehow related to this, to the end of his life Grothendieck believed that an entity from outer space visited him and revealed him a message of truth and peace, described in "La clef des songes", a very long manuscript where he applied his mathematical ideas to his dreams in order to understand consciousness.

[@MrNoremac99]

For those wondering how he knew all the other digits of the speed of light, it’s defined as exactly 299 792 458 m/s. In fact, the metre is defined in terms of the speed of light, rather than the other way around, which is what gives us this nice round number.

[@adrianwright8685]

A more sensible and convenient definition might have been 300,000,000 m/s but I suppose this would have made the metre 0.07% smaller than it used to be!

[@ginomorales8989]

@@adrianwright8685 c=3•10^8=pi•g^8, according to an engineer friend of mine

[@adrianwright8685]

@@ginomorales8989 c is approx 3.10^8 but exactly 299792458 m/s as this comment made clear. ( pi.g^8 ? )

[@Idk-hp3oo]

@@adrianwright8685 its a bad Engineering joke : pi=3 and g=10(instead of 9.81 m/s^2 (which is the acceleration due to gravity on earth without any air

[@Chalisque]

As for 3*10^8, I would imagine that that would mess up with research which relied on the pre-existing value of c. In 1982, I imagine the value of c was known to more than 3.s.f, so research prior to the 1983 redefinition of the metre would make calculations assuming the error in the value of c was smaller than 3.s.f. If you made c=3e8, then many of those calculations would need to be re-done.

[@ravivaradhan4956]

One can feel the deep devotion that Vakil has for Grothendieck @ 33:36

[@pairadeau]

Marvelous lecture. Bless.

[@swavekbu4959]

The "light" that turns on when "everything makes sense" is not restricted to mathematics, it can occur in any discipline or hobby. Musicians, for instance, experience that too after so much time and experience with their instrument.

[@olenaerhardt7725]

Thank you for this spectacular lecture. I've never thought about unification in Mathematics, I thought it is only in Physics, when we are trying to unify 4 forces. The sound is very low though, something should be done about it in the future. Thank you again.

[@travelgalaxy8291]

great teacher

[@pectenmaximus231]

Argh. Each time I want the talk to settle on one of the incredible points raised, we’re already veering off somewhere else. I guess it’s an awful lot of interesting stuff to put in a single talk but dang. Guess this is my AG apperitif

[@KaliFissure]

And what is the distribution of Pythagorean triples on the circle? Is the density even or shaped?

[@wesleyfortney2339]

Even. Every tangent is being made the hypotenuse of a right triangle with the diameter as leg. An interesting analysis would be the ratio of length of the diameter leg it's perpendicular, as the tangent is shifted through all positions. Like a clock hand that ticks centered on the circumference. The cool thing about math is that the calculation would not be needed for the entire circumference: operational formula is used to flip coordinates &/or determine coordinates for quantized patterns. The circle is mathematically considered a pattern of infinite differentiation, pi has the never ending remainder as demonstration. By bounding analysis functions with operational limits, calculus translates math into materially functional results, meaning: material & operational precision is defined for functional processes. Machined tolerances is the best example for material analogy: the precision requirements are dictated by tool capabilities & material characteristics, which are determined & integrated as limiting factors for engineering design & processes as well as the parameters for calculations required by control software. Hehe

[@NothingMaster]

When the universe (of anything, numbers included) comes in chunks--whenever you could identify distinct objects (real or conceptual/concrete or abstract)--then space, time, points, digits, sets, curves, vectors, tensors, geometries, fields, shapes, smoothness, patterns, operators, functions, constants, graphs, relations, operations, actions, evolutions, conservations, invariants, laws, symmetries, transformations, and even the continuum, the imaginary, the transcendental, the boundless, and the transfinite are all bound to happen. Without the ‘chunks’ only nothing is possible. At the heart of it all is the question: What makes it all chunky; why the quantum universe (not just of matter and dimensions, but also of thought, numbers, and the graininess of imagination)?

[@user-cb9gg4zp7o]

A fundamental difference between the speed of light and pi is that the speed of light is a dimensionful quantity, so its numerical value depends on the units used and the only values that don't depend on the units used is c=1, c=infinity and c=0. c=1 is relevant for special relativity, c=infinity describes the non-relativistic approximation and c=0 is the so-called Carrollian limit. One can choose units in each case so that the speed of light takes the appropriate value for each of the three cases. Whereas pi is a dimensionless quantity, its nvumerical value does not depend on any convention (units are conventions, dimensions are not). That's why the digits of pi have a significance that the digits of the speed of light don't have.

[@meofamily4]

It stopped being comprehensible at 22 minutes and 15 seconds. Of course that's more testimony to my own limitations than to the clarity of the lecture. I have no way to see how the real solutions ought to lie on a closed curve in two-dimensional space. Let alone where the complex plane (well, perhaps four-dimensional space, I'm not sure) came from.

[@qart-hadasht]

We want integer solutions of x^n + y^n = z^n, this is roughly equivalent to rational solutions of (x/z)^n + (y/z)^n = 1. So the solutions we are looking for should be rational points (rational coordinates) on the graph of X^n + Y^n = 1 with X=x/z, Y=y/z. The graph of X^n + Y^n = 1 is just a closed curve in 2D space (try graphing it). Where does the complex plane come from? Well, if we look for complex solutions of X^n + Y^n = 1, that is just the graph of X^n + Y^n = 1 with X and Y both complex numbers, we get a curve in C^2 (curve because one dimension in C in a C^2 space, if you insist on sounting dimension using R instead, you get a surface in C^2=R^4 space but this not how dimension is defined in alg. geom). Why did we do this? Believe it or not, finding complex solutions is easier than finding rational solutions. In very vague terms, if we look at all complex solutions, then problems become essentially geometric and are 'easier' but when you look for rational solutions things are harder because there's algebra and arithmetic shenanigans in addition to geometry that you have to deal with. Here's an easy example that should provide some intuition: What are solutions to x^2 + y^2 = 1? Over reals it's easy to compute: it's all points on a circle so (cos(t), sin(t)) for t from 0 to pi. What are solutions over rationals? (The answer is not so easy, but still doable, look up Pythagorean triples)

[@Micky_Tick]

Does someone know from where the quote from André Weil at 36:40 is from ?

[@colinmclarty8082]

André Weil, «De la métaphysique aux mathématiques», Œuvres, t. II, p. 408.

[@KaliFissure]

If a black hole event horizon is the exterior membrane of a true perforation in spacetime doesn’t that mean that our manifold must itself have at least one perforation? Which means our universe is not a hyper sphere but a hyper toroid but because gravity it must have lopsided manifold density. 3d cardioid? The navel (event horizon) mapping onto an entire area on the least curved portion of surface. Is this loop quantum gravity? The neutrons which invert at moment of collapse of neutron star into black hole are accelerated by gravity to c. They emerge in least dense regions, decay into hydrogen and do the journey again. Diffuse hydrogen>nebulae>galaxies/stars>neutron star>black hole. The inversion of the circle.

[@Anonymous95202]

seek psyciatric help

[@KaliFissure]

Instead..... I did some geometry study since that post and found the radially symmetric Klein bottle which solves all the topological issues of the manifold and gives us neutron decay cosmology. A homeostatic universe maintained by the reciprocal processes of electron capture at event horizons and free neutron decay in deep voids. DM is decayed Neutrons DE is the expansion caused by that decay from 0.6fm³ neutron to 1m³ of hydrogen gas.

[@KaliFissure]

Because primes have no clean divisors this makes each of them a new unity. They have no harmonics yet. They are sine/root.

[@taibanganbakonjengbam6902]

👍👍

[@milanstevic8424]

....aye aye aye aayy....

[@xyzct]

It's such a bizarre mannerism; talk really fast, but stutter in search for basic, everyday words, thereby taking three times as long as necessary to say the simplest thing.

[@forheuristiclifeksh7836]

9:11

[@shivb707]

Lit

[@diktakt1187]

20 07 Tricks

[@VeteranVandal]

I think saying pi is something empirical, at least up to a very precise (but not arbitrarily precise) sense, is completely valid - not that it matters for your argument, to be honest, but I'll tell you that pi isn't so much more abstract than the speed of light. It's obviously true that we don't need to determine pi physically or empirically, but we sure CAN do it, and it appears so frequently that this means that pi is fundamental to our world, physically, beyond what a lot other numbers can. Mathematics abstracts the measuring part out of it, but that doesn't make the number itself less important. In fact, if the metric we experience was completely different, then it wouldn't be empirical and likely not as important. A circle and a sphere are inherently empirical, while a hypersphere isn't. It doesn't matter pi relates to all of them, what matters is that pi matters way before the abstractions come in. Pi is a completely material, empirical object, so much so we stumbled on it EXACTLY to solve practical problems, instead of abstract ones. Now, if you want to say other numbers are purely abstract, that I can get behind. Or that we don't need to use empirical methods to determine it. I also disagree with Wigner. I think it's perfectly reasonable that a language can explain something. What is sometimes surprising are the coincidences that are unexpected, and that are, at times, very hard to connect or to discover. That's usually unreasonable.

[@KaliFissure]

Pi has to be irrational so that a wave traveling in space doesn’t self resonate and destroy everything. A wave traveling through the diameter will be out of phase (to an inharmonic way) to that wave traveling on the surface.

[@Robert-er5wq]

mph by a Canadian at a European university...

[@vtrandal]

At 2:50 he is quoting Andrew Wiles.

[@LeTerrarien]

Thank you!

[@kenichimori8533]

Abelian define lecture.

[@vremiavremiavremiavremiasasi17]

🇷🇺🇷🇺🇷🇺🇷🇺🇷🇺🚩🚩🚩🚩🚩🚩🚩🚩❤️❤️❤️❤️❤️❤️❤️❤️❤️🤗🤗🤗🤗

[@kenichimori8533]

Solution = Y = 0

[@kenichimori8533]

Twin Algebraic Geometry 0 = 2y = = x2

[@ass09tube]

This guy is smart, competent mathematician. But he is not at level of Manjula Bhargava, Akshay Venkatesh, Vardhan, Vardarajan n others. 12 years back, I remember vividly on his Stanford web page, he wrote : if you are from Stanford n email I will respond, else not. That summed it up for his mindset : any query, email from someone outside Stanford is beyond range of existence for Prof Ravi Vakil. He was associated with Barry Mazur n some other greats but he has been concerned with brand and lobbying rather than deep research. I consider his best is history. Talented n promising till 2013 but disappointing thereafter.

[@vaibhavkrupakar240]

Same can be said for Kiran kedlaya, but this guy is still a top mathematician in algebraic geometry Yes he is no match to other indian greats like manjul bhargava, Akshay venkatesh, harish Chandra, SR Srinivasa varadhan, and most recently Bhargav bhatt

[@AspartameBoy]

Sad he said little about AG

[@ass09tube]

@@sergelangfan your reply is nauseating ..

[@StevenNess]

quite a few issues with this lecture and flaws of understanding of science. should do more listening and less talking.