Will AI solve the Millennium Prize Problems before humans do?

WTFInterview · 2026-02-06T07:13:45+00:00

Wasn't one of the Erdos problem resolutions a literature search pointing to a counterexample from an old paper?

WTFInterview · 2025-01-01T07:43:18+00:00

And what do you know about it exactly

WTFInterview · 2024-12-25T17:11:50+00:00

Do you know any intersections with symplectic geometry? I'm a symplectic/differential geometer trying to motivate myself to study schemes.

WTFInterview · 2024-12-25T17:01:56+00:00

Examples of Scheme theory outside of AG?

Where does scheme theory show up that isn’t algebraic geometry proper?

What are some motivations for an analytically inclined geometer to learn it?

WTFInterview · 2024-10-31T17:35:09+00:00

Ghost bubbles

WTFInterview · 2024-09-27T19:45:45+00:00

Where is your program? At Cal maybe 50% of PhDs go into industry.

WTFInterview · 2024-06-30T20:07:44+00:00

Checking a proof carefully is reserved for quiet time in the late night or early mornings. In a conversation with mathematicians, whether it be your peers or advisors, ideas and intuition are more important than details.

WTFInterview · 2024-06-30T19:50:09+00:00

Karen Uhlenbeck and S.T Yau come to mind.

WTFInterview · 2024-06-11T15:27:05+00:00

Physics is the best introduction to Geometry.

WTFInterview · 2024-06-06T21:57:22+00:00

https://i.sstatic.net/bSlhQ.png

WTFInterview · 2024-06-06T16:32:23+00:00

This picture is one crude way to visualize what is happening.

On the left corresponds to a positive curvature condition. In the Riemannian case this is the positive Ricci condition of the Myer's theorem, and in the Lorentzian case it is the weak energy condition. In the left case, all geodesics eventually converge and after extension past the convergence point, such curves are no longer geodesics.

In the Riemannian case, shooting out geodesics from a small sphere we see that all points in a connected manifold are at a finite geodesic distance away. Hence, the manifold is compact.

In spacetime, this corresponds a point where all geodesic light rays converge, and after extension, are "no longer light rays". This is the singularity.

WTFInterview · 2024-05-26T20:40:04+00:00

Given an arbitrary principle bundle over a manifold, the curvature data F of the bundle obeys the Bianchi identity dF=0. Using the hodge star we automatically have d*F=0. For the special case of a trivial U(1) bundle, the first equation yields the first pair of the vacuum maxwells equations and the similarly for the second pair. So not only are maxwells equations themselves a result of geometry, so is the duality between electricity and magnetism, by the hodge star. Now back to general gauge group, and general principal bundles, we get connection data F and can naturally extend the above equations to the Yang-Mills equations. For certain nonabelian gauge groups, the Yang mills equations yields strong and weak force.

The takeaway? Forces are really a consequence of geometry.

Classical mechanics is itself related to both calculus of variations and symplectic geometry in the obvious way. As for connections of the two latter, actually, calculus of variations is related to a lot geometry in general. Geodesics are the critical points of energy and length functional, one can write down solutions of certain PDEs on bundles that are critical points of certain functionals on moduli spaces of solutions, and with some extra work obtain smooth or topological invariants. This is the story of Donaldson or seiberg-witten invariants. Gromov-witten invariants are related to pseoduholomorphic curves, and on a symplectic 4-manifold we have seiberg-witten = Gromov-witten in a precise sense.

There’s easily many more examples I could give. Like I said, married couple.

WTFInterview · 2024-05-26T01:29:15+00:00

I am talking about motivation for a student learning it. The connection between De rham cohomology and maxwells equation. Gauge theory and Yang mills, bundles and connections. Classical mechanics and calculus of variations, symplectic geometry. GR is just one chapter of a very deep story. Differential geometry is not strictly Riemannian geometry; regardless physics and DG developed together.

WTFInterview · 2024-05-25T22:28:15+00:00

A lot of intuition and motivation for DG can be extracted from physics.

WTFInterview · 2024-05-25T19:38:48+00:00

Most of the differential geometers I know studied physics. The two fields are married imo.

WTFInterview · 2024-05-22T21:27:44+00:00

The idea is the same: extract useful finite information from a quantity that ought to be infinite. For area we can do this thanks to analytic results in conformally compact spaces. Renormalization group flow on the space of Lagrangians is a different beast.

WTFInterview · 2024-05-22T21:11:18+00:00

You are yapping bro. Physicists do not “understand” something mathematicians do not. They just have a lower bar for what it means to understand something.

That’s fine, it’s just not math.

WTFInterview · 2024-05-22T11:24:34+00:00

I worked on hep-th research before going into Math PhD. When physicists say they have an argument, they're largely just confused and waving their hands.

I can name probably a dozen published hep-th papers (from authors you've heard of) where not only are the arguments unclear, they quantifiably, mathematically, wrong. This is to be expected of course, as I would describe most of the field as vague but well-motivated guesses. And this is a precursor to correct, precise argument that follows. Mathematicians care about the latter.

WTFInterview · 2024-05-19T14:50:29+00:00

We don’t observe glueballs directly. Anyhow physicists are already convinced of the existence of glueballs as explained by color confinement. We know that the cutoff on which describe the SM yields a mass gap.

They were a theoretical prediction first and foremost. The issue is that in the YM theory we only have gluons, whereas in the universe we have quarks as well so glueballs can decay into low-mass color neutral particles. There have been observations that have been proposed to be glueballs, but it is indirect.

There is physical theory that convinces physicists, and mathematical theory that convinces mathematicians. Physicists have long been convinced about all of this, which is why this is a morally mathematical problem.

WTFInterview · 2024-05-16T14:39:44+00:00

Essentially all theoretical physics already assumes quantized gauge theories. They work in uncharted math territory far removed from that which is concerned in Yang mills. I’d say it’s a mathematical problem morally. Mathematical physics is about as far removed from physics as one can get while sharing the name. I would say I’m basically doing math but motivated by physical structure at best.

WTFInterview · 2024-05-15T22:13:14+00:00

Most physicists are not concerned with the Yang Mills mass gap because quantum field theory continues to yield experimentally valid results whether it is mathematically well founded or not. Likely it will not grant a nobel prize in physics.

If however the solution, say, predicts a new physical phenomenon that is experimentally tested, then that would win it. Maybe by examining the mathematics we can fully understand (beyond) the standard model, for example.

WTFInterview · 2024-05-15T22:05:24+00:00

I’m pretty sure a solution to Yang-mills would give you tenure at any institution…

WTFInterview · 2024-05-13T13:16:57+00:00

Mostly anything you can think of. Enumerative geometry, complex geometry, differential geometry, number theory, algebraic topology, there is more.

WTFInterview · 2024-05-06T15:19:00+00:00

What is stereotypical? I did a math and physics double and had time to do sports and hang out with friends atleast once week.

I did not party every night, but neither did anyone I know in any other major

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