It feels great that a field medalist discussing all these topics here.
@ChronusZed
3 жыл бұрын
I'd love to see more colloquium style talks like this!
@x_gosie
3 жыл бұрын
It so fascinating for me how he present these kind of ideas! I'm your new subscriber! Keep the good content, I love it.
@888Xenon
3 жыл бұрын
Excellent video! I fear you've opened yourself up to taking requests for videos now, would love to see some comments regarding polyomino enumeration.
@pamelahernandez6955
3 жыл бұрын
The fact that i more interested in listening to this professor rather than my own mathematics teacher
@Syobnaf
3 жыл бұрын
Suggested topic: Frankl's Union Closed Sets conjecture Love these videos!
@annaclarafenyo8185
3 жыл бұрын
Formalizing the knot diagram can be done by a computer with a scanner, the path of the knot can be easily turned into a marked triangulation of a two-sphere by adding arbitrary points in each connected 2d region and connecting these to the knot and to each other with straight lines. In the resulting triangulation, the knot itself is a simplicial path, a sequence of edges, which intersects itself. Each intersection is then labelled as an overcrossing or undercrossing. There are obvious reductions then which eliminate redundant triangles from the abstract simplicial complex, to reduce the size to a standard form (although obviously non-unique for each knot type). This is not prohibitive although I don't think anyone has written such code.
@maxmuller445
3 жыл бұрын
A problem that remains is how to would you check the correctness of calculations. One would probably need tho formalize them into a sequence of homeomorphisms (and possibly other transformations) and corresponding proofs and check their correctness?
@annaclarafenyo8185
3 жыл бұрын
@@maxmuller445 There is a standard computational formalism for simplicial transformations, they predate and are easier than CW complexes. Nowadays there is even also a computational formalism now for CW complexes, although I am not familiar with it beyond knowing it exists and that the buzzword is "Quillen Model Category".
@tesset8828
3 жыл бұрын
Wow this is very cool.
@maxmuller445
3 жыл бұрын
@@annaclarafenyo8185 Interesting, didnt know about that
@kmill31415
3 жыл бұрын
I've written a program that does some of this, though rather than a simplicial complex the knot is represented as a combinatorial map with degree-4 and degree-2 vertices (a combinatorial map is essentially an oriented surface made of polygons -- no need to restrict ourselves to only triangles). It's a variant on something known to knot theorists as a PD code. It can calculate various invariants, such as the Conway potential and the Jones polynomial. There's also a feature to beautify a knot diagram by collapsing degree-2 edges, subdividing the diagram into squares, and using a spring force model to produce a Tutte embedding. The program's online; look up "KnotFolio." (Eventually I hope to write a new version in Lean where all the transformations and computations are proved correct. That'll be a while though.)
@betabenja
3 жыл бұрын
I subscribed to your videos because they were interesting and clear. I did not know who you were. please forgive me.
@abillyparviz
3 жыл бұрын
Excellent:)
@migarsormrapophis2755
3 жыл бұрын
yeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
@vinkozoric1283
3 жыл бұрын
Him trying his best to explain hard math problems in a simple language. Me 9:42 , Hey, I know that one, that's a puppy...
@tizcoloko
3 жыл бұрын
its clearly a amoeba irritated, puppy needs a tail at least, owww my taco de macisa is now cold thanks buddy to steal mt attention.
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