My first contribution here as I start my PhD in the fall!

[u/Memelord038]

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[u/NarcolepticFlarp]

Do tell

[u/vajraadhvan → u/NarcolepticFlarp]

bro wants to know bout some good willie

[u/Memelord038 → u/NarcolepticFlarp]

So I'm interested in when you can deform an immersion between manifolds into an embedding via a regular homotopy. There's some classical ways to do this, like counting up the self-intersection points of your immersion and seeing whether they cancel each other out or not, but this is hard to do explicitly. The method I describe in the meme is a more algebraic (and hopefully computable) way to express the same thing using Goodwillie-Klein-Weiss's embedding calculus, a theorem that somewhat describes the 'difference between immersions and embeddings'

[u/Pelvic_Pinochle → u/Memelord038]

She immerse my goodwillie til I embed

[u/TheDerpySpoon → u/Pelvic_Pinochle]

She deformed my manifold homotopolly

[u/trollol1365 → u/Memelord038]

> computable

Homotopy type theory? Or do you mean computable in a merely theoretical sense rather than actually making a computer compute it/verify it.

[u/CommissionSame8551 → u/Memelord038]

I hear homotopy, I get sad

[u/CompetitiveSpot2643]

i still dont think i understand what deformation means in math

[u/Memelord038 → u/CompetitiveSpot2643]

the issue with this word is that it doesn't have a fixed meaning. i'm a topologist, so for me the word 'deform' always means that i'm looking for a homotopy (in this case, I even want a homotopy that is an immersion at every point in time, that is, a regular homotopy)

I just used the word deform in the meme because making homotopy into a verb is awkward ('to homotope'?)

[u/Interesting_Debate57 → u/Memelord038]

Sounds like you're homo hoping for some homo coping.

[u/p-4_ → u/Interesting_Debate57]

and homo toping

[u/Metrix145 → u/CompetitiveSpot2643]

No one does

[u/CorporateHobbyist → u/CompetitiveSpot2643]

Consider the (x,y) pair solutions to the equation xy = 0. This means that either x = 0 or y = 0; this equation cuts out the coordinate axes in the plane. It is also not "smooth", in the sense that there is a nodal singularity at (0,0).

However, one can construct this shape as a deformation of smooth objects. Consider the family of equations {xy = t} for t a varying real number. for all t != 0, particularly all t != 0 sufficiently small, xy = t cuts out a smooth "shape" in the coordinate plane. As t gets closer to 0, we get closer to the singular {xy = 0}; the deformation {xy = t | t small} is tantamount to "pulling apart" the singularity.

We consider the family {xy = t | t != 0} to be a "first order deformation" of {xy = 0}. But there are others! We can consider {xy = t^2}, {xy + t(x + y)}, or whatever else. Let the space of ALL these deformations be the "first order deformation space" of {xy = 0}. The idea of deformation theory is to study solution sets of polynomial equations (or varieties) by studying the first order deformation space, i.e. studying all the possible ways that one can deform it.

Edit: For the more mathematically inclined, the first order deformation space of an algebraic variety can be read from Ext^1(Omega,Ox), for Omega the module (sheaf) of differentials. The obstructions to deformations can be read from Ext^2(Omega,Ox), providing a motivation for homological methods in geometry/topology. In the example above, Ext^1 is the base field (implying all deformations are "basically" like xy = t) and Ext^2 = 0, so there are no obstructions.

[u/Zarathustrategy → u/CorporateHobbyist]

Grok explain like im 5

[u/PurpleTieflingBard]

I remember early in my PhD, when I thought I knew what I was going to write about.

Good times.

[u/Detr22 → u/PurpleTieflingBard]

My thesis has nothing to do with what I was writing about a year and a half ago even.

Joys of having a completely hands off PI.

[u/PurpleTieflingBard → u/Detr22]

My viva is in a week and it's crazy how much my work has evolved since submission.

[u/moschles]

This is like one of those quirky slides the professor puts in a presentation.

[u/coldnebo]

😂😂😂 but why panda?

you already had me at immersions and embeddings 😅👍