mathematicalrosebudthorn

[mathematicalrosebudthorn]

2 april 2019

Rose:

rational cells = infinite mapping cylinders

Thorn:

whitehead manifold is contractible

Bud:

to extend map homotopically, one needs to imagine its image on the target

[mathematicalrosebudthorn]

31 march 2019

Rose:

even if one rescales Nijenhuis norm, not obvious that there is some J that realizes the new Nijenhuis norm with its own nijenhuis tensor

projection formula

Thorn:

espace etale has discrete fibers ... picture of a sheaf

scalar curvature is a function (trace of Ricci), Ricci curvature is a 2-tensor (trace of sectional) ... contract two vectors into sectional, take trace of matrix from bilinear form B(e_i, e_j) over all j. dim x dim matrix

Bud:

Hopf fibration

[mathematicalrosebudthorn]

15 march 2019

Rose:

J[X,Y] = [JX,Y] if X, Y holomorphic therefore DD is a well defined complex subbundle (since J-invariant)

Thorn:

calabi eckmann manifolds have complex tangent bundles that don’t split?

Bud:

almost complex manifolds are (almost) complex

[mathematicalrosebudthorn]

9 march 2019

Rose:

dual of taut is k-ample

no subbundles of tautological quotient over flag (pull back over fiber)

Thorn:

leray spectral sequence

a map corresponds to complete linear system iff projective space has dim(global sections) - 1

Bud:

projections are well defined, given an orthogonal complement

[mathematicalrosebudthorn]

6 march 2019

Rose:

tautological of n-planes in n+k space has no subbundles if n <= k by characteristic class argument

Thorn:

microflexibility

O(1) and O(-1) are complex duals ie Hom_C(, C)

a hermitian metric identifies V with V* BAR 

Bud:

euler class

[mathematicalrosebudthorn]

5 march 2019

Rose:

k-ample for k = (r-1)(s-r)

map to projective space is projection to point

straightening diffeomorphism will make you lose epsilon-closeness, since |df(x) - F(x)| < epsilon does not mean |df o h^-1(x)- F(x)| is less than epsilon; h could send x far away from where it is close with F(x)

however will remain transverse to D, immersion, totally real, etc. since those are conditions on planes rather than points

Thorn:

microflexible

subbundle of tautological

Bud:

sections = maps to fiber

[mathematicalrosebudthorn]

3 march 2019

Rose:

regular integral curve

Thorn:

what is eliashberg logic in argument for h principle of contact transverse?

microflexibility logic

Bud:

obstruction theory

[mathematicalrosebudthorn]

2 march 2019

Rose:

can push around whole Lie algebra germs :) therefore D is bracket generating

Thorn:

assuming tautological quotient of 3n in 4n has no subbundles :L

Bud:

difference cochain is a homotopy element since f, g FIXED and agree on lower skeleton

[mathematicalrosebudthorn]

27 february 2019

Rose:

how to use openness + microflexibility to get microflexibility of whole relation? 

Thorn:

D is bracket generating

Bud:

Lie group action is by diffeomorphisms (inverses exist)

[mathematicalrosebudthorn]

25 february 2019

Rose:

sommese theorem true without smoothness assumption on zero locus :)

no formal obstructions up to n < 77

holonomic homotopy parameter = points nearby is due to derivatives being defined on open sets

Thorn:

D bracket generating

“span vector bundle” = vector fields span vector space at each point

Bud:

obstruction theory

[mathematicalrosebudthorn]

24 february 2019

Rose:

Sommese assumes B compact

but relative homotopy vanishing theorem for E k-ample and globally generated ... which we have

Thorn:

D is bracket generating

Bud:

obs theory -- if target fibers already homotopic, obs well defined

need local trivialization for obs to be well defined (up to homotopy)

[mathematicalrosebudthorn]

23 february 2019

Rose:

microflexible : S_0 and S_epsilon agree on ends of band, so one can interpolate (bump picture with singularity). .. weird that homotopy parameter = holonomic sections nearby

Thorn:

D is bracket generating

Bud:

J invariant subspace exists

[mathematicalrosebudthorn]

22 february 2019

Rose:

I is not smooth; it is not homogeneous

Thorn:

microflexible extension ... why does s_0 and s_epsilon agree on band?

D is bracket generating

Bud:

pi_1 of (X - codim 2) = pi_1 (X) :) 

[mathematicalrosebudthorn]

21 february 2019

Rose:

modify relation to be totally real in horizontal direction; locally integrable and microflexible should still go through, since open condition

Thorn:

how to glue vector field extensions together? go through whole holonomic approximation argument inductively? yes

how to holonomically interpolate for something microflexible?

is I homogeneous? no D bracket-generating?

Bud:

not every cell in k-skeleton is a wedge of k-spheres or bounding

[mathematicalrosebudthorn]

19 february 2019

Rose:

MICROFLEXIBILITY = GLUE LOCAL SOLUTIONS TOGETHER

seen as extending holonomic homotopies for a little bit of time (restriction maps of sheaves are serre fibrations)

use this + interpolation to patch together global solution 

Thorn:

why is I homogeneous

therefore why can we define subbundle using group action

Bud:

e(E) = e(F)e(B) for any fibration (CW complex)

contractible manifold V immersed into W has no obstructions to normal line field; therefore can immerse V x \R into W

[mathematicalrosebudthorn]

4 february 2019

Rose:

D is totally non-integrable; its quotient /complement is a universal bundle therefore no subbundles (both real and complex)

Thorn:

why is I homogeneous / GL(n,C) acting on the vector to give a subbundle in complement

microflexible / locally integrable => holonomic approximation

Bud:

OBSTRUCTION THEORY

[mathematicalrosebudthorn]

1 february 2019

Rose:

obstructions to dyamical systems

Thorn:

why doesn’t argument prove general h-principle for transverse distributions?

why is D totally non-integrable?

implicit/inverse function theorems

Bud:

littlewood’s three principles