braids, morse theory,

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braids, morse theory, parabolic pdes

• robert ghrist
• department of mathematics
• university of illinois
• urbana-champaign, usa

may 2005
joint work with rob vandervorst mathematics vu
amsterdam
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the many emanations of morse
(in low-dimensional topology include...)
lagrangian floer theory
contact homology / symplectic field theory
heegaard floer homology
today a forcing theory for the dynamics of
parabolic pdes
a simpler setting in which to see some similar
phenomena...
inspired by problems in applied pdes
reaction-diffusion eqs...
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parabolic pdes
consider uniformly parabolic pdes on circle
ut uxx f(x,u,ux)
assumptions
periodicity x in S1
regularity f satisfies blah, blah, blah to
preclude blowups
we care about bounded solutions u(x,t) which are
either
stationary (t-independent)
periodic (in t)
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a flow on space of curves
we will consider solutions u(x,t) as a
time-varying graph
u
x
multiple solutions evolve independently (product
flow)
curves with integer period are permissible (lift
to a cover of the circle)
to a topologist these curves look like...
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braids
lift to the 1-jet extension of the graphs in
(x,u,ux)
u
x
assuming no tangencies, the graphs lift to braids
which are
closed (since on S1)
positive (since these are holonomic or
legendrian)
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the comparison principle
what happens at tangencies between strands?
assume u(x,t) v(x,t) and ux(x,t) vx(x,t),
some x,t
(u-v)t ut - vt
uxx f(x,u,ux) - vxx - f(x,v,vx)
uxx -vxx
tangencies resolve according to curvature, or,
linking
several authors (angenent, fiedler, matano) have
used the number of intersections of graphs as a
weak gradient function (lap numbers)
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a cartoon
Cn
u
Cn space of all n-strand holonomic periodic
curves
Sn space of singular curves (exhibiting
tangencies)
braid class u path component of u in Cn-Sn
comparison principle the dynamics on Cn is
transverse to Sn
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notation
Cn holonomic curves on n strands (periodic,
positive)
Sn singular curves in Cn (tangencies)
Cn-Sn nonsingular braid diagrams
Sncoll collapsed singular braids (image in Ck
for kltn)
lemma the product flow of any parabolic pde on
Cn is transverse to Sn - Sncoll in a direction
which decreases the word length of the braid
tangencies to all orders separate immediately
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problem
all braid classes border Scoll
this comes as no surprise, since we know we
cannot have a theory which works for all
parabolic pdes, some of which heat are too
trivial...
definition a relative braid class is the set of
braid pairs uv (u,v) (u,v) is
isotopic to (u,v) fixing v
these will be used in forcing theory v
skeleton of stationary curves
we avoid the above two problems by focusing on
relative braid classes which are proper
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proper braid classes
uv is proper if no free strands can collapse
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forcing via a braid index
idea given the braid class of a skeleton of
stationary curves, v, build a relative braid
class u v with free strands u
if uv proper, the braid class has boundary
transverse to the flow
this has the flavor of a morse theoretic
scenario... there should be an index whose
nonvanishing is indicative of a solution...
what is the appropriate notion of an index for
this system?
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morse theory a la conley...
S0
S1
S2
µ 2
µ 1
µ 0
morse index dimension of unstable manifold
(gradient, nondegenerate)
conley index homotopy type of unstable set in
an isolating block
B isolating block
conley index of B h(B) B/E,E
(pointed homotopy type)
works for any type of vector field and forces
invariant sets in B
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the homotopy index for braids
theorem to each proper relative braid class
uv there exists a well-defined conley index
h(uv)
topological invariant independent of the choice
in uv
idea of proof
discretize the x-coordinate and work in the
category of piecewise-linear braids, fixed
period
i
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the homotopy index for braids
Cnd space of period d piecewise linear braids,
n strands
(homeomorphic to Rnd)
Snd space of period d piecewise linear singular
braids
(separates Cnd into discretized braid classes)
for fixed period d, we consider Bdudvd
Edsingular braids on ?Bd of decreasing
length
definition h(udvd) Bd/Ed,Ed , as in the
conley theory
question what happens under refining the
discretization of uv?
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stabilization theorem
theorem the homotopy index h(udvd)
Bd/Ed,Ed stabilizes
d gt length of braid in word metric suffices...

• show that for d gt word-length of braid, discrete
  braid equivalence
• corresponds to topological braid equivalence.

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stabilization theorem
theorem the homotopy index h(udvd)
Bd/Ed,Ed stabilizes
d gt length of braid in word metric suffices...

• show that for d gt word-length of braid, discrete
  braid equivalence
• corresponds to topological braid equivalence.

2. define stabilization operator ECnd ? Cnd1
inserts trivial period-1 braid at end of the
braid
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stabilization theorem
theorem the homotopy index h(udvd)
Bd/Ed,Ed stabilizes
3. place parabolic lattice dynamics on Cnd
d(ui)/dt Ri(ui-1,ui,ui1) where Ri is
semi-monotone
?Ri /?ui-1 0 and ?Ri /?ui1 gt 0
show that (Bd,Ed) is well-behaved for any choice
of Ri
4. choose the right dynamics to analyze action of
ECnd ? Cnd1
Rd (ud-1,ud,ud1) (ud1 - ud)/e
when e?0, the space Cnd1 collapses to the image
E(Cnd)
fact conley index is well-behaved under singular
perturbations
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back to parabolic pdes
theorem let v set of stationary solutions and
let uv be a (bounded) proper braid class. If
h(uv) is nonzero, then there exist
periodic/stationary solutions in the braid class
uv.
proof discretize the x coordinates and show that
the dynamics of the parabolic recurrence
relation converge as the step size goes to zero.
lots of extensions to multiplicity and type...
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a simple example
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a simple example
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a simple example, extended
take a cover of the S1 domain and lift skeleton
weave free strand through the braid
these project down to multiple-period curves of
nonzero index
exponential growth in stationary solutions of
fixed period
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exact, dissipative pdes
things are much cleaner when the equations are
exact
uxxf(x,u,ux) a(x,u,ux)d/dx(?uxL - ?uL)
some L
as the dynamics are gradient
all periodic solutions are stationary...
we also consider dissipative equations
u f(x,u,0) ? -8 as u ? 8
all braid classes are bounded...
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general forcing theorem
theorem any exact, dissipative equation which
has stationary solutions in a nontrivial braid
must possess an infinite collection of
stationary solutions of distinct braid types.
ingredients of proof
beginning with skeleton, add free strand and
compute index
add the free strands to the skeleton and repeat
take covers of the base circle lift, compute,
project
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time-periodic case
we can apply a similar method to understanding
systems with time-periodic solutions
(rotating waves, etc.)
theorem consider the equation ut uxx ug(u)
uxh(x,u,ux) where
1. g(0)gt0 and g has at least one positive and one
negative root
then there exists an infinite set of
nonstationary time-periodic solutions, all in
different braid classes.
example ut euxx u(1-d2u2) uxu2
shown by angenent-fiedler to have rotating waves
for e small
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a few generalizations
1. different boundary conditions
2. inhomogeneous curvature terms
ut f(x,u,ux,uxx) with uxx terms bounded from
0
3. degenerate parabolicity
ut (uxp-1ux)x g(x,u,ux) p-laplacian
4. improper braid classes
what does this braid index mean? what topological
content?
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a discrete duality
for discrete braids of even period, there is a
notion of a dual braid...
D
this has the effect of reversing the direction of
the flow on singular braids
theorem for braids of period 2p, the homology is
2np-dual Hk(h(DuDv)) H2np-k(h(uv))
(n free strands)
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shifting and twisting
this duality gives a slick proof of the following
theorem appending a full positive twist to a
braid pair shifts the homological index by
degree 2n (n free strands)
proof given discretized braid pair (uv) with
period 2p, we can append a full twist by acting
via DE2D for sufficiently high discretization
period
D
E2
D
we could also prove this on the homotopy level
(multiple suspension)
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beyond legendrian braids
is it possible to extend the index to
non-positive braids?
garsides theorem any braid has normal form ---
a positive braid followed by some number of
(full) negative twists
idea for an arbitrary braid pair (uv), define
the homology index in terms of the garside normal
form then shift the homology backwards by the
number of twists
conjecture this is a well-defined relative floer
homology for braid pairs which agrees with our
index for positive braids and gives a forcing
theory for hamiltonian systems