| galathaea 2007-05-08, 4:09 am |
|
in any category C
a flow is a pair (X, t)
where X is an object of C
and t:X->X a morphism
if every morphism in the category
factors as the product of a mono and an extremal epi
the category is called well-behaved
and a flow can be called cyclic if
n
the join of the equalisers of t and id
X
over all n
is X itself
http://tac.mta.ca/tac/volumes/10/15/10-15.pdf
similarly
in the stone category of
compact
hausdorff
totally connected spaces
we may define a flow (X, t) as nearly cyclic when
given two elements x , x of X
1 2
and a neighborhood N of x then
2
there exists a clopen c with
x elementOf c subobject N
2
and there exists a nonempty collection {m , m , ...}
1 2
which is a subcollection of {0, 1, ... n-1} for some n
m
so that t (x ) elementOf c <=> m equiv m (mod n) for some i
1 i
^^^^^^^^^^^^^^..^^^^^^^^^^^^^^^
the resolution of the geometrisation conjecture
was accomplished in ricci flow
ricci flow on a manifold is defined as
2
del g + 2 R - - R g = 0
t ij ij d ij
a flow of metric that
through singularities and solitons
eventually brings any 3-manifold to one of three types
- diffeomorphic to a graph manifold
- a closed hyperbolic manifold
- or can be split by a finite collection
of disjoint incompressible tori
into parts each diffeomorphic to graph manifolds
or complete noncompact hyperboic manifolds of finite volume
http://arxiv.org/pdf/math/0211159
http://arxiv.org/pdf/math/0303109
http://arxiv.org/pdf/math/0307245
by considering cycles in oriented 3-manifolds
with no aspherical factors
it can be shown that the ricci flow
will always cause extinction in finite time
~~~~~~~~~~~~~..~~~~~~~~~~~~~~~
of the unsolved problems of physics
one of the longest unresolved issues
is the resolution of the second law of thermodynamics
with the microscopic mechanics of particle action
standard mechanical interpretations
ascribe evolution to symplectomorphisms
which are volume preserving flows over symplectic manifolds
such that
the pullback of the real closed nondegenerate 2-form
after the flow
is identical to the preflow 2-form
now
poincare proved
given such a hamiltonian flow
where additionally the symplectic manifold
is of finite volume at the evolution constant E
then in any finite neighborhood of the manifold
a point in the neighborhood
will return to the neighborhood
after a finite number of applications of the flowmorphism
this implies that entropy cannot strictly increase
but must also recurr on such H-finite flows
##########..#########
recent work
on the existence and uniqueness
of navier-stokes solutions
has made progress on decomposing solutions geometrically
although penny smith's programme
encountered unforseen difficulties
( with christy sormani's inevitable attempt
to grab some of the attention )
gill and zachary have been making some interesting progress
http://arxiv.org/pdf/math-ph/0701064
http://arxiv.org/pdf/math-ph/0701038
in their work
m-dissipative conditions provide near cyclicity
used to great advantage in providing classes
of time-global solutions
and sufficiency conditions for class membership
%%%%%%%%%%%%%..$$$$$$$$$$$$$$
peter selinger
using the fibrational models of the lambda-mu calculus
was able to prove a syntactic duality
between the call-by-name and call-by-value calculi
this duality preserves the operational interpretation
the fiberings are constructed via
a classic strong exponential
much as the standard local translational flows
on a lie manifold
induced by the local algebra
&&&&&&&&&..&&
alain connes has does some out-of-the box work
connecting his well-known work on noncommutative geometry
with the riemann hypothesis
following work by hilbert, polya, and selberg
he has developed spectral interpretations of the zeroes
from lefshetz trace formulas
over the riemannian flow
where he is able to extract data from the cyclic flows
cf.
" trace formula in noncommutative geometry
and the zeros of the riemann zeta function "
older work by b bagchi
like
" recurrence in topological dynamics
and the riemann hypothesis "
similarly extracts this data
from both cyclic and nearly cyclic flows
(((((((((((((!!))))))))))))))
there is something of a convergence here
scattered strands of a tale
grasping for solid ground
struggling to contain these
the most famous mathematical and physical problems of our time
but still slipping away as flows often do
will this next century be the century of flows?
the cyclic
the nearly cyclic
and everything that remains?
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
|