Morse theory
In differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology.
Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography. Morse originally applied his theory to geodesics (critical points of the energy functional on paths). These techniques were used in Raoul Bott's proof of his periodicity theorem.
The analogue of Morse theory for complex manifolds is Picard–Lefschetz theory.
Basic concepts
Consider, for purposes of illustration, a mountainous landscape M. If f is the function M → R sending each point to its elevation, then the inverse image of a point in R (a level set) is simply a contour line. Each connected component of a contour line is either a point, a simple closed curve, or a closed curve with a double point. Contour lines may also have points of higher order (triple points, etc.), but these are unstable and may be removed by a slight deformation of the landscape. Double points in contour lines occur at saddle points, or passes. Saddle points are points where the surrounding landscape curves up in one direction and down in the other.
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Wishful Thinking
by: Mary EstherVerse 1
standing on the edge
i wonder why it had to be
i guess that we wearnt soul mates
and youre no good for me
saying i may be crazy
but im not and you were here
with me until the day came
and you know i was left in tears
i thought we were
meant to be together
but i guess we were
meant to be whatever
we used to say forever
but now we are saying never
Chours
that was my wishful thinking
i didnt see your headlights blinking
on and on
on and on
i thought we could always be
but you wearnt meant for me
i was wrong
i was wrong
all i hoped would be
that was my wishful thinking
Verse 2
turning the page over
flipt the script and what ima do now?
i bow my head to pray
please lord, help me to smile
people look at me
they talk in this world
but i am only one person
i am only one girl
i thought we were
meant to be together
but i guess we were
meant to be whatever
we used to say forever
but now we are saying never
Chours
that was my wishful thinking
i didnt see your headlights blinking
on and on
on and on
i thought we could always be
but you wearnt meant for me
i was wrong
i was wrong
all i hoped would be
that was my wishful thinking
Bridge
i wished we would be forever
and i thought we could be
i wished we would be together
and i thought we were meant to be
all of the things i ever knew
i thought i was right for you
but you were not right for me
and i guess we were never meant to be
that was my wishful thinking
i didnt see your headlights blinking
i thought we could always be
but you wearnt meant for me
Chours
that was my wishful thinking
i didnt see your headlights blinking
on and on
on and on
i thought we could always be
but you wearnt meant for me
i was wrong
i was wrong
all i hoped would be
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