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'''Ev Teel Urizen''' is a [[fictional character]] in the [[Marvel Universe]]. It was created by writer [[Mike Carey]]. Ev Teel Urizen first appeared in issue 197 of [[X-Men]] (May, 2007).
:''A '''Morse function''' is also an expression for an [[anharmonic oscillator]]''
Ev Teel Urizen is a [[mummudrai]], a mental parasite constructed of energy latices that exists on the astral plane. Mummudrai are "born" with every living being in the womb. In [[Shi'ar]] legend, they are described as the a living being's first contact with the idea of "the other". While many die in the womb, some choose to remain in the host's mind, harmless and dormant. This was the case with Urizen, who stayed harmlessly with its host Ul'var Urizen.
In [[differential topology]], the techniques of '''Morse theory''' give a very direct way of analyzing the [[topological space|topology]] of a [[manifold]] by studying [[differentiable function]]s on that manifold. According to the basic insights of [[Marston Morse]], a differentiable function on a manifold will, in a ''typical'' case, reflect the topology quite directly. Morse theory allows one to find [[CW complex|CW structure]]s and [[handle decomposition]]s on manifolds and to obtain substantial information about their [[homology (mathematics)|homology]].
Urizen fled from the edge of Shi'ar space following a trail of psychic energy to the X-Men's Mansion, seeking help from the X-Men's powerful psychics to stop Hecatomb before "he" (as Urizen refers to it) consumed everything.
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 celebrated [[Bott periodicity theorem|periodicity theorem]].
Like all mummudrai, Ev Teel Urizen possesses the powers of telekenisis and telepathy and can inhabit and posess nearly anyone by performing jumps on the astral plane.
==Basic concepts==
[[Image:Saddle_point.png|thumb|right|A saddle point]]
==Fictional Character Biography==
Consider, for purposes of illustration, a mountainous landscape ''M''. If ''f'' is the [[function (mathematics)|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.
===Origins===
Ev Teel Urizen was born together with its original host, Ul'var Urizen. Mummudrai can meet a variety of fates, most dieing in the womb, while others survive by consuming the minds of their hosts or choosing to peacefully and dormantly reside in the host's mind. Urizen chose the latter and peacefully existed within Ul'var.
===Extraction and War===
[[Image:Saddlecontour.png|thumb|left|Contour lines around a saddle point]]
During a Shi'ar war with another planet, the Heptarchy, Urizen was foreceably extracted from Ul'var and psychically disected by Shi'ar scientists and engineers whose goal was to create an artificial mummudrai from its template that could be harnessed into a controllable weapon. The weapon, termed the "Hecatomb" could be turned on and off and would ideally be placed near the planet. There it would consume and erase the minds of anything nearby, killing the planet's population while leaving the planet intact and inhabitable.
Imagine flooding this landscape with water. Then, assuming the ground is porous, the region covered by water when the water reaches an elevation of ''a'' is f<sup>−1</sup> (-∞, ''a''<nowiki>]</nowiki>, or the points with elevation less than or equal to ''a''. Consider how the topology of this region changes as the water rises. It appears, intuitively, that it does not change except when ''a'' passes the height of a [[Critical point (mathematics)|critical point]]; that is, a point where the [[gradient]] of ''f'' is 0. In other words, it does not change except when the water either (1) starts filling a basin, (2) covers a saddle (a [[mountain pass]]), or (3) submerges a peak.
A flaw in Hecatomb's design, however, caused the weapon to malfunction. Hecatomb could not contain the vast amount of mental energy absorbed from the planet's population at one time and as a result, merged all the minds of the dead together with itself. Now consisting of the many horrified minds of an entire planet's population, the Shi'ar technology to shut it off failed and only served to anger it and further exacerbate its insatiable hunger for thoughts and minds.
[[Image:3D-Leveltorus.png|thumb|right|The torus]]
To each of these three types of critical points - basins, passes, and peaks (also called minima, saddles, and maxima) - one associates a number called the index. Intuitively speaking, the index of a critical point ''b'' is the number of independent directions around ''b'' in which ''f'' decreases. Therefore, the indices of basins, passes, and peaks are 0, 1, and 2, respectively.
===Flight and "Supernova"===
Define M<sup>a</sup> as f<sup>−1</sup>(-∞, ''a''<nowiki>]</nowiki>. Leaving the context of topography, one can make a similar analysis of how the topology of M<sup>a</sup> changes as ''a'' increases when ''M'' is a torus oriented as in the image and ''f'' is projection on a vertical axis, taking a point to its height above the plane.
Urizen eventually broke free from its Shi'ar containment fields, but found itself chased by the Hecatomb. Urizen believes this to be due to the fact that it's "scent" was familiar to Hecatomb, since Hecatomb was modeled off of Urizen.
Following a trail of psychic energy, Urizen fled from Shi'ar space to Earth, a journey that took centuries. Following behind it, Hecatomb consumed any mind that it encountered along its path.
[[Image:3D-Cylinder_and_disk_with_handle.png|thumb|left|These figures are homotopy equivalent]]
[[Image:3D-Cylinder_with_handle_and_torus_with_hole.png|thumb|right|These figures are homotopy equivalent]]
When ''a'' is less than 0, M<sup>a</sup> is the empty set. After ''a'' passes the level of ''p'' (a critical point of index 0), when 0<''a''<''f''(''q''), then M<sup>a</sup> is a disk, which is homotopy equivalent to a point, (a 0-cell) which has been "attached" to the empty set. Next, when ''a'' exceeds the level of ''q'' (a critical point of index 1), and ''f''(''q'') <''a''<''f''(''r''), then M<sup>a</sup> is a cylinder, and is homotopy equivalent to a disk with a 1-cell attached (image at left). Once ''a'' passes the level of ''r'' (a critical point of index 1), and ''f''(''r'')<''a''<''f''(''s''), then M<sup>a</sup> is a torus with a disk removed, which is homotopy equivalent to a cylinder with a 1-cell attached (image at right). Finally, when ''a'' is greater than the critical level of ''s'' (a critical point of index 2) M<sup>''a''</sup> is a torus. A torus, of course, is the same as a torus with a disk removed with a disk (a 2-cell) attached.
Eventually arriving at Earth, Urizen took possession of the deceased body of an old men, Paul Brunner, who had been hit by a truck. Piloting the dead body, Urizen reached the gates of the X-Mansion. Upon arrival, it jumping into the mind of the recently rescued and comatose [[Lady Mastermind|Reagen Wyngarde]], leaving the X-Men to puzzle about the nature of an animated corpse of no one in partucular suddenly stopping outside their gate. They were distracted, however, by the more pressing issue at hand--the [[Children of the Vault]]. During the conflict, Reagen regained consciousness and was recruited by [[Rogue (comics)|Rogue]] onto her new team of [[X-Men (vol. 2)|X-Men]].
We therefore appear to have the following rule: the topology of M<sup>α</sup> does not change except when α passes the height of a critical point, and when α passes the height of a critical point of index γ, a γ-cell is attached to M<sup>α</sup>. This does not address the question of what happens when two critical points are at the same height. That situation can be resolved by a slight perturbation of ''f''. In the case of a landscape (or a manifold [[embedding|embedded]] in [[Euclidean space]]), this perturbation might simply be tilting the landscape slightly, or rotating the coordinate system.
==="Infection"===
This rule, however, is false as stated. To see this, let ''M'' equal ''R'' and let ''f''(''x'')=''x''<sup>3</sup>. Then 0 is a critical point of ''f'', but the topology of M<sup>α</sup> does not change when α passes 0. In fact, the concept of index does not make sense. The problem is that the second derivative is also 0 at 0. This kind of situation is called a degenerate critical point. Note that this situation is unstable: by rotating the coordinate system under the graph, the degenerate critical point either is removed or breaks up into two non-degenerate critical points.
After defeating the Children of the Vault, Rogue's squad next came into conflict with Pandemic. Pandemic kidnapped Rogue to perform experiments on her to see if mutations could be caught and exploited like a virus. The team went to rescue her and during a conflict, the as of yet unrevealed Urizen protected Reagen by attacking a soldier behind her, causing [[Iceman (comics)|Iceman]] to suspect something was up with Reagen, though she appeared unaware of anything strange. Later, after Rogue had been rescued while in critcal condition, [[Sabretooth (comics)|Sabretooth]] stated that Reagen doesn't smell right and that there is something inside of her that "stinks like it's dead." Reagen also ignored these comments.
==Formal="Red developmentData"===
In ''X-Men'' #197, While the team rushed Rogue to a hospital on Providence, an island created by Cable, Reagen demonstrated that she had been fully aware of something residing in her mind and was determined to remove it. Using her own powers of illusion on herself and some assisted hypnosis, Reagen discovered where Urizen was hiding. This is technically the first time Urizen was revealed/introduced in totallity. Urizen tried to hide behind a mental wall and even took Reagen's own form to trick her, but Reagen was hardly fooled and angrily expelled Urizen from her mind. However, Urizen forefully took over Reagen's body and made her way to Cable, whom it recognized as someone powerful enough to help it and to whom it explained the grave situation of Hecatomb's imminent arrival on Earth.
Urizen offered to bond with Cable and restore his lost telepathy and telekenisis, making the two fo them stronger as one to face Hecatomb. Remembering the previous threat caused by another mummudrai, [[Cassandra Nova]], the X-Men were distrustful of Urizen's motives and Cable intially refused. However, Hecatomb arrived as stated and began slaughtering the population of Providence to feed while searching for Urizen, eventually wielding the power of 8 billion minds as offensive weapons. Seeing that there was no other choice, Cable reluctantly agreed. Together, Cable and Urizen battled Hecatomb, but began losing ground when Hecatomb adapted its attacks to separate the two.
For a real-valued [[smooth function]] ''f'' : ''M'' → '''R''' on a [[differentiable manifold]] ''M'', the points where the [[exterior derivative]] of ''f'' vanishes are called [[critical point (mathematics)|critical point]]s of ''f'' and their images under ''f'' are called [[critical value]]s. If at a critical point ''b'', the matrix of second partial derivatives (the [[Hessian matrix]]) is non-singular, then ''b'' is called a '''non-degenerate critical point'''; if the Hessian is singular then ''b'' is a '''degenerate critical point'''.
===Death===
For the functions
In a last ditch effort, Cable/Urizen telepathically woke up Rogue, who was in critical condition and infected with Pandemic's "Strain 88", a virus that caused Rogue to instantly absorb a person's entire mind with one touch, effectively killing them. Before Rogue could get to the scene of the battle and eventually defeat it, Hecatomb managed to separate the mind meld between Cable and Urizen with a mind blast. In its final moments, Urizen screamed that it was dying and frantically begged Cable not to let Hecatomb assimilate it before Hecatomb finally consumed and "killed" Urizen.
:<math>f(x)=a + b x+ c x^2+d x^3+\cdots</math>
from '''R''' to '''R''', ''f'' has a critical point at the origin if ''b''=0, which is non-degenerate if ''c''≠0,(''f'' is of the form ''a''+''cx''<sup>2</sup>+...) and degenerate if ''c''=0,(''f'' is of the form ''a''+''dx''<sup>3</sup>+...).
A less trivial example of a degenerate critical point is the origin of the [[monkey saddle]].
Before Hecatomb could kill Cable next, Rogue managed to defeat it by absorbing all 8 billion of Hecatomb's merged minds into herself. Though Rogue now contains all the memories of every dead mind that Hecatomb absorbed, it is unknown whether or not Urizen's is among them.
The '''[[index_(mathematics)|index]]''' of a non-degenerate critical point ''b'' of ''f'' is the dimension of the largest subspace of the [[tangent space]] to ''M'' at ''b'' on which the Hessian is negative definite. It is easy to see that this corresponds to the intuitive notion that the index is the number of directions in which ''f'' decreases.
===ThePowers Morseand lemma=Abilities==
As a mummudrai, Ev Teel Urizen possesses significant telepathic and telekinetic abilities. It is able to posess anyone with a sentient mind (though things such as Cable's forcefield can stop it) and can exploit that individual's abilities (such as using Lady Mastermind's illusions or Mystique's morphing abilities). It can also reside dormantly within a host's mind, though its ability to be detected is clear from Reagen's own psychic abilities and Sabretooth's enhanced sense of smell.
When mind-melded with Cable, Urizen's power (as a hybrid with Cable) is increased, giving them immense telekenisis (the ability to throw the Children of the Vault's ''The Conquistador'', a UCC-Type Tanker and the ability to levitate the entire island of Providence). After they are separated in ''X-Men''#199, Cable says that his telekenisis and telepathy died with Urizen.
Let ''b'' be a non-degenerate critical point of ''f'' : ''M'' → '''R'''. Then there exists a [[chart (topology)|chart]] (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>) in a [[neighborhood (topology)|neighborhood]] ''U'' of ''b'' such that ''x''<sub>i</sub>(''b'')=0 for all ''i'' and
{{Uncategorized|date=June 2007}}
:''f''(''x'') = ''f''(''b'') − (''x''<sub>1</sub>)<sup>2</sup> − ... − (''x''<sub>α</sub>)<sup>2</sup> + (''x''<sub>α+1</sub>)<sup>2</sup> + ... + (''x''<sub>''n''</sub>)<sup>2</sup> + higher order terms
throughout ''U''. Here α is equal to the index of ''f'' at ''b''. As a corollary of the Morse lemma we see that non-degenerate critical points are [[isolated point|isolated]].
For functions from '''R'''<sup>2</sup> to '''R''' with a critical point at the origin, the Morse lemma implies that after rotation of coordinates ''f'' will be of the form
:<math>f(x,y)=a+(Ax^2+By^2)/2+ \mbox{higher order terms},</math>
which will be degenerate if ''A'' = 0 or ''B'' = 0.
A smooth real-valued function on a manifold ''M'' is a '''Morse function''' if it has no degenerate critical points. A basic result of Morse theory says that almost all functions are Morse functions. Technically, the Morse functions form an open, dense subset of all smooth functions ''M'' → '''R''' in the ''C''<sup>2</sup> topology. This is sometimes expressed as "a typical function is Morse." or "a [[generic]] function is Morse".
As indicated before, we are interested in the question of when the topology of ''M''<sup>''a''</sup> = f<sup>−1</sup>(-∞, ''a''] changes as ''a'' varies. Half of the answer to this question is given by the following theorem.
:'''Theorem.''' Suppose ''f'' is a smooth real-valued function on ''M'', ''a'' < ''b'', f<sup>−1</sup>[''a'', ''b''] is [[compact space|compact]], and there are no critical values between ''a'' and ''b''. Then ''M''<sup>''a''</sup> is [[diffeomorphic]] to ''M''<sup>''b''</sup>, and ''M''<sup>''b''</sup> [[deformation retract]]s onto ''M''<sup>''a''</sup>.
It is also of interest to know how the topology of ''M''<sup>''a''</sup> changes when ''a'' passes a critical point. The following theorem answers that question.
:'''Theorem.''' Suppose ''f'' is a smooth real-valued function on ''M'' and ''p'' is a non-degenerate critical point of ''f'' of index γ, and that ''f''(''p'') = ''q''. Suppose ''f''<sup>−1</sup>[''q'' − ε, ''q'' + ε] is compact and contains no critical points besides ''p''. Then for ε sufficiently small ''M''<sup>''q'' + ε</sup> is [[homotopy equivalent]] to ''M''<sup>''q'' − ε</sup> with a γ-cell attached.
These results generalize and formalize the 'rule' stated in the previous section. As was mentioned, the rule as stated is incorrect; these theorems correct it.
Using the two previous results and the fact that there exists a Morse function on any differentiable manifold, one can prove that any differentiable manifold is a CW complex with an ''n''-cell for each critical point of index ''n''. To do this, one needs the technical fact that one can arrange to have a single critical point on each critical level.
===The Morse inequalities===
Morse theory can be used to prove some strong results on the homology of manifolds. The number of critical points of index γ of ''f'': ''M'' → '''R''' is equal to the number of γ cells in the CW structure on ''M'' obtained from "climbing" ''f''. Using the fact that the alternating sum of the ranks of the homology groups of a topological space is equal to the alternating sum of the ranks of the chain groups from which the homology is computed, then by using the cellular chain groups (see [[cellular homology]]) it is clear that the [[Euler characteristic]] is equal to the sum
:<math>\sum(-1)^{\gamma}C^{\gamma}\,</math>
where ''C''<sup>γ</sup> is the number of critical points of index γ. Also by cellular homology, the rank of the n<sup>th</sup> homology group of a CW complex ''M'' is less than or equal to the number of n-cells in ''M''. Therefore the rank of the γ<sup>th</sup> homology group is less than or equal to the number of critical points of index γ of a Morse function on ''M''. These facts can be strengthened to obtain the '''Morse inequalities''':
:<math>C^\gamma -C^{\gamma -1}+-\cdots \pm C^0 \ge {\rm{Rank}}[H_\gamma (M)]-{\rm{Rank}}[H_{\gamma -1}(M)]+- \cdots \pm {\rm{Rank}}[H_0 (M)]</math>
===Morse homology===
[[Morse homology]] is a particularly perspicuous approach to the [[Homology (mathematics)|homology]] of [[smooth manifold]]s. It is defined using a [[generic choice]] of Morse function and [[Riemannian metric]]. The basic theorem is that the resulting homology is an invariant of the manifold (i.e. independent of the function and metric) and isomorphic to the singular homology of the manifold; this implies that the Morse and singular Betti numbers agree and gives an immediate proof of the Morse inequalities. An infinite dimensional analog of Morse homology is known as [[Floer homology]].
[[Ed Witten]] developed another related approach to Morse theory in 1982 using [[harmonic function]]s.
===Morse-Bott theory===
The notion of a Morse function can be generalized to consider functions that have degenerate critical manifolds. That is, the kernel of the Hessian at a critical point equals the tangent space to the critical submanifold. A Morse function is the special case where the critical manifolds are zero-dimensional. The index is most naturally thought of as a pair
:(i<sub>−</sub>, i<sub>+</sub>),
where i<sub>−</sub> is the dimension of the unstable manifold at a given point of the critical manifold, and i<sub>+</sub> is i<sub>−</sub> plus the dimension of the critical manifold. If the Morse-Bott function is perturbed by a small function on the critical locus, the index of all critical points of the perturbed function on a critical manifold of the unperturbed function will lie between i<sub>−</sub> and i<sub>+</sub>).
Morse-Bott functions are useful because generic Morse functions are difficult to work with; the functions one can visualize, and with which one can easily calculate, typically have symmetries. They often lead to positive-dimensional critical manifolds. [[Raoul Bott]] used Morse-Bott theory in his original proof of the [[Bott periodicity theorem]].
See [[Round function]], for an instance.
[[Morse homology]] can also be formulated for Morse-Bott functions; the differential in Morse-Bott homology is computed by a [[spectral sequence]]. Frederic Bourgeois developed a neat approach in the course of his work on a Morse-Bott version of [[symplectic field theory]].
==See also==
*[[Sard's lemma]]
*[[Lyusternik-Schnirelmann category]]
==References==
* Bott, Raul (1988). [http://www.numdam.org/item?id=PMIHES_1988__68__99_0 Morse Theory Indomitable.] ''Publications Mathématiques de l'IHÉS.'' '''68''', 99–114.
* Bott, Raoul (1982). ''Lectures on Morse theory, old and new.'', Bull. Amer. Math. Soc. (N.S.) 7, no. 2, 331–358.
* Cayley, Arthur (1859). On Contour and Slope Line. ''The Philosophical Magazine'' '''18''' (120), 264-268.
* Matsumoto, Yukio (2002). An Introduction to Morse Theory
* Maxwell, James Clerk (1870). [http://www.maths.ed.ac.uk/~aar/surgery/hilldale.pdf On Hills and Dales.] ''The Philosophical Magazine'' '''40''' (269), 421–427.
* {{cite book | last=Milnor | first=John | title=Morse Theory | publisher= Princeton University Press | year=1963 | id=ISBN 0-691-08008-9 }} A classic advanced reference in mathematics and mathematical physics.
* Milnor, John (1965). Lectures on the [[h-Cobordism theorem]] - scans available [http://www.maths.ed.ac.uk/~aar/surgery/hcobord.pdf here]
* Morse, Marston (1934). "The Calculus of Variations in the Large", ''American Mathematical Society Colloquim Publication'' '''18'''; New York.
* Seifert, Herbert & Threlfall, William (1938). Variationsrechnung im Grossen
[[Category:Morse theory| ]]
[[Category:Smooth functions]]
[[Category:Lemmas]]
[[fr:Théorie de Morse]]
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