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| In [[algebraic geometry]], a '''moduli space''' is a geometric space (usually a [[scheme (mathematics)|scheme]] or an [[algebraic stack]]) whose points represent algebro-geometric objects of some fixed kind, or [[isomorphism class]]es of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth [[algebraic curve]]s of a fixed [[genus (topology)|genus]]) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects.
| | It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>For those who have almost any issues regarding where as well as how you can make use of [http://www.youtube.com/watch?v=90z1mmiwNS8 Best Dentists in DC], you can e mail us from our web site. |
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| ==Motivation==
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| Moduli spaces, quite broadly, are geometric solutions to geometric classification problems. That is, given a geometric object that we want to describe (such as lines, surfaces, or [[elliptic curves]]), moduli spaces can tell us which geometric objects should be considered isomorphic and how the family of geometric objects can modulate (vary throughout the family).
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| [[File:Real projective line moduli space example.pdf|thumb|Constructing '''P'''<sup>1</sup>('''R''') by varying 0 ≤ θ < π or as a quotient space of '''S'''<sup>1</sup>.]]
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| As a simple motivating example, consider how we can describe the collection of lines in '''R'''<sup>2</sup> which intersect the origin. We want to assign a quantity, a modulus, to each line ''L'' of this family which can uniquely identify it, so it is natural to suggest a positive angle θ(''L'') with 0 ≤ θ < π, which will yield all lines in '''R'''<sup>2</sup> which intersect the origin. The set of lines ''L'' just constructed is known as '''P'''<sup>1</sup>('''R''') and is called the [[real projective line]].
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| We can also describe the collection of lines in '''R'''<sup>2</sup> which intersect the origin by means of a topological construction. That is, consider '''S'''<sup>1</sup> ⊂ '''R'''<sup>2</sup> and notice that to every point ''s'' ∈ '''S'''<sup>1</sup> that we can identify a line ''L''(''s'') in the collection if the line intersects the origin and ''s''. Yet, this map is two-to-one, so we want to identify ''s'' ~ −''s'' to yield '''P'''<sup>1</sup>('''R''') ≅ '''S'''<sup>1</sup>/~ where the topology on this space is the [[quotient topology]] induced by the [[quotient map]] '''S'''<sup>1</sup> → '''P'''<sup>1</sup>('''R''').
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| Thus, when we consider '''P'''<sup>1</sup>('''R''') as a moduli space of lines that intersect the origin in '''R'''<sup>2</sup>, we capture the ways in which the members of the family (lines in the case) can modulate by continuously varying 0 ≤ θ < π.
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| ==Basic Examples==
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| ===Projective Space and Grassmannians===
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| The [[real projective space]] '''P'''<sup>''n''</sup> is a moduli space which parametrizes the space of lines in '''R'''<sup>''n''+1</sup> which pass through the origin. Similarly, complex projective space is the space of all complex lines in '''C'''<sup>''n''+1</sup> passing through the origin.
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| More generally, the [[Grassmannian]] '''G'''(''k'', ''V'') of a vector space ''V'' over a field ''F'' is the moduli space of all ''k''-dimensional linear subspaces of ''V''.
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| ===Chow Variety===
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| The [[Chow ring|Chow Variety]] '''Chow'''(d,'''P'''<sup>''3''</sup>) is a projective algebraic variety which parametrizes degree ''d'' curves in '''P'''<sup>''3''</sup>. It is constructed as follows. Let C be a curve of degree ''d'' in '''P'''<sup>''3''</sup>, then consider all the lines in '''P'''<sup>''3''</sup> that intersect the curve C. This is a degree ''d'' divisor ''D_C'' in '''G'''(''2'', ''4'') the Grassmannian of lines in '''P'''<sup>''3''</sup>. When ''C'' varies, by associating ''C'' to ''D_C'', we obtain a parameter space of degree ''d'' curves as a subset of the space of degree ''d'' divisors of the Grassmannian: '''Chow'''(d,'''P'''<sup>''3''</sup>).
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| ===Hilbert Scheme===
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| The [[Hilbert scheme]] '''Hilb'''(''X'') is a moduli scheme. Every closed point of '''Hilb'''(''X'') corresponds to a closed subscheme of a fixed scheme ''X'', and every closed subscheme is represented by such a point.
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| ==Definitions==
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| There are several different related notions of what it means for a space ''M'' to be a moduli space. Each of these definitions formalizes a different notion of what it means for the points of a space to represent geometric objects.
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| ===Fine Moduli Spaces===
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| This is the most important notion.{{fact|date=June 2013}} Heuristically, if we have a space ''M'' for which each point ''m''∈ ''M'' corresponds to an algebro-geometric object ''U<sub>m</sub>'', then we can assemble these objects into a topological family ''U'' over ''M''. (For example, the Grassmannian '''G'''(''k'', ''V'') carries a rank ''k'' bundle whose fiber at any point [''L''] ∈ '''G'''(''k'', ''V'') is simply the linear subspace ''L'' ⊂ ''V''.) ''M'' is called a '''base space''' of the family ''U''. We say that such a family is '''universal''' if any family of algebro-geometric objects ''T'' over any base space ''B'' is the [[Pullback (category theory)|pullback]] of ''U'' along a unique map ''B'' → ''M''. A fine moduli space is a space ''M'' which is the base of a universal family.
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| More precisely, suppose that we have a functor ''F'' from schemes to sets, which assigns to a scheme ''B'' the set of all suitable families of objects with base ''B''. A space ''M'' is a '''fine moduli space''' for the functor ''F'' if ''M'' [[representable functor|corepresents]] ''F'', i.e., the functor of points '''Hom'''(−, ''M'') is naturally isomorphic to ''F''. This implies that ''M'' carries a universal family; this family is the family on ''M'' corresponding to the identity map '''1'''<sub>''M''</sub> ∈ '''Hom'''(''M'', ''M'').
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| ===Coarse Moduli Spaces===
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| Fine moduli spaces are very useful, but they do not always exist and are frequently difficult to construct, so mathematicians sometimes use a weaker notion, the idea of a coarse moduli space. A space ''M'' is a '''coarse moduli space''' for the functor ''F'' if there exists a natural transformation τ : ''F'' → '''Hom'''(−, ''M'') and τ is universal among such natural transformations. More concretely, ''M'' is a coarse moduli space for ''F'' if any family ''T'' over a base ''B'' gives rise to a map φ<sub>''T''</sub> : ''B'' → ''M'' and any two objects ''V'' and ''W'' (regarded as families over a point) correspond to the same point of ''M'' if and only if ''V'' and ''W'' are isomorphic. Thus, ''M'' is a space which has a point for every object that could appear in a family, and whose geometry reflects the ways objects can vary in families. Note, however, that a coarse moduli space does not necessarily carry any family of appropriate objects, let alone a universal one.
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| In other words, a fine moduli space includes ''both'' a base space ''M'' and universal family ''T'' → ''M'', while a coarse moduli space only has the base space ''M''.
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| ===Moduli stacks===
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| It is frequently the case that interesting geometric objects come equipped with lots of natural [[automorphism]]s. This in particular makes the existence of a fine moduli space impossible (intuitively, the idea is that if ''L'' is some geometric object, the trivial family ''L'' × [0,1] can be made into a twisted family on the circle '''S'''<sup>1</sup> by identifying ''L'' × {0} with ''L'' × {1} via a nontrivial automorphism. Now if a fine moduli space ''X'' existed, the map '''S'''<sup>1</sup> → ''X'' should not be constant, but would have to be constant on any proper open set by triviality), one can still sometimes obtain a coarse moduli space. However, this approach is not ideal, as such spaces are not guaranteed to exist, are frequently singular when they do exist, and miss details about some non-trivial families of objects they classify.
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| A more sophisticated approach is to enrich the classification by remembering the isomorphisms. More precisely, on any base ''B'' one can consider the category of families on ''B'' with only isomorphisms between families taken as morphisms. One then considers the [[fibred category]] which assigns to any space ''B'' the groupoid of families over ''B''. The use of these ''categories fibred in groupoids'' to describe a moduli problem goes back to Grothendieck (1960/61). In general they cannot be represented by schemes or even [[algebraic space]]s, but in many cases they have a natural structure of an [[algebraic stack]].
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| Algebraic stacks and their use to analyse moduli problems appeared in Deligne-Mumford (1969) as a tool to prove the irreducibility of the (coarse) [[moduli of algebraic curves|moduli space of curves]] of a given genus. The language of algebraic stacks essentially provides a systematic way to view the fibred category that constitutes the moduli problem as a "space", and the moduli stack of many moduli problems is better-behaved (such as smooth) than the corresponding coarse moduli space.
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| ==Further Examples==
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| ===Moduli of curves===
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| {{details|Moduli of algebraic curves}}
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| The moduli stack <math>\mathcal{M}_{g}</math> classifies families of smooth projective curves of genus ''g'', together with their isomorphisms. When ''g'' > 1, this stack may be compactified by adding new "boundary" points which correspond to stable nodal curves (together with their isomorphisms). A curve is stable if it has only a finite group of automorphisms. The resulting stack is denoted <math>\overline{\mathcal{M}}_{g}</math>. Both moduli stacks carry universal families of curves. One can also define coarse moduli spaces representing isomorphism classes of smooth or stable curves. These coarse moduli spaces were actually studied before the notion of moduli stack was invented. In fact, the idea of a moduli stack was invented by Deligne and Mumford in an attempt to prove the projectivity of the coarse moduli spaces. In recent years, it has become apparent that the stack of curves is actually the more fundamental object.
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| Both stacks above have dimension 3''g''−3; hence a stable nodal curve can be completely specified by choosing the values of 3''g''−3 parameters, when ''g'' > 1. In lower genus, one must account for the presence of smooth families of automorphisms, by subtracting their number. There is exactly one complex curve of genus zero, the Riemann sphere, and its group of isomorphisms is PGL(2). Hence the dimension of <math>\mathcal{M}_0</math> is
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| :dim(space of genus zero curves) - dim(group of automorphisms) = 0 − dim(PGL(2)) = −3. | |
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| Likewise, in genus 1, there is a one-dimensional space of curves, but every such curve has a one-dimensional group of automorphisms. Hence the stack <math>\mathcal{M}_1</math> has dimension 0. The coarse moduli spaces have dimension 3''g''-3 as the stacks when ''g'' > 1 because the curves with genus g > 1 have a only finite group as its automorphism i.e. dim(group of automorphisms) = 0. Eventually, in genus zero the coarse moduli space has dimension zero, and in genus one, it has dimension one.
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| One can also enrich the problem by considering the moduli stack of genus ''g'' nodal curves with ''n'' marked points. Such marked curves are said to be stable if the subgroup of curve automorphisms which fix the marked points is finite. The resulting moduli stacks of smooth (or stable) genus ''g'' curves with ''n''-marked points are denoted <math>\mathcal{M}_{g,n}</math> (or <math>\overline{\mathcal{M}}_{g,n}</math>), and have dimension 3''g''−3+''n''.
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| A case of particular interest is the moduli stack <math>\overline{\mathcal{M}}_{1,1}</math> of genus 1 curves with one marked point. This is the stack of [[elliptic curve]]s, and is the natural home of the much studied [[modular form]]s, which are meromorphic sections of bundles on this stack.
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| ===Moduli of varieties===
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| In higher dimensions, moduli of algebraic varieties are more difficult to construct and study. For instance, the higher dimensional analogue of the moduli space of elliptic curves discussed above is the moduli space of abelian varieties. This is the problem underlying [[Siegel modular form]] theory. See also [[Shimura variety]].
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| ===Moduli of vector bundles===
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| Another important moduli problem is to understand the geometry of (various substacks of) the moduli stack Vect<sub>''n''</sub>(''X'') of rank ''n'' [[vector bundle]]s on a fixed [[algebraic variety]] ''X''. This stack has been most studied when ''X'' is one-dimensional, and especially when n equals one. In this case, the coarse moduli space is the [[Picard scheme]], which like the moduli space of curves, was studied before stacks were invented. Finally, when the bundles have rank 1 and degree zero, the study of the coarse moduli space is the study of the [[Jacobian variety]].
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| In applications to [[physics]], the number of moduli of vector bundles and the closely related problem of the number of moduli of [[Fiber bundle|principal G-bundles]] has been found to be significant in [[gauge theory]].
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| ==Methods for constructing moduli spaces==
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| The modern formulation of moduli problems and definition of moduli spaces in terms of the moduli functors (or more generally the [[fibred category|categories fibred]] in [[groupoid]]s), and spaces (almost) representing them, dates back to Grothendieck (1960/61), in which he described the general framework, approaches and main problems using [[Teichmüller space]]s in complex analytical geometry as an example. The talks in particular describe the general method of constructing moduli spaces by first ''rigidifying'' the moduli problem under consideration.
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| More precisely, the existence of non-trivial automorphisms of the objects being classified makes it impossible to have a fine moduli space. However, it is often possible to consider a modified moduli problem of classifying the original objects together with additional data, chosen in such a way that the identity is the only automorphism respecting also the additional data. With a suitable choice of the rigidifying data, the modified moduli problem will have a (fine) moduli space ''T'', often described as a subscheme of a suitable [[Hilbert scheme]] or [[Quot scheme]]. The rigidifying data is moreover chosen so that it corresponds to a principal bundle with an algebraic structure group ''G''. Thus one can move back from the rigidified problem to the original by taking quotient by the action of ''G'', and the problem of constructing the moduli space becomes that of finding a scheme (or more general space) that is (in a suitably strong sense) the quotient ''T''/''G'' of ''T'' by the action of ''G''. The last problem in general does not admit a solution; however, it is addressed by the groundbreaking [[geometric invariant theory]] (GIT), developed by [[David Mumford]] in 1965, which shows that under suitable conditions the quotient indeed exists.
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| To see how this might work, consider the problem of parametrizing smooth curves of genus ''g'' > 2. A smooth curve together with a [[complete linear system]] of degree ''d'' > 2''g'' is equivalent to a closed one dimensional subscheme of the projective space '''P'''<sup>''d−g''</sup>. Consequently, the moduli space of smooth curves and linear systems (satisfying certain criteria) may be embedded in the Hilbert scheme of a sufficiently high-dimensional projective space. This locus ''H'' in the Hilbert scheme has an action of PGL(''n'') which mixes the elements of the linear system; consequently, the moduli space of smooth curves is then recovered as the quotient of ''H'' by the projective general linear group.
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| Another general approach is primarily associated with [[Michael Artin]]. Here the idea is to start with any object of the kind to be classified and study its [[deformation theory]]. This means first constructing [[infinitesimal]] deformations, then appealing to '''prorepresentability''' theorems to put these together into an object over a [[formal scheme|formal]] base. Next an appeal to [[Alexandre Grothendieck|Grothendieck's]] [[Grothendieck existence theorem|formal existence theorem]] provides an object of the desired kind over a base which is a complete local ring. This object can be approximated via [[Artin's approximation theorem]] by an object defined over a finitely generated ring. The [[spectrum of a ring|spectrum]] of this latter ring can then be viewed as giving a kind of coordinate chart on the desired moduli space. By gluing together enough of these charts, we can cover the space, but the map from our union of spectra to the moduli space will in general be many to one. We therefore define an [[equivalence relation]] on the former; essentially, two points are equivalent if the objects over each are isomorphic. This gives a scheme and an equivalence relation, which is enough to define an [[algebraic space]] (actually an [[algebraic stack]] if we are being careful) if not always a scheme.
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| ==In Physics==
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| {{details|moduli (physics)}}
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| The term moduli space is sometimes used in [[physics]] to refer specifically to the moduli space of vacuum expectation values of a set of scalar fields, or to the moduli space of possible [[string background]]s.
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| Moduli spaces also appear in physics in [[topological field theory|cohomological field theory]], where one can use [[Feynman path integral]]s to compute the [[intersection number]]s of various algebraic moduli spaces.
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| ==References==
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| {{reflist}}
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| *{{Cite journal
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| * [[David Mumford|Mumford, David]], ''Geometric invariant theory''. [[Ergebnisse der Mathematik und ihrer Grenzgebiete]], Neue Folge, Band 34 Springer-Verlag, Berlin-New York 1965 vi+145 pp {{MathSciNet|id=0214602}}
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| * Mumford, David; Fogarty, J.; Kirwan, F. ''Geometric invariant theory''. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin, 1994. xiv+292 pp. {{MathSciNet|id=1304906}} ISBN 3-540-56963-4
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| * Papadopoulos, Athanase, ed. (2007), Handbook of Teichmüller theory. Vol. I, IRMA Lectures in Mathematics and Theoretical Physics, 11, European Mathematical Society (EMS), Zürich, {{doi|10.4171/029}}, ISBN 978-3-03719-029-6, MR2284826
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| * Papadopoulos, Athanase, ed. (2009), Handbook of Teichmüller theory. Vol. II, IRMA Lectures in Mathematics and Theoretical Physics, 13, European Mathematical Society (EMS), Zürich, {{doi|10.4171/055}}, ISBN 978-3-03719-055-5, MR2524085
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| * Papadopoulos, Athanase, ed. (2012), Handbook of Teichmüller theory. Vol. III, IRMA Lectures in Mathematics and Theoretical Physics, 17, European Mathematical Society (EMS), Zürich, {{doi|10.4171/103}}, ISBN 978-3-03719-103-3.
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| == External links ==
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| * J. Lurie, [http://math.harvard.edu/~lurie/papers/moduli.pdf Moduli Problems for Ring Spectra]
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| [[Category:Moduli theory]]
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| [[Category:Invariant theory]]
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It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.
Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.
At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.
Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.
Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.
Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.
Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.
Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.
In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.
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