People first encountering algebraic geometry are often disconcerted. What is all the fuss about? Systems of polynomial equations? XIXth century theorems about 27 lines?
That does not seem to jive with algebraic geometry being a core subfield of pure mathematics.
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That does not seem to jive with algebraic geometry being a core subfield of pure mathematics.
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Here is one of the not-so-secret but under-emphatized reasons why algebraic geometry is great:
The mathematical world is surprisingly algebraic.
As in: there are many interesting mathematical objects which are non-obviously connected to algebraic geometry.
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The mathematical world is surprisingly algebraic.
As in: there are many interesting mathematical objects which are non-obviously connected to algebraic geometry.
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Here are some examples of this slogan; there are many more all over the place.
a) Compact Riemann surfaces
b) Compact Lie groups
c) Shimura varieties
d) Most finite simple groups
e) Rational polytopes
f) Many interesting constants
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a) Compact Riemann surfaces
b) Compact Lie groups
c) Shimura varieties
d) Most finite simple groups
e) Rational polytopes
f) Many interesting constants
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g) Complex oriented cohomology theories
h) ...
Let's go for a stroll through this garden of algebraic surprises.
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h) ...
Let's go for a stroll through this garden of algebraic surprises.
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a) Compact Riemann surfaces.
A Riemann surface is a 1-dimensional complex manifold; a space which looks locally like a piece of the complex plane, on which one can do complex analysis.
General Riemann surfaces are quite wild, with complicated behaviour "at infinity".
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A Riemann surface is a 1-dimensional complex manifold; a space which looks locally like a piece of the complex plane, on which one can do complex analysis.
General Riemann surfaces are quite wild, with complicated behaviour "at infinity".
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Riemann proved that *compact* Riemann surfaces are algebraic curves; they can always be cut out by a finite system of polynomial equations (several variables, complex coefficients).
(Riemann's proof was not quite complete, but this story is not for today).
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(Riemann's proof was not quite complete, but this story is not for today).
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This theorem tells us that 1-dimensional complex analysis, 1-dimensional algebraic geometry and 2-dimensional riemannian geometry are intertwined.
Moreover, algebraicity is contagious: many objects attached to compact riemann surfaces also turn out to be algebraic!
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Moreover, algebraicity is contagious: many objects attached to compact riemann surfaces also turn out to be algebraic!
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E.g. the solutions of certain non-linear PDEs on Riemann surfaces, discovered by Hitchin, are actually algebraic objects. The study of those "Higgs bundles" draws together people from mathematical physics, symplectic geometry, representation theory, number theory...
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b) Compact Lie groups
A compact Lie group is... a Lie group which is compact! So a bounded group of continuous symmetries.
Again this is a very analytic definition of objects which come up naturally in differential geometry and physics.
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A compact Lie group is... a Lie group which is compact! So a bounded group of continuous symmetries.
Again this is a very analytic definition of objects which come up naturally in differential geometry and physics.
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Surprise! Compact Lie groups are algebraic. More precisely, they can all be realised as subgroups of a matrix group GL_n(C), cut out by polynomial equations with real coefficients.
Similarly, all their finite-dimensional representations are algebraic in a precise sense.
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Similarly, all their finite-dimensional representations are algebraic in a precise sense.
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Finally, other objects attached to compact Lie groups, like generalised flag varieties, are also algebraic varieties.
This is in a way the start of *geometric representation theory* : algebraic geometry in the service of representation theory.
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This is in a way the start of *geometric representation theory* : algebraic geometry in the service of representation theory.
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c) Shimura varieties
A bounded symmetric domain is an open subset U in C^n such that, for every x in U, there is an holomorphic involution of U with x as isolated fixed point. In other words, lots of holomorphic symmetries. Ex: open unit ball B_n in C^n.
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A bounded symmetric domain is an open subset U in C^n such that, for every x in U, there is an holomorphic involution of U with x as isolated fixed point. In other words, lots of holomorphic symmetries. Ex: open unit ball B_n in C^n.
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A more complicated example: the set S_n of symmetric complex n by n matrices Z such that Id- Z* Z is positive definite (Z* conjuguate-transpose).
There is a real algebraic group acting on U by holomorphic automorphisms. In this group, there are many discrete subgroups.
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There is a real algebraic group acting on U by holomorphic automorphisms. In this group, there are many discrete subgroups.
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Arithmetic subgroups are particular discrete subgroups which (roughly) can be written with matrices with integral coeffs. Let G be such a (torsion-free) arithmetic subgroup acting on U. Then U/G is a complex manifold.
Thm(Baily-Borel): U/G is an algebraic variety.
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Thm(Baily-Borel): U/G is an algebraic variety.
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The theorem can be made a lot more precise: U/G is a quasi-projective variety, there is a natural projective compactification built from U, the complex algebraic variety U/G is actually defined over a number field.
U/G is called a Shimura variety (small lie here!)
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U/G is called a Shimura variety (small lie here!)
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Ex: U=unit disk in C, G in SL_2(R) defined by integral matrices with congruence conditions. Then U/G is called a modular curve.
Shimura varieties are central objects in modern number theory, in a way that's a bit difficult to summarize.
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Shimura varieties are central objects in modern number theory, in a way that's a bit difficult to summarize.
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Basically, complex analysis on Shimura varieties is (a part of) the theory of modular and automorphic forms.
Automorphic forms are a blend of representation and number theory which gives rise to well-understood L-functions (generalising the Riemann zeta function).
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Automorphic forms are a blend of representation and number theory which gives rise to well-understood L-functions (generalising the Riemann zeta function).
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The fact that Shimura varieties are algebraic and defined over a number field connects automorphic forms with the arithmetic of reduction of Shimura varieties over finite fields. Shimura varieties are often moduli spaces of abelian varieties and this is useful as well!
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Shimura varieties, with their triple analytic/algebraic/arithmetic nature, feature in the proofs of Mordell's conjecture by Faltings and of Fermat's last theorem by Wiles, among many other results in modern number theory.
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