User:Z423x5c6/secreto

Richard E. BORCHERDS

Modular forms

Intro
1. Monster moonshine
2. Almost integer (-163, Heegner number)
3. Sphere packing (dim 8, 24)
4. Langland's program
Modular forms
1. Modular function is a function from the moduli space of elliptic curve to $\mathbb {C}$
2. $f(\sigma \tau )=f(\tau )$
3. Modular forms
  1. (Def) $f(\sigma \tau )=(c\tau +d)^{k}f(\tau )$
  2. "Invariant forms" $\sigma (f(\tau )(\mathrm {d} \tau )^{k/2})=f(\tau )(\mathrm {d} \tau )^{k/2}$
  3. $g$ : homogeneous function on lattice: $g(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-k}g(\omega _{1},\omega _{2}),f(\tau )=g(1,\tau )$
  4. Ratio of modular forms of same weight gives modular functions
4. Eg. Weierstrass P function, Laurent expansion
5. Eisenstein series
  1. Meromorphic function approach
  2. "Invariant forms" approach
  3. differ by a factor of $\zeta (k)$
Fourier coefficients
1. Take the sum row by row to show explicit formula of $E_{4}$
2. Need to use the function ${\frac {\pi }{\tan \pi z}}$ , Bernoulli numbers.
3. 691 is a special number.
4. $E_{4}^{2}=E_{8}$ , $E_{4}\times E_{6}=E_{10}$ . Mysterious relations between $\sigma _{3},\sigma _{7},\ldots$
5. 'First' interesting modular function: j-invariant (need weight-12 / weight-12).
6. Dimensions of spaces of modular forms are small.
Fundamental domain
1. Elliptic points ( $e^{i\pi /3}$ , $e^{i\pi /2}$ ) corresponds to lattices with extra symmetries
2. Cusp point
3. For other levels, fundamental domain has a different shape (but similar in nature, with elliptic points, cusp points...)
Classification of level 1 forms
1. Correct way to count zeros of forms on fundamental domains:
  1. Need to count zero at $i\infty$
  2. Count zeros at elliptic points 'as fractional zeros'
  3. This way of counting also generalizes to other levels
  4. Number of zero $={\frac {k}{12}}={\frac {\text{Area of FD}}{4\pi }}$
2. Use discriminant form to do induction $k\to k-12$
3. Conclusion: Forms are generated by $E_{4}$ and $E_{6}$ as polynomials
Modular functions
1. j is the fundamental modular function. Every modular function is a ration function of j.
2. "Black magic" application: prove Picard theorem.
$\Delta$ $\Delta$ and $E_{2}$ $E_{2}$
1. ${\frac {\Delta '}{\Delta }}=2\pi iE_{2}$
2. $E_{2}$ "could be holomorphic, or modular, but not both."
3. Summing the series for $E_{2}$ by rows or by columns differ by a constant, because not absolute convergenct.
4. $\eta ^{24}=\Delta$
5. (A lot of complex analysis)
$\theta$ $\theta$
1. Modular condition only holds wrt a subgroup of $SL_{2}(\mathbb {Z} )$ generated by ${\begin{pmatrix}1&2\\0&1\end{pmatrix}}$ and ${\begin{pmatrix}0&1\\-1&0\end{pmatrix}}$ (of index 3) (and up to a root of unity).
2. Functional equation of $\theta$ can prove that of $\zeta$ .
3. Convergence issues are solved by chopping off $\theta$ and some analytic continuation.
Intro to Hecke operators
1. Symmetries that $f(2\tau )$ has? Need to consider $f(2\tau )+f(\tau /2)+f((\tau +1)/2)$ . Index 3 subgroup of $SL_{2}(\mathbb {Z} )$
2. Sum of functions of sub lattices of index n.
3. Sum over matrices
4. 2 and 3 are the same. 1 is a little bit not so nice.
5. Action on Fourier coefficients: both expanding and contracting.
Product formula for j
1. (the formula)
2. related to Weil denominator formula
3. Monster Lie algebra
4. Modular functions are determined by coefficients of $q^{0},q^{-1},\ldots$
5. (Magical proof of taking log, Hecke operator then some property of modular functions)
Hecke operators on forms
1. Similar to 9
2. Applied on $\Delta$ , get information on Ramanujan's $\tau$ function
3. L series, similar functional equation as $\zeta$
4. $|\tau (p)|\leq 2p^{11}$
Peterson inner product
1. If space of forms has dimension $>1$ , it is still spanned by 'eigen functions' of Hecke operators
2. Hecke operators are self adjoint (under Peterson inner product) and commutative
3. To define an $SL_{2}(\mathbb {Z} )$ -invariant inner product

AG1: Varieties

Intro
1. Motivation by example: parametrizing the circle
2. Important POV: equation = functor from Ring to AlgGp/AlgVar
Examples
1. $y^{2}=x^{3}+x^{2}$ $y^{2}=x^{3}+x^{2}$
  1. Resolution of singularities - blow up
2. $x^{3}+y^{3}=9$ $x^{3}+y^{3}=9$
  1. Elliptic curve, group law
Theorems: Bezout, Pappus, Pascal
Kakeya set on finite fields (with proof), and example of 27 lines on a cubic
Affine space, Zariski topology
1. Eg: determinantal variety
Noetherian space
1. Hilbert basis theorem
2. Noetherian + Hausdorff = finite
3. Noetherian = finite union of irr.s
Weak Nullstellensatz
1. Maximal ideal <=> points
Strong Nullstellensatz
1. Rabinowitsch trick
2. Algebraic set <=> radical ideals
3. Egs: nilpotent matrices, commuting matrices
Lasker-Noether theorem
1. Primary, coprimary modules
2. For $\mathbb {Z}$ -modules, it becomes the structure theorem
Proof (sketch)
1. All ideals are intersection of irr ideals
2. irr is primary
3. (xy, y²) illustrating the idea of embedded point
Quotient of varieties by group (with 3 eg)
1. Taking quotient of algebraic sets by group could be difficult (how can one describe the set in affine space?)
2. Do it on the ring side (use invariant functions)
3. eg: $A^{n}/S_{n}$
4. eg: $A^{n}/GL_{n}$
5. eg: $SL_{n}$ $SL_{n}$ on $a_{n}x^{n}+a_{n-1}x^{n-1}y+\ldots +a_{0}y^{n}$ $a_{n}x^{n}+a_{n-1}x^{n-1}y+\ldots +a_{0}y^{n}$
  1. invariant function example: discriminant (b²-4ac)
Hilbert's proof
1. Reynolds operator (taking average over the group action)
2. Can generalize to compact groups (could still integrate)
3. Also $SL$ , by Weyl's unitary trick
4. Nagata's counterexample (use unipotent group, which is 'counterexample to everything (?????)')
3 examples of quotients
1. cyclic group $\sigma (x)=\xi x,\sigma (y)=\xi y$ , invariant functions generated by $x^{n},x^{n-1}y,\ldots ,y^{n}$
2. parameter space of cyclohexane, 1 point (chair) + 1 1-d space (boat)
3. moduli space of elliptic curve $y^{2}=x(x-1)(x-\lambda )$ , j-invariant
Many definitions of dimension
Projective space
1. Hopf fibration
2. Fano plane
Desargues' theorem
1. corr. to associativity of multiplication (similar to Pappus <-> commutativity of multiplication)
2. Duality in projective geometry <-> dual space in linear algebra
Affine and projective varieties
1. Don't just homogenize one set of generators of the ideal, may add extra curves on it.
Product of varieties
1. Serge embedding
2. Eg: $P^{1}\times P^{1}\hookrightarrow P^{3}$
3. Spheres have (complex) lines in it:
  1. x=1, y=iz on x²+y²+z²=1
Veronese surface and variety of lines in space
1. Veronese surface: $(x,y,z)\to (x^{2},xy,y^{2},xz,yz,z^{2})$
2. Generalize to $A^{m}\to S^{n}(A^{m})$
3. lines in P³ = 2-dim subspaces of A⁴ = points in G(2,2)
4. Plücker embedding
5. Decompose G(2,2) into affines give cohomology groups over $\mathbb {C}$ and number of points over finite fields
6. Weil conjecture?
Grassmannians
1. idea: $V\to W\rightsquigarrow \wedge ^{m}V\to \wedge ^{m}W$ which is just projective space
2. Plücker relation
3. Covered by affines
4. Littlewood-Richardson rule
5. Line complexes
6. Quotient of affine varieties by affine varieties may be projective (maybe not even projective)
7. Hilbert scheme
8. The embedding $P^{3}\to P^{5}$ is 'natural'
Projective bundles
1. Hirzebruch surface: $(\lambda ,\mu )(x_{0},x_{1},x_{2},x_{3})=(\lambda x_{0},\lambda x_{1},\mu x_{2},\mu \lambda ^{-n}x_{3})$
2. Scroll (higher dimensional generalization)
3. Abstract variety
Toric varieties
1. Drawing/gluing cones on the lattices/dual lattices
Category
1. Affine varieties form a category (in two ways: with regular/rational functions)
Regular functions
1. local regularity=regularity
2. proof: 'partition of unity'
3. regular functions on P1=constant
Morphisms of varieties
1. Geometric space = locally ringed space
2. Different functions = different geometry, eg, top, C1, rational functions
3. Eg: A1->cuspidal cubic, morphism+homeo, but not isomorphism
Algebraic varieties and fg algebra w/o nilpotent
1. Equivalent of (opposite) category
2. Group varieties
3. Coproduct
4. Hopf algebra
Twisted cubic
1. isomorphic to P1, but they corr to different graded rings
2. A2-origin is not affine, but A1-origin is affine
3. The main difference is that one is of codim 1, the other is of codim 2.
(Verified that Serge embedding is the product, satisfying universal property...)
Automorphism of An and Pn
1. Aut A1 = ax+b group
2. Aut A2 is already much larger, for example, x->x, y->y+p(x)
3. Jacobian conjecture
4. Aut P1 = PGL(2)
5. Image of morphism from An to An could be crazy, for example, (x, y)->(x, xy)
Ax-Grothendieck thm
1. Prove the finite field case, then use some crazy ideas from model theory.
Rational map
1. Category of varieties, dominant rational maps
2. Elliptic curve is not rational
Elliptic function
1. Weierstrass p function -> elliptic curve 'is' torus
2. P1 'is' sphere
3. so elliptic curve is not rational
Cubic surfaces
1. Cubics vanishing on 6 points in P2 provide a cubic surface
2. Dimension argument provides heuristic reason most cubic surfaces can be constructed this way
3. Cubic surface 'is' blowing up 6 points on P2
Blow up
1. Some nice examples
2. Cone x2+y2=z2 becomes cylinder
3. Whitney umbrella
  1. Blowing up at origin provides again a Whitney umbrella
  2. Blowing up a line is good
More on blow up
1. blow up one point on R2=Mobius band
2. blow up 1 point on P1xP1=blow up points on P2
3. blow up (x^2,y^2) in k[x, y] -> Whitney umbrella
Atiyah flop
1. xy=zw, blow up at origin and the exceptional locus is P1xP1
2. This singularity has 2 'minimal' way to resolve, namely the two P1.
3. If we want a minimal model, we must allow 'terminal singularities'
Singular points
1. Singular points are those with tangent space of 'wrong dimension'
2. Set of singular points is closed (for varieties)
3. Set of nonsingular points is open dense (for varieties)
Zariski (co)tangent space
1. m/m2
2. Dual of m/m2 = map from R to k[ε]/(ε2)
3. Use module to define: W=module generated by dx1, dx2,..., dxn with Leibniz rule
4. A local ring is called regular if the dimension of Zariski tangent space equals the dimension of the local ring
Du Val singularities
1. Some singularities are called Du Val singularities
2. Eg: x2+y3+z5=0
3. Blow up 8 times to resolve the singularity
4. The intersection pattern of those exceptional locus looks like the Dynkin diagram of E8
Examples of resolutions
1. x4+y4=z2
  1. Blow up once -> singular line -> blow up again
2. Blow up can make things worst:
  1. x2-yz=0
  2. Blow up on y=z=0
  3. Get a Whitney umbrella
3. Number theory
  1. $\mathbb {Z} [{\sqrt {-3}}]$ , not UFD
  2. Has a 'singular' point
  3. Normalization $\rightsquigarrow \mathbb {Z} \left[{\frac {{\sqrt {-3}}+1}{2}}\right]$
4. Application
  1. To analytic continue $\int |f(x_{1},\ldots ,x_{n})|^{s}\varphi (x_{1},\ldots ,x_{n})dx$
Completion
1. $k[x]\rightsquigarrow k[[x]]$
2. $\mathbb {Z} _{p}$
3. $R\to {\hat {R}}$ may not be injective (if not Noetherian)
4. May produce zero divisors/nilpotents (eg $y^{2}=x^{3}+x^{2}$ )
5. $R\to {\hat {R}}\to \ldots \to R/m^{3}\to R/m^{2}\to R/m=k$ is like focusing on smaller and smaller neighborhood
Resultant
Proper map
1. The 'correct' analog of 'compactness'
2. $\mathbb {P} ^{n}\to \{{\text{point}}\}$ $\mathbb {P} ^{n}\to \{{\text{point}}\}$ are proper
  1. Proof: induction
  2. Using 'blowup $\mathbb {P} ^{n}$ at a point is a $\mathbb {P} ^{1}$ -bundle over $\mathbb {P} ^{n-1}$
Survey of curves
1. 3 viewpoints of algebraic curves over $\mathbb {C}$ $\mathbb {C}$ :
  1. Nonsingular projective curves
  2. Compact Riemann surfaces
  3. Finitely generated field of tran deg 1
2. Classify by genus:
  1. Genus 0: $\mathbb {P} ^{1}$
  2. Genus 1: elliptic curves (j-invariant, λ-line/S3)
  3. Genus 2: hyper-elliptic
    1. $y^{2}=(x-a_{1})\ldots (x-a_{6})$
    2. 6 points in $\mathbb {P} ^{1}$ / $PSL_{2}(\mathbb {C} )$
    3. $\mathbb {A} ^{3}/(\mathbb {Z} /5\mathbb {Z} )$
  4. Genus 3:
    1. hyper-elliptic: 5-dim family
    2. deg 4 nonsingular curves in plane: 6-dim family
    3. Eg(Trott curve): $144(x^{4}+y^{4})-225(x^{2}+y^{2})+350x^{2}y^{2}+81=0$
    4. 28 bi-tangents
  5. Genus 4: intersection of cubic and quadric in $\mathbb {P} ^{3}$
  6. Genus 5: intersection of 3 quadrics in $\mathbb {P} ^{4}$
Hurwitz curve
1. Most symmetric curve of genus g
2. g=0, 1: infinite group
3. g>1: $|G|\leq 84(g-1)$ (in char 0)
4. Idea of proof: use orbitfold Euler characteristic
5. Must be 3 conical points, of order 2, 3, 7
Examples of Hurwitz curve
1. Even for g=0, 1 we can get something interesting
2. Eg there is a group of order 60 acting on $\mathbb {P} ^{1}$ , some finite groups acting on elliptic curve
3. g=2? Cannot achieve 84.
  1. Remember during the proof, if $|G|\neq 84(g-1)$ then $|G|\leq 48(g-1)$
  2. 48 is achievable
4. g=3? Yes.
  1. Klein quartic: $x^{3}y+y^{3}z+z^{3}x=0$ in $\mathbb {P} ^{2}$
  2. Aut group = $PSL_{2}(F_{7})$
Resolution of curve singularities
1. Given a function field, recover the curve?
2. Field generated by x, y, with $f(x,y)=0$ . Possibly singular.
3. Can be resolved by repeatedly blowing up and changing variables.
4. Works for algebraic curve, but for scheme (if we allows multiple factors) this may not work. Also requires char=0
Newton's rotating ruler
1. Can solve algebraic equation in terms of Puiseux series
2. Field of Puiseux series is algebraically closed and 'quasi-finite' (similar Galois extensions)
3. Can resolve singularities using Puiseux series
Hilbert polynomial
Degree of a projective variety
1. Arithmetic genus $=(-1)^{\text{dim}}(\chi (V)-1)$
2. $\chi (V)=$ constant term of the Hilbert polynomial
3. $\chi (V)$ is 'better' in the sense that $\chi (V\times W)=\chi (V)\times \chi (W)$
4. Hilbert polynomial is the 'only' invariant (Hilbert scheme ???)
Bezout's theorem
1. Def of multiplicity: not always well defined, eg, multiplicity of $\mathbb {Z} /2\mathbb {Z}$ in $\mathbb {Z}$
2. Only well defined for minimal primes (maximal varieties)
3. Defined by length of $M_{p}$ over $R_{p}$ : idea is that localization is making an ideal maximal. This ideal is both minimal and maximal, so it is the 'only' ideal.
4. Finally, define the 'intersection multiplicity' of a subvariety, to make Bezout's theorem true.

AG2: Scheme

Introduction
1. Generalize to arbitrary rings
2. Introduction of sheaf
Sheaves
1. Ring elements are naturally 'functions' on Spec, but the codomain varies.
2. Sheaf of _ is just like _ (eg Sets, Abelian groups)
3. Sheaf morphism surjective doesn't mean surjective on open sets.

Commutative algebra

Introduction
1. Algebraic geometry
2. Number theory
3. Invariant theory
Rings, ideals and modules
1. Some examples of rings
  1. Analysis: with unity <-> compact space
  2. Adding unity = 1-point compactification
2. Ideals
3. Modules
  1. Modules are better, more general
Syzygy
1. Ring of invariants
  1. O(3): $x^{2}+y^{2}+z^{2}$
  2. Sn: generated by e1, ..., en
  3. An: generated by e1, ..., en, $\Delta$ , with a syzygy
  4. Z/3Z: order-2 syzygy
Invariant theory
1. Concept of Noetherian ring/modules
Noetherian rings
1. Many examples
2. Subring of Noetherian rings may not be Noetherian, but quotient is
R Noetherian => R[x] Noetherian
1. Proof
2. Similar proof, R Noetherian => R[[x]] also Noetherian
Proof of Hilbert's theorem
1. Noetherian + Reynold's operator
2. Can generalize to Sl(R), using Weyl's unitarian trick