Instructor: Brendon Rhoades
Instructor's Email: bprhoades (at) math.ucsd.edu
Instructor's Office: 7250 APM
Instructor's Office Hours: By appointment
Lecture Time: 1:00-1:50pm MWF
Lecture Room: 5402 APM
Lectures:
4/2: Administrivia. Chapter 1. Polynomials. k[x_1, ... ,x_n].
Ideals, ideal generation. Problems: Are ideals finitely generated,
ideal membership, ideal equality, ideal intersection in terms of generators.
Affine n-space k^n. The vanishing locus of a set of polynomials.
4/4: No class - Prof. Rhoades in Philadelphia.
4/6: Chapter 1. Varieties. Zariski topology. The ideal of a subset
of k^n. The division algorithm in k[x]. Chapter 2.2. Monomial orders: lex,
grlex, grevlex.
4/9: Chapter 2.3. The division algorithm in k[x_1,...,x_n]; examples
and properties. Chapter 2.4. Monomial ideals. Dickson's Lemma. Alternative
characterization of monomial orders.
4/11: Chapter 2.5. Groebner bases. Every ideal I in k[x_1, ... , x_n]
has a Groebner basis. Groebner bases are bases. Hilbert's Basis Theorem;
ascending chain condition; Noetherian rings. Ideal membership with
Groebner bases. Uniqueness properties of remainders with Groebner bases.
Chapter 2.6. S-polynomials. Statement of Buchberger's Criterion.
4/13: No class - Prof. Rhoades is in Nashville.
4/16: No class - Prof. Rhoades is ill.
4/18: Chapter 2.6. Proof of Buchberger's Criterion. Section 2.7.
Buchberger's Algorithm. Proof of finiteness and correctness. Minimal
Groebner bases. Example.
4/20: Section 2.7. Reduced Groebner bases. Every ideal has a unique
reduced Groebner basis given a monomial order on k[x_1, ... , x_n].
Standard monomial bases for quotients of polynomial rings.
Section 3.1. Solving systems of polynomial equations. The elimination method
and elimination ideals. The Elimination Theorem.
4/23: Section 3.1. The Extension Theorem. Example. Failure
over non-algebraically closed fields. Section 3.2.
Coordinate projections. The Geometric Extension Theorem.
The Closure Theorem. Proof of Closure Theorem (Part 2, in codimension 1)
assuming Part 1 and the Extension Theorem.
4/25: Section 3.3. Polynomial parameterization: twisted cubic
and its tangent surface. Implicitization problem. Solution with Groebner
bases. Proof of Polynomial Implicitization Theorem (over infinite field).
Section 3.5. Domains, irreducibles, and units. UFDs. Examples.
4/27: Section 3.5. GCD's in k[x] using resultants. Properties
of resultants. Section 3.6. Resultants in k[x_1, ... , x_n].
Proof of the Extension Theorem using resultants.
Section 4.1. The consistency problem for polynomial systems.
Statement of Weak Nullstellensatz.
4/30: Section 4.1. Proof of the Weak Nullstellensatz. The Strong
Nullstellensatz. Section 4.2. Radial ideals. Ideal-Variety correspondence.
Section 4.3. Ideal sum, product, and intersection; effect on varieties.
Bases for ideal intersections via elimination theory.
5/2: Section 4.4. Zariski Closure. Proof of the Closure Theorem.
Colon ideals and differences of varieties. Section 4.5. Irreducibility of varieties;
prime ideals. Points and maximal ideals. Section 4.6.
Irreducible decomposition.
5/4: Chapter 7. Action of matrix groups on polynomials. Invariant subrings.
Examples: G = {I}, G = GL_n(k), G = S_n, G = roots-of-unity.
5/7: Chapter 7. Graded vector spaces; Hilbert series. Hilbert series
of invariant rings. Two isomorphic matrix groups with different invariant rings.
Molien's Theorem. Example; proof.
5/9: Chapter 7. Applications of Molien's Theorem. Invariance of degrees.
Product of degrees equals size of group. Applications to dihedral groups and signed
permutation groups.
5/11: Chapter 7. Hilbert's Theorem: the invariant ring attached to
any finite matrix group G (when char k = 0) admits a finite set of homogeneous
generators. Noether's Theorem: In fact, the explicit set {R_G(x^a) : |a| <= |G|}
generates the invariant ring.
5/14: Hyperplanes and reflections. Reflection groups. Examples:
dihedral, symmetric group, signed permutations, type D_n. Symmetry groups of regular
polytopes.
5/16: Weyl groups. (Ir)reducible reflection groups; Cartan-Killing Classification.
Statement of Shephard-Todd-Chevalley. Divisibility Lemma. Strange Lemma. Chevalley proof
strategy.
5/18: Proof of Chevalley's Theorem. Invariant degrees. Sum of degrees.
5/21: Proof of sum-of-degrees formula. Jacobian Criterion for algebraic
independence. Examples: S_n. Shephard-Todd Theorem.
5/23: Reflection groups over fields other than R.
Invariant theory.
Classification of complex
reflection groups.
Fixed spaces. Solomon's Theorem statement.
5/25: Stirling numbers of the first kind; Solomon's Theorem for S_n.
The exterior algebra E; computations.
5/30: GL_n action on E; connection to determinants. Polynomial forms.
GL_n-action on polynomial forms. The total derivative df of a polynomial f.
Connection to Jacobians. Proof the the Jacobian of a basic invariant system
of a reflection group G is alternating. Statement of the structure theorem for
G-invariant polynomial forms.
Homework Assignments:
Homework 1, due 4/20/2018.
Homework 2, due 5/4/2018.
Homework 3, due 5/18/2018.
Lectures Notes:
Lectures 1-11.
Lectures 12-14.
Lectures 15-18.