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.