Math 583 - Applied Algebraic Geometry

Brief summary of each class and the corresponding homework exercises.

Upcoming Events

10 Monday - 5/29/2017

• No class -- Memorial Day. Slides due on Tuesday by 6:00pm!

10 Wednesday - 5/31/2017

• Presentations

10 Friday - 6/2/2017

• Presentations

Past Events

9 Friday - 5/26/2017

• Based on Chapter 7 Section 4 and some of Chapter 8.
• Group orbits and affine varieties. Intro to projective varieties.

9 Wednesday - 5/24/2017

• Based on Chapter 7, Sections 3 and most of 4.
• Exercises: Chapter 7, Section 3: 1, 5,6 10.
• Exercises: Chapter 7, Section 4: 3, 4, 6, 10, 11, and 16.

9 Monday - 5/22/2017

• Based on Chapter 7, Sections 1 and 2.
• Exercises: Chapter 7, Section 2: Graham said 1-16 plus Sean's questions: "Say G is a finite group of diagonal matrices in Gl(n,k). Is the ring of G-invariants simply a polynomial ring in certain monomials? If the ring of invariants is a polynomial ring in monomials is G a subgroup of the diagonal matrices?"
• Exercises: Chapter 7, Section 1: 1,10,11,12,14.
• Exercises: Chapter 7, Section 2: 4,5,6,7.

8 Wednesday - 5/17/2017

• Based on Chapter 6, Section 4.
• Automatic geometric theorem proving.
• Exercises: Chapter 6, Section 4: 2, 8, 12

8 Monday - 5/15/2017

• Tentatively based on Chapter 6, Sections 1 and 2.
• Geometric description of Robots
• Exercises: Chapter 6, Section 1: 3,4,5.
• Exercises: Chapter 6, Section 2: 3,4,6,7.

7 Friday - 5/12/2017

• Based on Chapter 5, Section 6
• Noether normalization
• Exercises: Chapter 5, Section 6: 1,2, 4, 6,18

7 Wednesday - 5/10/2017

• Based on Chapter 5, Sections 5.
• Rational Functions on a variety, birational equivalence.
• Exercises: Chapter 5, Section 5: 2,3,4,13

7 Monday - 5/8/2017

• Based on Chapter 5, Section 4 Maybe a bit more.
• Subvarieties, pencil's, pullbacks, and isomorphic varieties.
• Exercises: Chapter 5, Section 4: 2,3,5,10,14,16.

6 Friday - 5/5/2017

• Based on the material toward the end of Chapter 5, Sections 3.
• Finite varieties and enumerative algebraic geometry
• Exercises: Chapter 5, Section 3: 2, 3, 6, 8, 12, 13

6 Wednesday - 5/3/2017

• Tentatively based on Chapter 5, Sections 1 and 2. Maybe a bit more.
• Intro to polynomial and rational function on a variety
• Exercises: Chapter 5, Section 1: 1,2,3,6,9,10
• Exercises: Chapter 5, Section 2: 5,9,10,11,14,18

6 Monday - 5/1/2017

• Based on Chapter 4, Sections 4, 5 and 6.
• More on quotient ideals and saturation. Going on to irreducible varieties and prime ideals.
• Exercises: Chapter 4, Section 5: 2,3,7,10,12,

5 Friday - 4/28/2017

• Based on Chapter 4, Sections 3 and 4.
• More on intersections of ideals, then going to to Zariski Closures and quotients.
• Exercises: Chapter 4, Section 4: 1,2,3,13,15.

5 Wednesday - 4/26/2017

• Based on Chapter 4, Sections 2 and 3.
• "The Nullstellensatz" and Sums,Products and Intersections of Ideals
• Exercises: Chapter 4, Section 2: 1,2,6,7,15,16.
• Exercises: Chapter 4, Section 3: 1, 5, 9, and 12, plus Josh's question from class "Does LT(I intersect J) =LT(I) intersect LT(J)?".

5 Friday - 4/28/2017

• Based on Chapter 3, Section 4 (reprive) and Chapter 4 Section 1.
• More on Envelopes and moving on to Hilbert's Nullstellensatz and radical ideals.
• Exercises: Chapter 4, Section 1: 1,2,4,6,10

4 Friday - 4/21/2017

• Based on Chapter 3, Sections 5.
• Exercises: Chapter 3, Section 5: 4,7
• Skim Chapter 3, Section 6 on Resultants on your own.

4 Wednesday - 4/19/2017

• Based on Chapter 3, Sections 3 and 4.
• Implicitization, Singular points and Envelopes
• Exercises: Chapter 3, Section 3: 4,6,7,9,12
• Exercises: Chapter 3, Section 4: 1,3,8,11, 15

4 Monday - 4/17/2017

• Based on Chapter 3, Sections 1 and 2.
• Elimination Theory
• Exercises: Chapter 3, Section 1: 2,3,5,7
• Exercises: Chapter 3, Section 2: 2,3,4

3 Friday - 4/14/2017

• Tentatively based on Chapter 2, Sections 9 and 10.
• Improvements on Buchberger's Algorithm
• Exercises: Chapter 2, Section 9: 1,2,3
• Exercises: Chapter 2, Section 10: 1,2, Bonus: 12.

3 Wednesday - 4/12/2017

• Based on Chapter 2, Sections 7 and 8.
• Buchberger's Algorithm
• First Applications of Gröbner Bases
• Exercises: Chapter 2, Section 7: 2,3,5,7,10,11
• Exercises: Chapter 2, Section 8: 1,3,8,11

3 Monday - 4/10/2017

• Based on Chapter 2, Sections 6.
• Properties of Gröbner Bases
• Exercises: Chapter 2, Section 6: 1,2,3,4,5,6,9,10,13 (sorry so many, but they are all useful!)

2 Friday - 4/7/2017

• Based on Chapter 2, Sections 4 and 5.
• Monomial Ideals and Dickson's Lemma
• Hilbert's Basis Theorem and Gröbner Bases
• Exercises: Chapter 2, Section 4: 4,5,7,8,10,11.
• Exercises: Chapter 2, Section 5: 1,7,8,9,13,18.

2 Wednesday - 4/5/2017

• Read Chapter 2, Section 1 in advance. This lecture is based on Chapter 2, Sections 2 and 3.
• Orderings on Monomials including lex, grlex, grevlex
• Division Algorithm for Polynomials in Any Number of Variables
• Exercises: Chapter 2, Section 1: 1,2,3,4,5.
• Exercises: Chapter 2, Section 2: 2,3,7,10,11,13
• Exercises: Chapter 2, Section 3: 1,2,3,5,6,7,8,11

2 Monday - 4/3/2017

• Introduction to Sage by Andrew Ohana
• Bring laptops to class if you wish.
• Please sign up for a SageMathCloud account, preferably with your UW net id. Look for worksheet in your new class project.

1 Friday - 3/31/2017

• Bezier equations would make a nice project!
• Ideals
• Division algorithm for polynomials in one variable
• Reading Assignment: Chapter 1, Sections 4 and 5. In particular, study the Euclidean algorithm for finding the gcd of two polynomials in k[x].
• Exercises: Chapter 1, Section 4: 1,3,5, 8, 12, 13.
• Exercises: Chapter 1, Section 5: 2, 3, 8, 9, 10, 11, 15, 16.

1 Wednesday - 3/29/2017

• Parametrizations of affine varieties
• Reading Assignment: Chapter 1, Section 3.
• Exercises: Chapter 1, Section 3: 1,2,5,8,10,11, (Bonus 16)

1 Monday - 3/27/2017

• Introduction to affine varieties
• Reading Assignment: Chapter 1, Sections 1 and 2.
• Exercises: Chapter 1, Section 1: 1,2,3,4,5,6.
• Exercises: Chapter 1, Section 2: 1,4,7,10,12,15.