Last Day of Class: Thursday, December 3.

Midterm Exam: Thursday, October 15.

Term Papers Due: Thursday, December 17.

Some familiarity with mathematical software (e.g. SAGE, Maple, or Mathematica)

by David Cox, John Little, and Donal O'Shea, Springer Undergraduate Texts in Mathematics, 2007.

2. Gröbner Bases

3. Elimination Theory

4. The Algebra-Geometry Dictionary

5. Polynomial and Rational Functions on a Variety

6. Robotics and Automatic Geometry Theorem Proving

7. Invariant Theory of Finite Groups

8. Projective Algebraic Geometry

9. The Dimension of a Variety

The sections to be covered in each lecture are listed below. Please read these before coming to class.

**Grading:**
There will be weekly homework sets and a midterm exam (in-class).
The final is a term paper (take-home).

The grading scheme is:
**Homework 35%,
Midterm 30%,
Term Paper 35%**.

**Homework:**
There will be a weekly homework assignment,
to be handed in on Tuesdays at 9:30am, at the end of class.

Late homework will not be accepted. No exceptions.
The assignments, posted below, refer to the text book.

No homework after November 10, so you can focus on your term paper.

**Final Exam:** You will write a term paper on a topic
of their choice related to the class. You may work on this by yourself

or in teams of two. Please submit a proposal for your project
on Tuesday, October 27. This should fit on
one page and contain:

names of author(s), title,
sources, and a brief description.
The final version of the paper is due on
Wednesday, December 16.

** DAILY SCHEDULE: **

Aug 27 (ER): 1.1 Polynomials and Affine Space, 1.2 Affine Varieties,
1.3 Parametrizations of Affine Varieties

Sep 1: 1.4 Ideals, 1.5 Polynomials of One Variable, 2.1 Introduction
to Gröbner Bases

Sep 3: 2.2 Orderings on Monomials, 2.3 A Division Algorithm

Sep 8: 2.4 Monomial Ideals and Dickson's Lemma, 2.5
The Hilbert Basis Theorem and Gröbner Bases

Sep 10: 2.6 Properties of Gröbner Bases, 2.7 Buchberger's Algorithm,
2.8 First Applications

Sep 15 (MM): 3.1 The Elimination and Extension Theorem,
3.2 The Geometry of Elimination

Sep 17 (MM): 3.3 Implicitization, 3.4 Singular Points and Envelopes

Sep 22 (MM): 3.5 Unique Factorization and Resultants, 3.6
Resultants and the Extension Theorem

Sep 24: 4.1 Hilbert's Nullstellensatz, 4.2 Radical Ideals and the
Ideal-Variety Correspondence

Sep 29: 4.3 Sums, Products and Intersections of Ideals,
4.4 Zariski Closure and Quotients of Ideals

Oct 1: 4.5 Irreducible Varieties, 4.6 Decomposition of a Variety

Oct 6 (MH): 5.1 Polynomial Mappings, 5.2 Quotients of Polynomial Rings

Oct 8 (MH): 5.3 Computing in K[x]/I, 5.4 The Coordinate Ring of an Affine Variety

Oct 13: Review for the Midterm

Oct 15: MIDTERM EXAM

Oct 20: The software Macaulay2, Discussion of Term Papers

Oct 22: 4.7. Primary Decomposition of Ideals, 9.1 The Variety of a Monomial Ideal

Oct 27: 8.1 The Projective Plane, 8.2 Projective Space and Projective Varieties

Oct 29: 8.3 The Projective Algebra-Geometry Dictionary,
8.4 The Projective Closure of an Affine Variety

Nov 3: 8.5 Projective Elimination Theory, 8.6 The Geometry of Quadric Hypersurfaces

Nov 5: 8.7 Bezout's Theorem, the software Bertini

Nov 10: 9.2 The Complement of a Monomial Ideal, 9.3
The Hilbert Function and the Dimension of a Variety

Nov 12: 9.4 Elementary Properties of Dimension,
9.5 Dimension and Algebraic Independence

Nov 17: 7.1 Symmetric Polynomials, 7.2 Finite Matrix Groups and Rings of Invariants

Nov 19: 7.3 Generators for the Ring of Invariants,
7.4 Relations Among Generators and the Geometry of Orbits

Nov 24: 9.6 Dimension and Nonsingularity, 9.7 The Tangent Cone

Dec 1: ** Presentation of Term Papers**

8:10 Liz Ferme: Borel-fixed Ideals

8:30 Marley Ummel: Robotics

8:50 Richard Adelstein: Bezout's Theorem

9:10 Albert Zheng: The Cayley-Bacharach Theorem

Dec 3: ** Presentation of Term Papers**

8:10 Joelle Lim: Elliptic Curves

8:30 Julio Soldevilla: N-Fold Integer Programming Games

8:50 Hannah Wheelen: Buchberger's Algorithm in Particle Physics

9:10 Claire Tiffany-Appleton and Meghan McConlogue:
Reverse Engineering of Gene Regulatory Networks

Dec 17: ** Presentation of Term Papers** (in **939** Evans Hall)

9:00 Nishant Pappireddi: Zerodimensional Varieties via Eigenvalues

9:20 Wei Cheng Ng: Phylogenetic Algebraic Geometry and Linguistics

9:40 Benjamin Chu: Ideals and Neural Codes

COFFEE BREAK

10:20 Vrettos Moulos: Real Algebraic Geometry and Optimization

10:40 Frank Ong: The Positivstellensatz

11:00 Mahrud Sayrafi: Semidefinite Optimization and Nonnegative Polynomials

COFFEE BREAK

11:40 Hui Yu Lu: The Quadratic Line Complex

12:00 Davis Foote: Algebraic Coding Theory

12:20 Sophia Elia: Modeling Surfaces with Surfex

LUNCH BREAK

13:40 Helen Zhenzheng Hu: Applications of Algebraic Geometry in Game Theory

14:00 Bryan Wang: The Polynomial Method in Graph Theory

14:20 Shensheng Chen: Gröbner Bases with a View towards Tropical Geometry

14:40 Edward Kim: The Hopkins-Levitzki Theorem

** Homework assignments: **

due Sep 1: Section 1.1: # 4; Section 1.2: # 4, 7, 10; Section 1.3: # 4, 6.

due Sep 8: Sec 1.4: # 3,8,12,15;Sec 1.5: # 2,10,12,17;
Sec 2.1: # 5; Sec 2.2: # 5,10,12; Sec 2.3: # 6,7.

due Sep 15: Sec 2.4: # 3, 11; Sec 2.5: # 10, 18; Sec 2.6: # 3, 10;
Sec 2.7: # 2, 11; Sec 2.8: # 3, 5.

due Sep 22: Sec 3.1: # 4, 5, 9; Sec 3.2: # 3, 5; Sec 3.3: # 9, 14; Sec 3.4: # 9, 12.

due Sep 29: Sec 3.5: # 7, 8, 9; Sec 3.6: # 2, 7; Sec 4.1: # 7, 10; Sec 4.2: # 2, 7.

due Oct 6: Sec 4.3: # 9, 11; Sec 4.4: # 2, 8, 10;
Sec 4.5: # 4, 6, 12; Sec 4.6: # 1, 4, 7.

due Oct 13: Sec 5.1: # 2, 9; Sec 5.2: # 5, 11, 16; Sec 5.3: # 5, 10, 13; Sec 5.4: # 9, 15.

due Oct 27: Sec 4.7: # 2, 3, 11, 12; Sec 9.1: 2, 5, 6;
submit your term paper proposal.

due Nov 3: Sec 8.1: 9, 11; Sec 8.2: 5, 16, 17, 19;
Sec 8.3: 3, 6; Sec 8.4: 5, 11, 12.

due Nov 10: Sec 8.5: 7, 9, 17; Sec 8.7: 7, 8, 9; Sec 8.6: 8, 14, 16.