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Computational Algebraic Geometry - Fall 2015

The lectures of this course will be based on a diverse selection of books and papers. However, the following books should constitute a great portion of the material for us. We'll be doing a good amount of theory mixed with coding and applications. My preferred software is Macaulay 2 but depending on the problem at hand, we might have to use other tools as well. The lectures will be mostly in English but students also are welcome to ask quesions, have discussions with their peers or give presentations in Spanish.

Download the coures outline here.

The course will be selfcontained. However, a solid background in undergraduate/graduate algebra is going to facilitate the process. Use Introduction To Commutative Algebra by Atiyah and McDonald or Commutative Algebra with a View Toward Algebraic Geometry by Eisenbud for the background. The asignments will be adjusted according to the level of the participating students.
The following list is an ambitious list of topics to include in this course. Although we might not be able to get through every topic, it's good to have a big picture to save us from getting trapped in the details.

Projects, Presentations during class time, Tues/Fri at 4 PM

The written documents are downloadable through the following links.
JERSON LEONARDO CARO REYES Tues, Oct 27
Rees and Blowup Algebras, Intgeral closure and Numerical Invariants
JUANITA DUQUE ROSERO Fri, Oct 30
Newton Polytopes, Bernstein's Theorem, Discriminants and the Seconadary polytope
JOSE MIGUEL CRUZ RANGEL Tues, Nov 3
Real Algebraic Geometry, Positivity and Optimization
GUSTAVO CHAPARRO SUMALAVE Fri, Nov 6
The Riemann-Roch Formula
JUAN RAFAEL ALVAREZ VELASQUEZ Tues, Nov 10
Computer Vision and Hilbert Schemes
DIEGO ANTONIO ROBAYO BARGANS Fri, Nov 13
D-modules and the Bernstein-Sato Polynomial

Homeworks and stuff

HW #1
HW #2
Midterm Test
HW #3
HW #4
HW #5
Final Exam

Practice Homework

These are some suggested homework questions to get you started. This list will be updated as we make progress.
From CLO
Read Chapter 1
2.2 4, 5, 6, 7
2.3 2, 4, 5, 9
2.4 5, 8, 10, 11
2.5 7, 9, 11, 15, 18
2.6 2, 3, 5, 9, 11
3.1 1, 4, 5, 6, 7
3.2 3, 4
3.3 2, 3, 6, 8, 12
3.4 2, 4, 11
3.5 5, 8, 9, 10, 11, 16
3.6 1, 3, 4, 6, 8
4.4 3, 8, 9, 10
4.7 4, 6, 8, 11, 12
5.3 5, 7 , 9, 13
5.4 4, 5, 6, 9, 13, 14
8.2 13, 14, 16, 17, 18
8.3 4, 6, 10
8.4 9, 11, 12, 13, 14
9.2 7, 8, 11, 14
9.3 8, 9, 10, 16, 19, 20
Chapter 19 and 20 Free resolutions and related topics: Commutative algebra with a view toward algebraic geometry by Eisenbud
Section 4.6 Local cohomology: An Intro to Local Cohomology by Weibel
Chapter 6 Grassmannians: Algebraic Geometry by Joe Harris
Chapter 18 Hilbert Schmes of Points on the plane: Combinatorial Commutative Algebra by Miller and Sturmfels