Topics to be Covered
Below is the list of topics to be covered.
- Computational Models & Complexity Classes (1 lecture)
- Depth and space complexity, complexity of linear algebra (1 lecture)
- Fundamental problems in commutative algebra and algorithms (2 lectures)
- Gröbner Basics (3 lectures)
- Degree Bounds for Groebner bases (5 lectures)
- Elimination Theory (3 lectures)
- Free Resolutions, Ext, Tor (2 lectures)
- Primary Decomposition (3 lectures)
- Nullstellensatz bounds (2 lectures)
- Koiran’s Nullstellensatz in PH (1 Lecture)
Lecture Schedule
|
Date |
Topics |
Notes |
| Lecture 0 |
May 5th |
Introduction & Overview |
|
| Lecture 1 |
May 5th |
Computational models & complexity classes |
lecture 1 |
| Lecture 2 |
May 7th |
Linear algebra in polylog space |
see rocketchat |
| Lecture 3 |
May 12th |
Solving homogeneous systems of polynomial equations & applications |
Lecture 3 |
| Lecture 4 |
May 14th |
Computing the dimension of an ideal |
Lecture 4 |
| Lecture 5 |
May 21st |
Gröbner basics I: monomial orders and division algorithm |
see rocketchat |
| Lecture 6 |
May 26th |
Gröbner basics II: monomial ideals, Hilbert basis theorem and Gröbner bases |
Lecture 6 |
| Lecture 7 |
May 28th |
Gröbner basics III: Buchberger’s algorithm |
see rocketchat |
| Lecture 8 |
June 2nd |
Numerical Invariants & Free resolutions |
Lecture 8 |
| Lecture 9 |
June 4th |
Dubé: Degree bounds for Gröbner bases |
see rocketchat |
| Lecture 10 |
June 9th |
Dubé: Degree bounds for Gröbner bases II |
see rocketchat |
| Lecture 11 |
June 11th |
Dubé: Degree bounds for Gröbner bases III |
see rocketchat |
| Lecture 12 |
June 16th |
Dubé: Degree bounds for Gröbner bases IV |
see rocketchat |
| Lecture 13 |
June 18th |
Dubé: Degree bounds for Gröbner bases V |
see rocketchat |
| Lecture 14 |
July 2nd |
Elimination Theory I |
see rocketchat |
| Lecture 15 |
July 7th |
Elimination Theory II |
see rocketchat |
| Lecture 16 |
July 9th |
Primary Decomposition - Theory |
[GP, Chapter 4.1] and
notes |
| Lecture 17 |
July 14th |
Computing Primary Decomposition I |
[GP, Chapter 4.2] and
notes |
| Lecture 18 |
July 16th |
Computing Primary Decomposition II |
[GP, Chapter 4.3] |
| Lecture 19 |
July 21st |
Noether Normalization & Nullstellensatz |
note1,
note2 and
note3 |
| Lecture 20 |
July 23rd |
Nullstellensatz degree bounds |
notes |
| Lecture 21 |
July 28th |
Koiran: polynomial system solving in PH |
notes |
Suggested Reading
|
Topics |
Suggested Reading |
| Lecture 1 |
Computational models & complexity classes |
[AB], [BCSS] and [SY] |
| Lecture 2 |
Linear algebra in polylog space |
Borodin’s 1977 paper and
Csanky’s paper or
Kozen’s Chapters 31-34 |
| Lecture 3 |
Solving homogeneous systems of polynomial equations & applications |
[S74] |
| Lecture 4 |
Computing the dimension of an ideal |
[S74] |
| Lecture 5 |
Gröbner basics I |
[CLO1, Chapter 2], [E, Chapter 15] |
| Lecture 6 |
Gröbner basics II |
[CLO1, Chapter 2], [E, Chapter 15] |
| Lecture 7 |
Gröbner basics III |
[CLO1, Chapter 2], [E, Chapter 15] |
| Lecture 8 |
Numerical Invariants and Free Resolutions |
[E, Chapter 1] |
| Lecture 9 |
Dubé I |
Dubé’s paper and
correction in HPS21 |
| Lecture 10 |
Dubé II |
Dubé’s paper and
correction in HPS21 |
| Lecture 11 |
Dubé III |
Dubé’s paper and
correction in HPS21 |
| Lecture 12 |
Dubé IV |
Dubé’s paper and
correction in HPS21 |
| Lecture 13 |
Dubé V |
Dubé’s paper and
correction in HPS21 |
| Lecture 14 |
Elimination Theory I |
[CLO1, Chapters 3 & 8] |
| Lecture 15 |
Elimination Theory II |
[CLO1, Chapters 3 & 8] |
| Lecture 16 |
Primary Decomposition - Theory |
[GP, Chapter 4.1] |
| Lecture 17 |
Computing Primary Decompositions I |
[GP, Chapter 4.2] |
| Lecture 18 |
Computing Primary Decompositions II |
[GP, Chapter 4.3] |
| Lecture 19 |
Noether Normalization & Nullstellensatz |
[G, Chapter 10], [H20, Chapter 4] |
| Lecture 20 |
Radical Ideals: Nullstellensatz degree bounds |
Jelonek’s paper |
| Lecture 21 |
Koiran: polynomial system solving in PH |
Koiran’s paper |