# Plan for Lectures

## Topics to be Covered

` These are tentative, and mostly in order`

- Introduction to Algebraic Complexity (2 lectures)
- Polynomial Identity Testing (1 lecture)
- Lower Bounds (1 lecture)
- Barriers to Lower Bounds and Algebraic Natural Proofs (1 lecture)
- Introduction to Commutative Algebra & Algebraic Geometry (1 lecture)
- Groebner Bases, Multivariate Polynomial Division Algorithm (3 lectures)
- Complexity of Ieal Membership Problem (1 lecture)
- Computational Invariant Theory (3 lectures)
- Optimization in Invariant Theory: Scaling Problems, Geodesic Convexity & Algorithms (3 lectures)
- Geodesic Convexity in Statistics: Sample Complexity for Matrix and Tensor Normal Models (1 lecture)
- Convex Algebraic Geometry (2 lectures)
- Hyperbolic Polynomials and Hyperbolic Programming (2 Lectures)
- Spectrahedral and Semidefinite Representations (2 lectures)
- Connections to Proof Complexity and Algorithms (1 lecture)
- Conclusion & Open Problems (1 lecture)

## Lecture Schedule

` These are tentative, and mostly in order`

Date | Topics | Slides | |
---|---|---|---|

Lecture 0 | January 11 | Introduction & Overview of Course | |

Lecture 1 | January 11 | Algebraic Circuits & Algebraic Complexity I | |

Lecture 2 | January 13 | Algebraic Circuits & Algebraic Complexity II | PDF and Baur-Strassen |

Lecture 3 | January 18 | Lower Bounds in Algebraic Complexity | |

Lecture 4 | January 20 | Polynomial Identity Testing | |

Lecture 5 | January 25 | Barriers to Lower Bound Techniques & Algebraic Natural Proofs | |

Lecture 6 | January 27 | Introduction to Commutative Algebra & Algebraic Geometry | |

Lecture 7 | February 1 | Multivariate Polynomial Division Algorithm & Monomial Ideals | |

Lecture 8 | February 3 | Groebner Bases & Buchberger’s Algorithm | |

Lecture 9 | February 8 | Elimination Ideals and Implicitization | |

Lecture 10 | February 10 | Complexity of Ideal Membership Problem | |

Lecture 11 | February 22 | Introduction to Invariant Theory | |

Lecture 12 | February 24 | Reynolds Operator & Finite Generation of Invariant Rings | |

Lecture 13 | March 1 | Primary and Secondary Invariants | |

Lecture 14 | March 3 | Introduction to Geometric Invariant Theory | |

Lecture 15 | March 8 | Scaling Algorithms | |

Lecture 16 | March 10 | Geodesic Convexity & General Scaling Algorithms | |

Lecture 17 | March 17 | Akshay - Geodesic Convexity in Statistics & Sample Complexity Bounds | No notes |

Lecture 18 | March 22 | Abhibhav - Hilbert Syzygy Theorem | |

Lecture 19 | March 24 | Introduction to Convex Algebraic Geometry I | |

Lecture 20 | March 29 | Introduction to Convex Algebraic Geometry II | |

Lecture 21 | March 31 | Hyperbolic Polynomials & Hyperbolicity Cones | |

Lecture 22 | April 5 | Spectrahedral & Semidefinite Representations | |

Lecture 23 | April 7 | Relating Hyperbolic and Semidefinite Programming: General Lax Conjecture | |

Lecture 24 | April 12 | Alex - Primary and Seconday Invariants | |

Lecture 25 | April 14 | Abhiroop - Completeness in VNP under different reductions | |

Lecture 26 | April 19 | Abhibhav - Noether Normalization Lemma | |

Lecture 27 | April 21 | Gian - Nagata’s counterexample |

## Suggested Reading

Topics | Suggested Reading | |
---|---|---|

Lecture 0 | Introduction & Overview of Course | notes |

Lecture 1 | Algebraic Circuits & Algebraic Complexity I | SY10 chapter 1 and R chapters 2 & 3 |

Lecture 2 | Algebraic Circuits & Algebraic Complexity II | SY10 chapter 2 and R chapter 5 |

Lecture 3 | Lower Bounds in Algebraic Complexity | SY10 chapter 3 and R chapters 7 & 8 |

Lecture 4 | Polynomial Identity Testing | SY10 chapter 4 |

Lecture 5 | Barriers to Lower Bound Techniques & Algebraic Natural Proofs | EGOW'18 and FSW'17 |

Lecture 6 | Introduction to Commutative Algebra & Algebraic Geometry | CLO'15 - chapter 1 |

Lecture 7 | Multivariate Polynomial Division Algorithm | CLO'15 - chapter 2 |

Lecture 8 | Monomial Ideals & Groebner Bases | CLO'15 - chapter 2 |

Lecture 9 | Buchberger’s Algorithm | CLO'15 - chapter 2 |

Lecture 10 | Complexity of Ideal Membership Problem | Madhu’s notes - lecture 14 |

Lecture 11 | Introduction to Invariant Theory | S'08 - chapters 1 & 2 |

Lecture 12 | Reynolds Operator & Finite Generation of Invariant Rings | S'08 - chapter 2 |

Lecture 13 | Primary and Secondary Invariants | S'08 - chapter 2 |

Lecture 14 | Introduction to Geometric Invariant Theory | W'17 Chapter 3 |

Lecture 15 | Scaling Algorithms | GO 18 |

Lecture 16 | Geodesic Convexity & General Scaling Algorithms | BFGOWW 18 |

Lecture 17 | Geodesic Convexity in Statistics & Sample Complexity Bounds | FORW 21 - coming soon |

Lecture 18 | Hilbert Syzygy Theorem | CLO'05 chapters 5 & 6 |

Lecture 19 | Introduction to Convex Algebraic Geometry I | BPT |

Lecture 20 | Introduction to Convex Algebraic Geometry II | BPT |

Lecture 21 | Hyperbolic Polynomials & Hyperbolicity Cones | Renegar’s notes |

Lecture 22 | Spectrahedral & Semidefinite Representations | BPT chapters 5 & 6 |

Lecture 23 | Relating Hyperbolic and Semidefinite Programming: General Lax Conjecture | O'20 |

Lecture 24 | Primary and Secondary Invariants | S'08 - chapter 2 |

Lecture 25 | Completeness in VNP under different reductions | |

Lecture 26 | Noether Normalization Lemma | |

Lecture 27 | Nagata’s counterexample |