School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
Office: Howard House 5.20
Email: fatemeh dot mohammadi at bristol dot ac dot uk
Office Hours: Mondays 2-3pm and by appointment
Course overview:
The "Combinatorial Commutative Algebra" field is concerned with the interplay between commutative algebra, combinatorics
and polyhedral geometry. The main goals are:
To translate conditions on ideals to equivalent
combinatorial conditions, and
Investigate combinatorial structures
in problems motivated by algebraic geometry.
A rough plan is to discuss one topic each week, so we may cover Chapters 1, 2, 3, 6, 7, 9 from Herzog-Hibi book and Chapter 1, 3, 4, 5, 7, 8, 14 from Miller-Sturmfels book.
The theory of Gröbner bases
Monomial ideals and simplicial complexes
Hilbert functions
Resolutions of monomial ideals
Multigraded Betti numbers
Cellular resolutions
Alexander duality
Toric varieties and lattice ideals
Flag varieties and plücker coordinates
Prerequisites for participants:
Some familiarity with the basic elements of modern algebra 2 (e.g., fields, groups, rings, ideals and modules) is
required.
The lectures will be self-contained and assume as little background knowledge as possible. The course is complementary to the TCC course taught last semester and you will be at an advantage if you have taken the previous course or seen the notes in advance.
Students are encouraged to experiment and explore the taught concepts in the computer algebra system Macaulay2 or Singular.
Homework:
Seven homeworks (one for each week) will be assigned based on audience background and their interests. Collaboration is both allowed and encouraged, but everyone must write up the solution by himself/herself.
Homeworks are listed at the end of each lecture note and they due a week after assignment.
Solutions to homeworks:
A selected set of solutions provided by students will be uploaded here.
All notes will be available to students and participants.
Lecture notes:
Lecture notes will be uploaded here after each class.