MATH 358.01: Combinatorial commutative algebra

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This course will cover a selection of topics from
the textbook Combinatorial commutative algebra,
by Miller and Sturmfels. It will provide ringtheoretic
viewpoints on combinatorial objects like
simplicial complexes (via Stanley-Reisner rings and
monomial ideals) and polyhedral cones (via semigroup
rings), but also combinatorial viewpoints on
algebraic objects like graded polynomial rings and
local cohomology. Students looking to specialize
in areas at the intersection of high-energy physics
and algebraic geometry will find the material useful,
since the algebra to be covered can be interpreted
geometrically or topologically, in terms of torus actions,
sheaf cohomology, or equivariant cohomology
and K-theory.
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Prerequisites: basic graduate algebra, including
rings, fields, and abstract commutative algebra, as
well as facility with algebraic topology, including
simplicial complexes, chain complexes, and homology.
Knowledge of polyhedra and their partially ordered
sets of faces is recommended (at the level of
Lectures 1 and 2 in Ziegler's book, Lectures on polytopes,
for example) but not strictly required. The
course is intended for second-year graduate students
and beyond, but it would be appropriate for firstyears
or undergraduates with extraordinary preparation
in commutative algebra or algebraic topology.