Abstract: Let $R$ be a polynomial ring, and $I$ a homogeneous ideal. Almost all algebraic and geometric information about $I$ is encoded in a related object called a minimal free resolution: a long exact sequence of free modules terminating in $I$. Finding these free resolutions is thus a central problem in modern commutative algebra.
I'll work some examples showing that the problem is prohibitively difficult even for monomial ideals, and discuss modern techniques that can (in good situations) describe the resolution of a monomial ideal in terms of a suitable topological object, such as a simplicial or CW complex.
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