Mathematics Colloquia and Seminars

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The Cohen-Macaulay condition plays a fundamental role in enumerative, algebraic, and geometric combinatorics. When one is interested in studying a generating function, it is common to recognize this generating function as the Hilbert series of an algebra or a polynomial ring quotient, then to deduce properties of the generating function from associated algebraic properties. A common argument of this type starts as follows: from your original algebra or quotient, find a SAGBI or Groebner degeneration that produces a semigroup algebra or monomial quotient. It is amazing how often the Cohen-Macaulay-ness (or lack thereof) of the resulting semigroup algebra or monomial quotient plays a significant role in what comes next, and there are several "standard" techniques now for studying the Cohen-Macaulay property in these settings. These techniques can often be applied as a black box, without the user knowing exactly what is happening; while this is very useful, it is somewhat unsettling over the long run. The reasons why the Cohen-Macaulay condition holds in such cases comes down to making arguments with local cohomology, which in my experience has a reputation among students in discrete mathematics as being a difficult topic. In this pair of talks, we will sketch an approach that students can use to develop their understanding of how Cohen-Macaulayness and local cohomology are related. We will focus in particular on identifying key ideas and accessible entry points to the different "pieces of the puzzle" in the literature.