Complete these exercises several months in advance of your anticipated research project with undergraduates. For example, if you are thinking of working with students over a summer, consider working on them between the fall and spring semesters.

The divisor sequence of an irreducible element (_atom_) $a$ of a reduced monoid $H$ is the sequence $(s_n)_{n \in \mathbb{N}}$ where, for each positive integer $n$, $s_n$ denotes the number of distinct irreducible divisors of $a^n$. In this work we â€¦

We introduce a recursive decomposition algorithm for the Betti diagram of a complete intersection using the diagram of a complete intersection defined by a subset of the original generators. This alternative algorithm is the main tool that we use to â€¦

We investigate decompositions of Betti diagrams over a polynomial ring within the framework of Boij-Soederberg theory. That is, given a Betti diagram, we decompose it into pure diagrams. Relaxing the requirement that the degree sequences in such pure â€¦

We give a complete description of the cone of Betti diagrams over a standard graded hypersurface ring of the form $k [x, y]/\langle q \rangle$, where $q$ is a homogeneous quadric. We also provide a finite algorithm for decomposing Betti diagrams, â€¦

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