| Fri, Oct 08, 2021
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OSU-AWM
3:30 PM
SSH 035 | | Beyond Vector Spaces: Syzygies and Resolutions of Powers of Monomial Ideals
Susan Morey, Texas State University
Host: Bella Tobin
This is part of the Distinguished Women in Mathematics Colloquium Series.
| | | Abstract: In linear algebra, a basis for a vector space is often used to obtain desired information. When expanding beyond vector spaces, a new approach is needed. A free resolution is one tool commonly used in commutative algebra and related fields to obtain information about an ideal. This talk will start with background information about what free resolutions are, how they can be viewed as generalizing linear algebra techniques, and why we care about them, and then progress to new results that show how to use combinatorial and graphical information to find minimal free resolutions for powers of a special class of ideals. Using combinatorial structures to obtain resolutions of monomial ideals is an idea that traces back to Diana Taylor’s thesis, where a simplex associated to the generators of a monomial ideal was used to construct a free resolution of the ideal. This concept has been expanded over the years, with various authors determining conditions under which simplicial or cellular complexes can be associated to monomial ideals in ways that produce a free resolution. This talk will focus on powers of square-free monomial ideals of projective dimension one. Trees and their graph products will be used to produce cellular complexes that support free resolutions of the powers of the ideal. Each of these resolutions will be minimal resolutions, and the cellular resolutions can also be viewed as strands of the resolution of the Rees algebra of the ideal. This research stems from a project initiated at a BIRS workshop “Women in Commutative Algebra†in Fall 2019, and contains joint work with Susan Cooper, Sabine El Khoury, Sara Faridi, Sarah Mayes-Tang, Liana Sega, and Sandra Spiroff. |
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