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Math & Stats Department Colloquium

Friday, March 3rd, 2023

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Nasrin Altafi (����ֱ��)

Title: On the shape of Hilbert functions of Gorenstein algebras

Abstract: The Hilbert function is a numerical invariant that measures the number of independent conditions for a given geometric object. A significant part of the beauty of Hilbert functions is derived from their ubiquity in commutative algebra and algebraic geometry. Characterizing the shape of the Hilbert function in Gorenstein algebras is a major problem in commutative algebra. A characterization of the Hilbert functions of the Artinian Gorenstein (AG) algebras with codimensions of less than four was provided in 1980. This characterization is no longer true for higher codimensions althouhg it characterizes the Hilbert functions of AG algebras with the so called "Lefschetz property”. The Lefschetz property concerns the rank of the multiplication map by a general linear form on a given Artinian algebra. The study of such properties originates from the Hard Lefschetz theorem, which was a breakthrough in algebraic topology and geometry. The Lefschetz properties are especially important in terms of their implications for the Hilbert function. During the last 25 years, investigations of these properties have been of great interest. Lefschetz properties encode information about the Artinian algebra and their Hilbert functions. I will attempt to provide an overview of this topic, as well as important results and questions that contribute to this subject.

Nasrin Altafi is a postdoctoral fellow at ����ֱ�� who is supported by a Swedish Research Council grant. She received her Ph.D. from the KTH Royal Institute of Technology in Sweden. Before moving to Canada, she was a postdoctoral fellow at Copenhagen University. Nasrin’s research interests lie in commutative algebra and algebraic geometry as well as their interactions with combinatorics and computational algebra.