Assistant Professor of Mathematics Courtney Gibbons was an invited speaker in seminars at Syracuse University and Adelphi University.
Gibbons spoke about her research in a talk titled “Parametrizing a family of indecomposable modules” in March 24 in an algebra seminar at SU. She explained that “a graded short Gorenstein ring R can be thought of as a ring with Hilbert series 1 + es + s^2, where e is the multiplicity of the ring.” She said that “over such a ring, it is known that nonlinear indecomposable modules are precisely the cosyzygies of the maximal ideal.”
In joint work with Avramov and Wiegand, Gibbons parametrized a family of indecomposable linear modules that are cosyzygies of ideals minimally generated by up to e-1 linear forms. The Hilbert series of these modules form the atoms of the semigroup of Hilbert series of linear modules and lead to a surprising result about the cone of Betti diagrams over R.
As a guest of Adelphi’s math and computer science seminar series on March 27, Gibbons presented “Blank Space.” She used a four-by-four sudoku as an example in her discussion focused on applying algebra to problems.
From the example problem, she developed the theory of ideals and varieties and stated Hilbert’s Nullstellensatz, a theorem from classical commutative algebra. She also touched upon the idea of primary decomposition. Gibbons said that “in addition to being beautiful results in pure mathematics, these algebraic tools have been used to solve problems from biology, physics and economics.”