On Thursday 11/06/26, 10:00 – 11:00 and 11:00 – 12:00 in M2, the Algebra and Topology Seminar will host two research talks by Kostas Karagiannis (NKUA) and Konstantia Manousou Sotiropoulou (NKUA). Titles and abstracts are below:
First Talk
Title: AN INTRODUCTION TO BOIJ-SÖDERBERGH THEORY
KONSTANTIA MANOUSOU SOTIROPOULOU
Abstract. Boij-Söderberg theory provides a geometric description of Betti diagrams of graded modules over a polynomial ring. Rather than classifying Betti tables one by one, the theory studies the rational cone spanned by all such diagrams and describes its extremal structure in terms of pure diagrams.
In this talk, I will give an introduction to the Cohen-Macaulay case. I will begin with minimal graded free resolutions, Hilbert functions, and Betti tables, and then introduce pure resolutions and the Herzog-Kühl equations. The main theorem states that every Cohen-Macaulay Betti diagram decomposes uniquely as a positive rational combination of pure diagrams, ordered along a chain of degree sequences. I will explain the geometric meaning of this result via simplicial fans and briefly discuss how the proof uses positivity statements coming from vector bundles on projective space. I will conclude with a brief example from canonical curves, illustrating how additional geometric structure, such as Gorenstein duality, can be visible at the level of Boij-Söderberg decompositions.
Second Talk
Title: EQUIVARIANT GENERALIZATIONS OF GORENSTEIN RINGS, BETTI TABLES AND DUALITY THEOREMS
KOSTAS KARAGIANNIS
Abstract. The study of syzygies and minimal free resolutions provides a powerful algebraic lens into the geometry of projective varieties. When a variety admits a finite group action, this algebraic data naturally elevates into an equivariant complex, refining classical Betti numbers into group representations. The refined homological data is elegantly encoded in equivariant Betti tables.
In this talk, I will introduce equivariant Betti tables and explore their underlying shape and structure. The focal point will be an equivariant generalization of the Gorenstein property for commutative graded algebras. I will illustrate how the group action on the canonical module dictates deep, structured symmetries within the equivariant resolution. Finally, I will characterize projective varieties whose coordinate rings are equivariantly Gorenstein. Establishing this correspondence requires proving equivariant generalizations of classical duality theorems in algebraic geometry, such as Grothendieck’s local duality and Serre’s projective duality, utilizing the framework of equivariant Serre functors.
All interested students, researchers, and faculty members are warmly invited to attend.