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From points to complexes: a concept of unexpectedness for simplicial complexes
Thiago Holleben
In 2018, Cook, Harbourne, Migliore and Nagel introduced the concept of unexpected hypersurfaces, which connects the study of Lefschetz properties of artinian algebras defined by po…
Planar ternary graphs, flag spheres, and Delannoy polynomials
Margaret Bayer, Richard Danner, Thiago Holleben +2
In 2022 Kim showed that when a graph is ternary (without induced cycles of length divisible by three), its independence complex is either contractible or homoto…
When are Morse resolutions polyhedral?
Louis Bu, Sara Faridi, Iresha Madduwe Hewalage +5
It is known that the chain complex of a simplex on vertices can be used to construct a free resolution of any ideal generated by monomials, and as a direct result, the Bett…
Spheres and balls as independence complexes
Susan M. Cooper, Sara Faridi, Thiago Holleben +2
The terms "whiskering", and more generally "grafting", refer to adding generators to any monomial ideal to make the resulting ideal Cohen-Macaulay. We investigate the independence…
Roller Coaster Gorenstein algebras and Koszul algebras failing the weak Lefschetz property
Thiago Holleben, Lisa Nicklasson
Inspired by the Roller Coaster Theorem from graph theory, we prove the existence of artinian Gorenstein algebras with unconstrained Hilbert series, which we call Roller Coaster alg…
Realizing resolutions of powers of extremal ideals
Trung Chau, Art M. Duval, Sara Faridi +3
Extremal ideals are a class of square-free monomial ideals which dominate and determine many algebraic invariants of powers of all square-free monomial ideals. For example, the $r^…
Coinvariant stresses, Lefschetz properties and random complexes
Thiago Holleben
Lefschetz properties and inverse systems have played key roles in understanding the -vector of simplicial spheres. In 1996, Lee established connections between these two algebra…
A Probabilistic Parking Process and Labeled IDLA
Pamela E. Harris, Thiago Holleben, J. Carlos Martínez Mori +3
In 1966, Konheim and Weiss [33] introduced a now classical parking protocol. The deterministic process and its resultant objects, known as parking functions, have since become a fa…