See also: Combinatorics seminar (Fridays 3:35) and Student Combinatorics and Algebra Seminar (Thursdays 4:40)
I'll survey multiple ways to answer the question, "When do ten points in the plane lie on a cubic curve?" My favorite answer involves just a straightedge but the story has many twists and turns, starting with the Cayley-Bacharach Theorem and revisiting some classical geometric results due to Carnot and Steiner. If time permits, I'll describe a second proof of the result using invariants and automated proof techniques. This is joint work with David Wehlau (Royal Military College of Canada).
We introduce the generalized Ishida complex over multigraded modules of affine semigroup rings. In addition, we propose the concept of degree space, Z^d, with a topology generated by the lattice points of polyhedral cones. Two applications are derived from these two concepts:
1) Cohen—Macaulayness criteria of quotient rings of polynomial rings by cellular/lattice binomial ideals; and
2) Duality between local cohomologies.
This is joint work with Laura Matusevich and Erika Ordog.
Given a complex of R-modules M, one can construct a variety that contains homological information about M, its cohomological support variety V_R(M). These have various homological applications – for example, Pollitz showed they can be used to characterize when R is a complete intersection. In this talk, we will discuss the following realizability question: given an appropriately chosen variety V, when can V be realized as V_R(M) for some M? To study this question, we give bounds on the dimension of such varieties. This is joint work with Ben Briggs and Josh Pollitz.
A numerical semigroup is a subset S of the natural numbers that is closed under addition. One of the primary attributes of interest in commutative algebra are the relations (or trades) between the generators of S; any particular choice of minimal trades is called a minimal presentation of S (this is equivalent to choosing a minimal binomial generating set for the defining toric ideal of S). In this talk, we present a method of constructing a minimal presentation of S from a portion of its divisibility poset. Time permitting, we will explore connections to polyhedral geometry.
No familiarity with numerical semigroups or toric ideals will be assumed for this talk.
A projective, normal variety, X , is called a Mori dream space (MDS) when its Cox ring, Cox(X), is finitely generated. While Mori dream spaces have nice behavior, no complete classification of them yet exists. Due to their combinatorial nature, one natural class of candidates for Mori dream spaces is projectivized toric vector bundles. In 2012, Jose Gonzalez proved that all rank-2 projectivized toric vector bundles are MDS. Kaveh and Manon (2019) gave a combinatorial description of toric vector bundles that we use to describe a family of rank-r toric vector bundles that are MDS. This description, along with a relationship with toric full flag bundles, also allows us to describe conditions under which a direct sum of MDS bundles are also MDS. We conclude with computational examples of bundles over products of projective space and directions for future research, including an algorithmic implementation.
The graded Möbius algebra of a matroid is graded commutative algebra that encodes the combinatorics of the lattice of flats of the matroid. As a special subalgebra of the augmented Chow ring of the matroid, these rings played an important role in the recent proof of the Dowling-Wilson Top Heavy Conjecture, which interpolates between the good algebraic properties of the augmented Chow ring and the combinatorics of the graded Möbius algebra. Recently, Mastroeni and McCullough proved that the Chow ring and augmented Chow ring of a matroid are Koszul. While not all graded Möbius algebras are Koszul or even quadratic, we will discuss joint work with Jason McCullough and Irena Peeva giving necessary and sufficient conditions for graded Möbius algebras to be quadratic and have quadratic Gröbner bases, drawing parallels with the much better studied case of Orlik-Solomon algebras. In the case of graphic matroids, we give necessary conditions for Koszulness that suggest a new characterization of strongly chordal graphs in terms of edge orderings.