The graded Möbius algebra (GMA) of a matroid is an algebra encoding the combinatorics of its lattice of flats. In 2016, Maeno and Numata showed that the GMA of a matroid is Gorenstein if and only if the matroid is modular. Joint with Jason McCullough, we show that modular matroids have Koszul GMAs by showing that their defining ideals have a quadratic Gröbner basis, and we show that the GMAs of nearly all affine geometries are quadratic.
Matrices have many factorizations which are amenable to certain applications, such as QR decomposition for least-squares problems. The singular value decomposition (SVD) is yet another factorization of a matrix, but which recovers other matrix factorizations and which has wide applicability to modern problems. I will talk about diagonalization of matrices, SVD, and showcase how these decompositions manifest in problems from computer vision and statistics.
The graded Möbius algebra of a matroid is an algebra encoding the combinatorics of the lattice of flats. In 2016, Maeno and Numata showed that the graded Möbius algebra of a matroid is Gorenstein if and only if the matroid is modular. We extend this result by showing that these algebras are also Koszul by showing that the defining ideals have a quadratic Gröbner basis. We also examine the Koszul property for graded Möbius algebras of affine geometries, showing that those of finite affine planes are Koszul using a filtration argument. This is joint work with Jason McCullough.
The graded Möbius algebra of a matroid is an algebra encoding the combinatorics of the lattice of flats. In 2016, Maeno and Numata showed that the graded Möbius algebra of a matroid is Gorenstein if and only if the matroid is modular. We extend this result by showing that these algebras are also Koszul by showing that the defining ideals have a quadratic Gröbner basis. We also examine the Koszul property for graded Möbius algebras of affine geometries, showing that those of finite affine planes are Koszul using a filtration argument. This is joint work with Jason McCullough.
For most rings, a lot of the data of the ring can be captured via its (minimal) free resolution. This can then be summarized with a Betti table which, in some sense, describes the complexity of the ring. If such a ring is also Artinian, the ring is said to have the weak Lefschetz property (WLP) if multiplication by some linear form is always full rank. Although Lefschetz properties are of interest to algebraists, many combinatorialists like to leverage constructions of Artinian algebras with the WLP to prove results about, for instance, log-concavity of sequences. Joint with Hal Schenck, we show that if the Betti table of an Artinian algebra has a certain substructure resembling a Koszul complex, then the Artinian algebra cannot have the WLP.
The stable Tamari lattice was introduced by Haiman through the lens of invariant polynomials, extended by Bergeron-Preville-Ratelle, and is a variation of the Tamari lattice. Stemming from the AMS MRC: Algebraic Combinatorics this past summer, I will describe some work in progress joint with Anna Pun, Herman Chau, Spencer Daugherty, and Juan Carlos Martínez Mori where we describe the combinatorics of this lattice. Namely, I will present properties of the lower order ideals of this lattice, the cover relations, and some enumeration problems (related to Catalan numbers and parking functions) we are tackling. I will discuss this data through some examples and Python code.
For most rings, a lot of the data of the ring can be captured via its (minimal) free resolution. This can then be summarized with a Betti table which, in some sense, describes the complexity of the ring. If such a ring is also Artinian, the ring is said to have the weak Lefschetz property (WLP) if multiplication by some linear form is always full rank. Although Lefschetz properties are of interest to algebraists, many combinatorialists like to leverage constructions of Artinian algebras with the WLP to prove results about, for instance, log-concavity of sequences. Joint with Hal Schenck, we show that if the Betti table of an Artinian algebra has a certain substructure resembling a Koszul complex, then the Artinian algebra cannot have the WLP.
In 1996, L'vovsky showed the Castelnuovo-Mumford regularity of the coordinate ring of a monomial curve is bounded by the sum of its semigroup's two largest gaps. We explore analogous results for toric surfaces embedded by incomplete linear systems, and show that for certain classes the regularity is controlled by the combinatorics of the associated semigroup.
In 1996, L'vovsky showed the Castelnuovo-Mumford regularity of the coordinate ring of a monomial curve is bounded by the sum of its semigroup's two largest gaps. We explore analogous results for toric surfaces embedded by incomplete linear systems, and show that for certain classes the regularity is controlled by the combinatorics of the associated semigroup.
In 1996, L'vovsky showed the Castelnuovo-Mumford regularity of the coordinate ring of a monomial curve is bounded by the sum of its semigroup's two largest gaps. We explore analogous results for toric surfaces embedded by incomplete linear systems, and show that for certain classes the regularity is controlled by the combinatorics of the associated semigroup.
In 1996, L'vovsky showed the Castelnuovo-Mumford regularity of the coordinate ring of a monomial curve is bounded by the sum of its semigroup's two largest gaps. We explore analogous results for toric surfaces embedded by incomplete linear systems, and show that for certain classes the regularity is controlled by the combinatorics of the associated semigroup.
In 1996, L'vovsky showed the Castelnuovo-Mumford regularity of the coordinate ring of a monomial curve is bounded by the sum of its semigroup's two largest gaps. We explore analogous results for toric surfaces embedded by incomplete linear systems, and show that for certain classes the regularity is controlled by the combinatorics of the associated semigroup.
In 1996, L'vovsky showed the Castelnuovo-Mumford regularity of the coordinate ring of a monomial curve is bounded by the sum of its semigroup's two largest gaps. We explore analogous results for toric surfaces embedded by incomplete linear systems, and show that for certain classes the regularity is controlled by the combinatorics of the associated semigroup, and is bounded by the area of the associated polygon.
The stable Tamari lattice was introduced by Haiman through the lens of invariant polynomials, extended by Bergeron-Preville-Ratelle, and is a variation of the Tamari lattice. Stemming from the AMS MRC: Algebraic Combinatorics this past summer, I will describe some work in progress joint with Anna Pun, Herman Chau, Spencer Daugherty, and Juan Carlos Martínez Mori where we describe the combinatorics of this lattice. Namely, I will present properties of the lower order ideals of this lattice, the cover relations, and some enumeration problems we are tackling. I will discuss this data through some examples and Python code.
For most rings, a lot of the data of the ring can be captured via its (minimal) free resolution, in turn summarized by a Betti table. If such a ring is also Artinian, the ring is said to have the weak Lefschetz property (WLP) if multiplication by some linear form is always full rank. Joint with Hal Schenck, we show that if the Betti table of an Artinian algebra has a certain substructure resembling a Koszul complex, then the Artinian algebra cannot have the WLP.
For most rings, a lot of the data of the ring can be captured via its (minimal) free resolution. This can then be summarized with a Betti table which, in some sense, describes the complexity of the ring. If such a ring is also Artinian, the ring is said to have the weak Lefschetz property (WLP) if multiplication by some linear form is always full rank. I will go through the background material necessary to be able to understand the follow-up talk in the Commutative Algebra Seminar.
Considered as an algebraic analog for the connected sum construction from topology, the connected sum construction introduced by Ananthnarayan, Avramov, and Moore is a method to produce Gorenstein rings. Joint with Nasrin Altafi, Roberta Di Gennaro, Federico Galetto, Rosa M. Miró-Roig, Uwe Nagel, Alexandra Seceleanu, and Junzo Watanabe, we determine the graded Betti numbers for connected sums and fiber products of Artinian Gorenstein algebras, where the fiber product in the local setting was obtained by Geller. We also show that the connected sum of doublings is the doubling of a fiber product ring. I will discuss these results through some examples and Macaulay2 code.
For most rings, a lot of the data of the ring can be captured via its (minimal) free resolution. This can then be summarized with a Betti diagram which, in some sense, describes the complexity of the ring. If such a ring is also Artinian, the ring is said to have the weak Lefschetz property (WLP) if multiplication by some linear form is always full rank. Although Lefschetz properties are of interest to algebraists, many combinatorialists like to leverage constructions of Artinian algebras with the WLP to prove results about, for instance, log-concavity of sequences. Joint with Hal Schenck, we show that if the Betti table of an Artinian algebra has a certain substructure resembling a Koszul complex, then the Artinian algebra cannot have the WLP.
In 1996, L'vovsky showed the Castelnuovo-Mumford regularity of the coordinate ring of a monomial curve is bounded by the sum of its semigroup's two largest gaps. We explore analogous results for toric surfaces embedded by incomplete linear systems, and show that for certain classes the regularity is controlled by the combinatorics of the associated semigroup, and is bounded by the area of the associated polygon.
Considered as an algebraic analog for the connected sum construction from topology, the connected sum construction introduced by Ananthnarayan, Avramov, and Moore is a method to produce Gorenstein rings. Joint with Nasrin Altafi, Roberta Di Gennaro, Federico Galetto, Rosa M. Miró-Roig, Uwe Nagel, Alexandra Seceleanu, and Junzo Watanabe, we determine the graded Betti numbers for connected sums and fiber products of Artinian Gorenstein algebras, where the fiber product in the local setting was obtained by Geller. We also show that the connected sum of doublings is the doubling of a fiber product ring. I will discuss these results through some examples and Macaulay2 code.
Starting with Lefschetz properties, moving on to toric varieties and Castelnuovo-Mumford regularity, and finishing with other miscellaneous projects, I will give an overview of the research I have conducted while at Auburn University. A common theme among all these projects is the strong presence (and necessity) of computation. As such, there will be many examples written in Macaulay2 and Python to help understand where these projects came from and how they were completed.
For most rings, a lot of the data of the ring can be captured via its (minimal) free resolution. This can then be summarized with a Betti diagram which, in some sense, describes the complexity of the ring. If such a ring is also Artinian, the ring is said to have the weak Lefschetz property (WLP) if multiplication by some linear form is always full rank. Although Lefschetz properties are of interest to algebraists, many combinatorialists like to leverage constructions of Artinian algebras with the WLP to prove results about, for instance, log-concavity of sequences. Joint with Hal Schenck, we show that if the Betti table of an Artinian algebra has a certain substructure resembling a Koszul complex, then the Artinian algebra cannot have the WLP.
Despite online homework's growing prevalence as a uniform component in coordinated mathematics courses, few studies have considered the connection, or lack thereof, between instructors of record and fixed online homework sets. In this presentation we will share the results of a mixed-methods study examining how 10 university mathematics educators assessed the quality of a sampling of online Calculus I homework assignments. In the course of this project, we introduced our educators to a novel instrument called the Course Alignment Analysis Tool (CAAT), which leverages graph theory to assess the alignment between the learning outcomes that an instructor feels should be prioritized and the learning outcomes most emphasized by an assignment or assessment. In this highly interactive presentation, we will share both our results and walk you through using CAAT. For this reason, you have homework: Please bring a set of homework problems from an undergraduate course in your discipline! (If you're in the math department or happen to really enjoy calculus, we will have the Calculus I homework sets available for you to use).
For most rings, a lot of the data of the ring can be captured via its (minimal) free resolution. This can then be summarized with a Betti diagram which, in some sense, describes the complexity of the ring. If such a ring is also Artinian, the ring is said to have the weak Lefschetz property (WLP) if multiplication by some linear form is always full rank. Although Lefschetz properties are of interest to algebraists, many combinatorialists like to leverage constructions of Artinian algebras with the WLP to prove results about, for instance, log-concavity of sequences. Joint with Hal Schenck, we show that if the Betti table of an Artinian algebra has a certain substructure resembling a Koszul complex, then the Artinian algebra cannot have the WLP.
For most rings, a lot of the data of the ring can be captured via its (minimal) free resolution. This can then be summarized with a Betti diagram which, in some sense, describes the complexity of the ring. If such a ring is also Artinian, the ring is said to have the weak Lefschetz property (WLP) if multiplication by some linear form is always full rank. Although Lefschetz properties are of interest to algebraists, many combinatorialists like to leverage constructions of Artinian algebras with the WLP to prove results about, for instance, log-concavity of sequences. Joint with Hal Schenck, we show that if the Betti table of an Artinian algebra has a certain substructure resembling a Koszul complex, then the Artinian algebra cannot have the WLP.
I plan on going through some software I have found useful for grad school; this includes Python, GitHub, Zotero, Overleaf, and Box. I plan on going through each and showcasing some features that I have found especially nice.
We study the Artinian reduction \( A \) of a configuration of a pointset \( X \subseteq \mathbb{P}^{n} \), and the relation of the geometry of \( X \) to Lefschetz properties of \( A \). Migliore-Zanello initiated the study of this connection, with a particular focus on the Hilbert function of \( A \), and further work appears in Migliore-Miró-Roig–Nagel. Our specific focus is on betti tables rather than Hilbert functions, and we prove that certain betti tables force the failure of the Weak Lefschetz Property (WLP); the corresponding Artinian algebras are typically not level, and the failure of WLP is not detected in terms of the Hilbert function.
The tropical semiring can be defined as the semiring over the extended real numbers where addition is defined by taking the maximum and multiplication is defined by classical addition. In this setting, there is a nice interplay between algebra and polyhedral geometry. In this talk, I plan on giving a brief introduction to some of the tropical analogs for results and ideas from classical algebraic geometry such as zeros of polynomials and Bezout's Theorem. These will mainly be explored through examples.
Tools from algebra and topology can be used to extract topological and geometric features of the data. The essence of topological data analysis (TDA) is to use a filtration on a finite point cloud to construct an algebraic complex modeling the data. The homology of this complex gives topological information about the data. Summaries of the topological information are shown through barcodes and persistence diagrams. This talk will give an overview of where TDA comes from, how it can be used, some examples, and potential research directions.
The tropical semiring can be defined as the semiring over the extended real numbers where addition is defined by taking the maximum and multiplication is defined by classical addition. In this setting, there is a nice interplay between algebra and polyhedral geometry. In this talk, I plan on giving a brief introduction to some of the tropical analogs for results and ideas from classical algebraic geometry such as zeros of polynomials and Bezout's Theorem. These will mainly be explored through examples.
Modules and rings are central topics of study in advanced algebra; in the special case where the ring is a field, a module is just a vector space. One much studied class of rings are Artinian quotients of polynomial rings over a field \( k \); Artinian means the ring itself is a finite dimensional vector space over \( k \). I'll discuss two interesting properties of Artinian rings—the Lefschetz property (which is about the behavior of multiplication in the ring), and free resolutions, which come about from doing "linear algebra with matrices of polynomials".