Plenary Speakers
Rebecca R.G. (George Mason)
Title: Classifying neural ideals
Abstract: The neural ideal was introduced by Curto, Itskov, et al in 2013 to study the firing patterns of a set of neurons (called a neural code), turning problems in neuroscience and coding theory into algebraic questions. They also introduced the canonical form of a neural ideal, a set of generators uniquely tied to the original neural code. Fortunately for commutative algebraists, the neural ideal can be turned into a squarefree monomial ideal by the process of polarization (Gunturkun, Jeffries, and Sun 2020). In this talk I will give an overview of neural ideals, describe a simple criterion for determining whether a neural ideal is in canonical form, and give an improved algorithm for computing the canonical form of a neural ideal. This work is joint with Hugh Geller.
Anna Weigandt (Minnesota)
Title: The Degrees of Determinantal Varieties
Abstract: The degree of an algebraic variety generalizes the usual notion of the degree of a polynomial. We will discuss degrees of determinantal varieties, exploring combinatorial and algebraic connections. Our main tool will be Gröbner geometry, which allows us to study properties of varieties using related unions of linear spaces.
Francesca Gandini (St. Olaf)
Title: Invariants three ways
Abstract: When studying a phenomenon, one of the most natural questions to wonder about is which quantifiable properties remain unchanged i.e., what are the invariants of the system. Historically, the subject of invariant theory has come to denote a branch of abstract algebra where we study invariant polynomials that are fixed under some prescribed change of variables or linear group action. Finding invariants is generally hard, but for some particularly nice actions, there are fast algorithms, some of which are implemented in the computer algebra system Macaulay 2. One way to improve on the speed of the algorithm would be to figure out in advance how large an invariant can be, and so provide an early stopping point for the search process. This is known as the problem of finding the Noether bound for the invariant ring. For finite group actions, one can use a homological property of some geometric objects (subspace arrangements) to determine this bound. Combinatorial properties of the subspace arrangement also encode features of the group action. Another perspective on the problem, is to consider skew commutative polynomials instead of commutative polynomials as working with square free objects can make computations easier and bounds on commutative and skew commutative invariants are tightly connected, suggesting ways to translate results from the commutative to the skew commutative world, and vice versa.
Senior Graduate Student Speakers
Karina Cho (Stony Brook)
Title: Approaching the Defining Ideal through Cones
Abstract: Given a smooth projective variety, how can we geometrically construct generators of its ideal? One way is to consider cones over the variety and look at the ideal generated by these cones. This cone ideal may not capture all of the polynomials through the variety, but in sufficiently high degree, the cone ideal and the defining ideal agree. I will describe my work in studying the degree in which cones generate the defining ideal.
Sasha Pevzner (Minnesota)
Title: Symmetric group fixed quotients of polynomial rings
Abstract: Let the symmetric group act on the polynomial ring S in n variables via variable permutation. We consider the quotient module M which sets a monomial equal to all of its images under the action. This is a module over the ring of invariants, with relatively little known about its structure. When using integer coefficients, we can embed M as an ideal inside the ring of symmetric polynomials. Doing so gives rise to a family of ideals - one for each n. Localizing the coefficient ring of S at a prime p reveals striking behavior in these ideals, which stay stable (in a sense) as n grows, but jump in complexity each time n equals a multiple of p. In this talk, we will discuss the construction of this family of ideals, as well as some results and conjectures on its structure.
Sandra Rodríguez Villalobos (Utah)
Title: We don’t talk about Frobenius: Redefining F-Thresholds
Abstract: The Frobenius endomorphism plays a very important role in studying singularities in prime characteristic. However, this map is not a homomorphism in other characteristics. To translate between characteristics and to be able to extend concepts to mixed characteristic, it is useful to describe invariants in prime characteristic without using the Frobenius homomorphism, and instead using tools like big Cohen-Macaulay algebras. In this talk, we will discuss two invariants used to study singularities in prime characteristic, test ideals and F-thresholds, and how to describe them without using the Frobenius map.