R 2:00-2:50 PM, Dolve 115
ORGANIZER: Sean Sather-Wagstaff

Schedule of talks

• Date: 28 August
  Speaker: Trevor McGuire
  Title: Very General Scarf Complexes and Their Motivation
  Abstract: In the theory of integer lattices, one often comes across the Scarf complex for a lattice. From a purely combinatorial standpoint, the Scarf complex is simply a set of special subsets of a lattice that has a defining property that is closed under taking further subsets; hence, it is a complex. Integer lattices carry with them an enormous amount of inherent structure, though. However, if we strip most of the structure away, the intrinsic feeling of the Scarf complex is still present. In this talk, we will cover the standard Scarf complex as it pertains to integer lattices, and we will then slowly strip away the dependence on the lattice to conclude with a suite of definitions culminating in a very general definition of a Scarf complex on posets. Many examples will be given running the gamut from nearly empty to incredibly complex.

• Date: 04 September
  Speaker: Jessica Striker
  Title: The generalized toggle group (or you can toggle anything!) - Part I
  Abstract: The toggle group on order ideals of a partially ordered set has been a fruitful research area in recent years. We extend this notion to the very general setting of any set of subsets. Interesting special cases of this general setting include chains, antichains, and interval closed sets of a poset, independent sets or vertex covers on a graph, and probably many more!
  No prior knowledge of any of these structures will be required for this talk.

• Date: 11 September
  Speaker: Jessica Striker
  Title: The generalized toggle group (or you can toggle anything!) - Part II
  Abstract: The toggle group on order ideals of a partially ordered set has been a fruitful research area in recent years. We extend this notion to the very general setting of any set of subsets. Interesting special cases of this general setting include chains, antichains, and interval closed sets of a poset, independent sets or vertex covers on a graph, and probably many more!
  No prior knowledge of any of these structures will be required for this talk.

• Date: 18 September
  Speaker: Susan Cooper
  Title: Counting Along with Macaulay
  Abstract: A standard strategy for comparing topological spaces is to consider associated vector spaces and their invariants. Given a homogeneous ideal in the polynomial ring with n variables, we group the elements of the ideal by degree which results in a collection of finite dimensional vector spaces. The collection of degree-by-degree dimensions is a function, called the Hilbert function, introduced by David Hilbert in his work in invariant theory. Hilbert functions have been extensively studied. The most celebrated result is Macaulay's Theorem which characterizes the functions both algebraically and in a simple combinatorial fashion. Much effort has gone into generalizing Macaulay's Theorem and studying consequences. In this talk we'll explore Macaulay's Theorem, see some applications, and consider a specialization to truncated polynomial rings.

• Date: 25 September
  Speaker: Susan Cooper
  Title: Counting Along with Macaulay, Part II
  Abstract: A standard strategy for comparing topological spaces is to consider associated vector spaces and their invariants. Given a homogeneous ideal in the polynomial ring with n variables, we group the elements of the ideal by degree which results in a collection of finite dimensional vector spaces. The collection of degree-by-degree dimensions is a function, called the Hilbert function, introduced by David Hilbert in his work in invariant theory. Hilbert functions have been extensively studied. The most celebrated result is Macaulay's Theorem which characterizes the functions both algebraically and in a simple combinatorial fashion. Much effort has gone into generalizing Macaulay's Theorem and studying consequences. In this talk we'll explore Macaulay's Theorem, see some applications, and consider a specialization to truncated polynomial rings.

• Date: 02 October
  Speaker: Sean Sather-Wagstaff
  Title: Some techniques for "understanding" algebraic objects, part 1
  Abstract: Given a algebraic object X, like a group or a ring, one would like to completely understand X. I will discuss what this might mean in various contexts, why it is impossible in most contexts, and how we adjust our expectations to find useful information and interesting problems.

• Date: 09 October
  no meeting

• Date: 16 October
  no meeting

• Date: 23 October
  Speaker: Sean Sather-Wagstaff
  Title: Some techniques for "understanding" algebraic objects, part 2
  Abstract: Given a algebraic object X, like a group or a ring, one would like to completely understand X. I will discuss what this might mean in various contexts, why it is impossible in most contexts, and how we adjust our expectations to find useful information and interesting problems.

• Date: 30 October
  Speaker: Catalin Ciuperca
  Title: Equations defining algebraic sets
  Abstract: How many equations does one need to define any algebraic curve in the three dimensional space? We address questions of this type and present an exposition of the general problem of finding the minimal number of equations that define an algebraic set in k^n (k algebraically closed field).

• Date: 06 November
  Speaker: Jason Boynton
  Title: Cartesian Squares and the Ring of Integer-Valued Polynomials
  Abstract: click here

• Date: 13 November
  no meeting

• Date: 20 November
  Speaker: Pye Aung
  Title: Regularity Theorem, Gorenstein Theorem and Beyond
  Abstract: The famous Regularity Theorem of Auslander, Buchsbaum and Serre states that a local ring $R$ is regular if and only if every $R$-module has a finite projective dimension. This idea of studying modules to characterize a property about the base ring has been successfully applied in various other cases. For example, Enoch and Jenda's Gorenstein Theorem states that a local ring $R$ is Gorenstein if and only if every (finitely generated) $R$-module has a finite Gorenstein injective dimension. We will explore this idea and discuss some of its applications in addition to the two famous ones above.

• Date: 28 November
  no meeting

• Date: 04 December
  Speaker: Jon Totushek
  Title: Detecting Gorenstein rings using homological dimensions
  Abstract: In this talk we will discuss a theorem of Foxby's that allows us to determine if a ring is Gorenstein using only flat and injective dimensions. Furthermore, we will define two other homological dimensions and consider a question posed by Takahashi and White.

• Date: 11 December
  Speaker: Hannah Altmann
  Title: Semidualizing modules over tensor products
  Abstract: Let R be a commutative, noetherian ring with identity. In some sense, the number of semidualizing modules gives a measure of the "complexity" of R. We are interested in that number. We will discuss constructing semidualizing modules over tensor products of rings over a field. In particular, this gives us a lower bound on the number of semidualizing modules over the tensor product.