MATH 790, Junior Algebra and Discrete Mathematics Seminar, 1 credit, Fall 2014
R 2:002:50 PM, Dolve 115
ORGANIZER:
Sean SatherWagstaff
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 degreebydegree 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 degreebydegree 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 SatherWagstaff
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 SatherWagstaff
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 IntegerValued 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.
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Last updated 06 Dec 2014.