"Minimal Prime Ideals in Rings "

For much of the talk, rings will be commutative with no non-zero nilpotent elements

"Algebraic Statistics, I - The Basics"

Ever wondered how to solve statistical problems like Maximum Likelihood using Groebner bases? Well, some other enterprising mathematicians have as well. This is the beginning lecture in an n-part series that describes these methods. The first lecture will focus on the interplay between independence ideals, markov properties, and graphical models.

"Algebraic Statistics, II - Directed Acyclic Graphs and Bayesian Networks"

Last time we covered how to convert an independence statement concerning random variables into an ideal generated by quadric polynomials. A joint probability distribution satisfies the independence statements if and only if it a point on the variety of this ideal intersected with the probability simplex.  Our hope was that we could garner information about the statistics related to these random variables merely by studying the ideals. However, this time we will show that one of the standard axioms for independence, the Contraction Axiom, fails in the algebraic setting.  This leads us to refine our approach by introducing directed acyclic graphs as "models" for our statistical experiment and working only with groups of independence statements that naturally arise from these models.  When we use a directed acyclic graph to represent a discrete statistical experiment we call it a Bayesian network (hence, the subtitle of this talk).

"Algebraic Statistics, II - Directed Acyclic Graphs and Bayesian Networks"

Last time we covered how to convert an independence statement concerning random variables into an ideal generated by quadric polynomials. A joint probability distribution satisfies the independence statements if and only if it a point on the variety of this ideal intersected with the probability simplex.  Our hope was that we could garner information about the statistics related to these random variables merely by studying the ideals. However, this time we will show that one of the standard axioms for independence, the Contraction Axiom, fails in the algebraic setting.  This leads us to refine our approach by introducing directed acyclic graphs as "models" for our statistical experiment and working only with groups of independence statements that naturally arise from these models.  When we use a directed acyclic graph to represent a discrete statistical experiment we call it a Bayesian network (hence, the subtitle of this talk).

"Algebraic Statistics, IV - The Distinguished Primary Component"

A Bayesian network, B, has its global Markov ideal, I, associated with it. This ideal may or may not be prime, however, there is always a minimal prime P of I that corresponds to the set of strictly positive probability distributions that satisfy the independence statements of B. We will discuss the dimension of this primary component and give a conjecture about the relationship between the degree-2 portion of P and the global Markov ideal I

"Algebraic Statistics, V - The Degree-2 Conjecture"

The Degree-2 Conjecture states that the global Markov ideal, I, is generated by the degree 2 portion of the distinguished prime ideal, P. We will focus on partially proving this conjecture. In particular, since a Bayesian network that is a forest has prime global Markov ideal this conjecture is trivially true in this case. We will prove the Degree-2 Conjecture for a larger class of Bayesian networks than forests.

"Algebraic Statistics, VI - The Degree-2 Conjecture: inclusion property"

"TBA"

"Interval Conjectures for level Hilbert functions"

The theory of Gorenstein and level (graded) algebras is an important topic of commutative algebra, because of both its intrinsic interest and the applications it has to several other fields - such as algebraic combinatorics, algebraic geometry, invariant theory, and even complexity theory. One fundamental invariant of graded algebras is the Hilbert function, which counts the dimension of such algebras in each graded piece. The goal of this talk is to present and discuss two conjectures I have recently formulated: the "Interval Conjecture" (IC) and the "Gorenstein Interval Conjecture" (GIC). These conjectures are inspired by the research performed in this area over the last few years. In particular, a series of recent results seems to indicate that it will be nearly impossible to characterize explicitly the sets of all Gorenstein or of level Hilbert functions. Therefore, the purpose of the IC and the GIC is to at least provide the existence of a very strong - and natural - form of "regularity" in the structure of such important and complicated sets. We are still far from proving my conjectures in full generality at this point, even if I have already succeeded in a few particular cases. In this talk I will also discuss the background and the main results obtained so far in this area, as well as the techniques I have employed to begin studying the two conjectures.

"The Eisenbud-Green-Harris Conjecture and Projective Space "

Macaulay's famous theorem compares homogeneous ideals to simpler monomial ideals. Many people have tried to generalize Macaulay's work.  Indeed, these generalizations have developed in many different directions.  One recent direction that has gained much attention is the Eisenbud-Green-Harris Conjecture (EGH Conjecture).  This conjecture restricts the ideals from Macaulay's Theorem to those containing regular sequences in fixed degrees. We will introduce the EGH Conjecture and explore it in a geometric setting by working with polynomials that vanish on point sets in projective space.

"Cellular Covers"

Let denote a category. By a cellular cover of an object is meant a pair where is an object in and is a morphism with the following property: for every morphism there exists a unique (!) morphism such that . This definition is, in a certain sense, dual to the categorical concept of localization. It can very well happen that the only cellular covers are trivial (i.e. is an isomorphism — this occurs e.g. for finite abelian groups), but it can also happen that the non-isomorphic cellular covers do not form a set (e.g. for certain rank one torsion-free abelian groups).

This concept has been introduced and nvestigated by Farjoun-G¨obel-Segev for groups and for divisible abelian groups. For abelian groups in general there is a more systematic study of this concept by Fuchs-G¨obel. For the kernels of there are substantial results by Buckner-Dugas.

We plan to survey the most elevant results on this subject concentrating on abelian groups, and consider extensions to modules as well as to other algebraic structures like totally ordered abelian groups. There are numerous unsolved problems.

"TBA"

"Examples of Linear Systems"

We will discuss several examples of the line bundles and divisors corresponding to maps from curves to projective space as defined last time. Then we will construct divisor class groups and use them to verify simple examples of the classical Brill-Noether theorem.

"The Semigroup of Right Ideals of a Ring"

Let R be a ring with identity. The set of right ideals of R forms a semigroup under right ideal multiplication, which we denote by . After presenting some general results on , we focus on rings R which have the property that every right ideal H satisfies ; such rings are called right weakly regular. For a right weakly regular ring R we present properties of R and

. We also present analogous results for semigroups. These results are from joint work with Henry Heatherly and Karl Kosler.

"TBA"