2014-2015

Wednesday, September 10th, 2014, 1:00-3:00 PM, Room 5489

Richard Gustavson

Introduction to Affine Varieties

Affine varieties are the building blocks of algebraic geometry. We will begin this introductory talk by discussing the geometry-algebra correspondence produced by looking at zero sets of polynomials, culminating in the proof of Hilbert's Nullstellensatz. We will then discuss rational and regular maps on affine varieties and observe some properties related to these maps. We will attempt to prove as many results in full detail as possible.

Wednesday, September 17th, 2014, 1:00-2:30 PM, Room 5489

Andreas Gittis

Introduction to Projective Varieties

We will begin this talk with a review of the commutative algebra underlying projective varieties in graded rings and homogeneous ideals. After introducing projective varieties with some examples and motivating interpretations, we will state and prove results of projective varieties that parallel the canonical results for affine varieties.

Wednesday, October 1st, 2014, 1:00-3:00 PM

Ryan Ronan

Abstract Affine Varieties and Prevarieties

The notion of an abstract affine variety will allow us to define affine varieties intrinsically as topological spaces with a corresponding sheaf of functions. In this talk, we will begin by defining regular functions on open subsets of affine varieties. We will then introduce abstract affine varieties, and highlight the important connection between this concept and our original definition for affine varieties. Finally, we will build upon these definitions to introduce the idea of an algebraic prevariety and discuss finite products of prevarieities.

Wednesday, October 8th, 2014, 1:00-2:30 PM

James Diotte

Introduction to Algebraic Varieties

Using the intrinsic categorical framework of prevarieties we can define abstract algebraic varieties and discuss their properties. We will begin by analyzing the important distinction between the Tikhonov and Zariski product topologies. With this distinction in mind it will be possible to define separation and other algebro-geometric analogues of familiar topological properties. Finally, with the theoretical groundwork laid we shall investigate concrete examples of varieties including the important class of quasi-projective varieties.

Wednesday, October 15th, 2014, 1:00-3:00 PM

James Diotte

Dimension and Further Properties of Varieties

This week we will begin by studying the concept of dimension, mostly from the pure commutative algebra viewpoint. Dimension theory will culminate in Krull's Hauptidealsatz, and how it relates to algebraic geometry. Next we'll cover constructability, properties of morphisms of varieties and Chevalley's theorem. If time permits we'll end with Zariski's main theorem.

Wednesday, October 22th, 2014, 1:00-2:30 PM

Peter Thompson

Tangent Spaces to Varieties

We will begin by defining the tangent space to an affine variety. Then we establish an isomorphism that allows us to generalize this definition to algebraic varieties. We divert to discuss rational maps between affine varieties and dominant morphisms between algebraic varieties. This groundwork will aid in proving several facts about dimensions of tangent spaces.

Wednesday, October 29nd, 2014, 1:00-3:00 PM

Richard Gustavson

Introduction to Algebraic Groups

Algebraic groups are a cornerstone of the theory of differential Galois theory. In this talk we will introduce the notion of both topological groups and algebraic groups, providing examples of both and discussing some of their basic properties. We will then prove one of the most fundamental results in algebraic group theory, that every affine algebraic group is isomorphic to a closed subgroup of some GL(n). If time permits, we will discuss quotients of linear algebraic groups.

Wednesday, November 5th, 2014, 1:00-2:30 PM

Richard Gustavson

Affine Group Schemes and Hopf Algebras

Algebraic groups contain much more structure than arbitrary varieties, since they are also groups. This extra structure occurs in the coordinate ring of the algebraic group, but can be difficult to observe if looked at from the basic theory of algebraic geometry. To get a better sense of this extra structure, we will introduce the notion of affine group schemes, which under certain circumstances are equivalent to affine algebraic groups. The fundamental result which allows us to observe the extra structure on the coordinate ring of an affine algebraic group is called Yoneda's Lemma. We will use this to show that the coordinate ring of an affine algebraic group is in fact a Hopf algebra, and prove the correspondence between affine group schemes and Hopf algebras.

Wednesday, November 12th, 2014, 1:00-3:00 PM

Matthew Cushman

The Lie Algebra of a Linear Algebraic Group

We will introduce Lie Algebras from an algebraic group perspective, drawing contrasts with the differential-geometric development.

Wednesday, November 19th, 2014, 1:00-2:30 PM

Eli Amzallag

Decomposition of Algebraic Groups

We continue our discussion of linear algebraic groups and their associated Lie algebras by investigating further connections between these two objects. In particular, we emphasize a characterization of the Lie algebra of a closed subgroup of a linear algebraic group in terms of the vanishing ideal of the closed subgroup. With this key piece of information, we show that Jordan decomposition can be done in any affine algebraic group. This will ultimately provide us with a way of writing any commutative linear algebraic group as a product of closed subgroups, producing a special case of Chevalley's structure theorem and motivation for studying Abelian varieties next semester.

Wednesday, December 3rd, 2014, 1:00-3:00 PM

Eli Amzallag

Solvable Algebraic Groups and the Lie-Kolchin Theorem

We continue our investigation of decomposing algebraic groups by turning to solvable ones. It turns out that connected solvable algebraic subgroups of GL(n,C) are conjugate to subgroups of the upper triangular group T(n,C). This is a version of the Lie-Kolchin theorem. With this result in hand, we will be able to decompose connected nilpotent affine algebraic groups in very much the same way that we decomposed commutative linear algebraic groups.

In the course of this discussion, we will also collect a criterion for deciding that the identity component of an algebraic group is solvable. Such information is useful in deciding if a differential equations is solvable by quadratures (the analogue of a polynomial equation being solvable by radicals).

Wednesday, December 10th, 2014, 1:00-3:00 PM

Eli Amzallag

The Correspondence Between Algebraic Groups and Lie Algebras

We continue to investigate the parallels between an algebraic group and its Lie algebra. In particular, we study the correspondence between closed subgroups of an affine algebraic group and Lie subalgebras of its Lie algebra. We will see that, because of this correspondence, a connected affine algebraic group is solvable if and only if its Lie algebra is.

Friday, January 30th, 2015, 4:30-6:30 PM, Room 4214.03

Matthew Cushman

Introduction to Differential Algebraic Structures

We introduce the concept of a ring with derivation, or differential ring, and the related concepts of differential ideal and differential field. We formalize the notion of a linear differential operator and consider differential modules over a field, K[d]. Finally, we look at differential field extensions and draw analogies with the algebraic field extensions of classical Galois theory.

Friday, February 6th, 2015, 4:30-6:30 PM, Room 4214.03

Peter Thompson

Picard-Vessiot Extensions

Under reasonable hypotheses, the Picard-Vessiot extension of a system of differential equations Y' = AY is the smallest differential field that contains a solution space. Automorphisms of PV extensions are the object of study of differential Galois theory, which provides insight into the study of differential equations. Given a system of differential equations Y' = AY over an ordinary differential field of characteristic zero whose field of constants is algebraically closed, we prove the existence and uniqueness of a Picard-Vessiot extension.

Friday, February 13th, 2015, 2:00-4:00 PM, Room 5489

Eli Amzallag

The Differential Galois Group of a Linear Differential Equation

We introduce the differential Galois group of a differential field extension and explore its properties in the Picard-Vessiot case. In particular, we show that the Galois group of a Picard-Vessiot extension can be realized as a linear algebraic group. We will also see that the dimension of the Galois group in the PV case is equal to the transcendence degree of the PV extension over the ground field.

Friday, February 20th, 2015, 4:30-6:30 PM, Room 4214.03

Richard Gustavson

The Fundamental Theorem of Differential Galois Theory

Differential Galois theory has many analogues with polynomial Galois theory. The most important of these is probably the fundamental theorem of differential Galois theory (the analogue of the fundamental theorem of Galois theory), which can be stated in two parts. The first part says that there is a natural one-to-one correspondence between intermediate fields of a Picard-Vessiot extension and closed subgroups of the differential Galois group. The second part gives a criterion for when an intermediate field is PV over the base field and provides a description of the Galois group of the intermediate field over the base field when this criterion is met.

Friday, February 27th, 2015, 4:30-6:30 PM, Room 4214.03

Eli Amzallag

Liouvillian Extensions

We conclude our study of differential Galois theory with a discussion of Liouvillian extensions, which parallel the radical extensions studied in classical Galois theory. We will see that there is a strong connection between a differential extension being Liouvillian and the identity component of the corresponding differential Galois group being solvable. This connection allows one to decide when a differential equation is solvable by quadratures, the analog of a polynomial equation being solvable by radicals.

Friday, March 6th, 2015, 4:30-6:30 PM, Room 4214.03

Richard Gustavson

The Problem with Projective Varieties and the Need for Sheaves

While morphisms between affine varieties can be thought of as restrictions of morphisms between then ambient affine space, the same luxury does not hold for morphisms between projective varieties. As a result, we need a way of defining morphisms that will apply to both affine and projective varieties. To do so, we introduce the notion of a sheaf. Sheaves will allow us to define morphisms locally, which is just the requirement needed for them to make sense on projective varieties. This talk will first examine some examples of projective varieties and why the naive concept of a morphism fails on them. It will then turn to the concept of sheaves. We will give a categorical definition of both a presheaf and a sheaf, and discuss examples of both. This is the first in a four part talk which aims to provide a firm basis for working with arbitrary varieties.

Friday, March 13th, 2015, 4:30-6:30 PM, Room 4214.03

Richard Gustavson

Stalks of Sheaves and the Structure Sheaf of an Affine Variety

Last time we saw the importance of being able to define morphisms of varieties locally, and introduced sheaves to help handle this hurdle. In this talk, we discuss one the notion of the stalk of a sheaf at a point, which will be extremely useful when we begin working with more arbitrary varieties. We then introduce the most important sheaf associated with an irreducible algebraic set, called the structure sheaf. This will allow us to finally give the definition of an affine variety, and prove some first properties of affine varieties.

Friday, March 20th, 2015, 4:30-6:30 PM, Room 5489

Richard Gustavson

Prevarieties: Definitions, Properties, and Examples

Affine varieties are too restrictive for many of the cases we want to deal with, so we must generalize our discussion to prevarieties, which are topological spaces that locally look like affine varieties. In this talk, we will discuss some basic properties of prevarieties, look at morphisms of prevarieties, and prove that every projective variety is in fact a prevariety.

Friday, March 27th, 2015, 4:30-6:30 PM, Room 4214.03

Richard Gustavson

Products of Prevarieties and the Hausdorff Axiom

We can finally bring our discussion to its full generality by introducing the notion of a variety, which is a prevariety with some extra requirements. First, we must introduce the notion of categorical products, and prove that affine varieties, projective varieties, and prevarieties all have products. We will then define a variety, observe that most of the examples of prevarieties we have looked at are actually varieties, and prove some basic facts about varieties.

Friday, April 17th, 2015, 4:30-6:30 PM, Room 4214.03

Richard Gustavson

Products of Prevarieties and the Hausdorff Axiom (Part 2)

We can finally bring our discussion to its full generality by introducing the notion of a variety, which is a prevariety with some extra requirements. Continuing our talk from last time, we first prove that projective varieties and prevarieties have categorical products. We will then define a variety, observe that most of the examples of prevarieties we have looked at are actually varieties, and prove some basic facts about varieties. With the foundations we have established over these five talks, we will be able to discuss deeper properties of varieties in the coming weeks.

Friday, April 24th, 2015, 4:30-6:30 PM, Room 4214.03

Eli Amzallag

Krull's Hauptidealsatz and Noether's Normalization Lemma Revisited

We introduce the notion of dimension of a variety and study how this quantity behaves on irreducible closed subsets of a variety. In the course of this discussion, two ways of obtaining our main result on dimension will be alluded to: Krull's Principal Ideal Theorem (algebraic) and Noether's Normalization Lemma (geometric). The latter, via the notion of finite morphisms, is the one that will be elaborated on.

Friday, May 1st, 2015, 4:30-6:00 PM, Room 4214.03