Graduate Student Algebra Seminar

Time: Wednesday 3:00-4:00pm

Location: SH 4519

Date	2007 - 2008 Schedule
May 14, 2008	Speaker: Ed Rehkopf Title: Quadratic Forms and Lattice Theory Abstract: The theory of Quadratic Forms is often thought of as the theory of inner-product modules over rings. This is closely related to the theory of inner-product spaces over fields. However, the differences between these objects lead to major gaps in our understanding. One example of such a gap is in the Gram-Schmidt process, whereby an inner product space is decomposed into a sum of one-dimensional spaces. However if we restrict ourselves to working over a subring rather than the entire field, then not every element can be expected to have an inverse; hence the Gram-Schmidt process is inadmissible. In this talk we will lay the ground work for quadratic forms and work towards various methods used to help fill in the aforementioned gaps. In particular we will look at inner-product modules, called lattices, over the polynomial ring k[x] where k is a finite field of odd order.
May 7, 2008	Speaker: Chris Nowlin Title: Quantum Group Stuff Abstract: Last week Garrett introduced R-matrices. We will pick up where he left off and see examples of noncommutative algebras related to these R-matrices that are deformations of polynomial rings. We will see a couple specific examples and see how these illustrate some general ideas in noncommutative algebra, such as Skew Polynomial Rings, Smash Products, Noetherian Rings and etcetera. This talk should be accessible to people who like algebra, though some people will likely get bored and tune out.
April 30, 2008	Speaker: Garrett Johnson Title: Examples of Quantum Matrix Groups Abstract: The coordinate rings of affine varieties are commutative algebras. If the variety is an algebraic group, such as GL(n), then the coordinate ring has the added structure of a Hopf algebra, and it will be commutative. I will give some examples of quantum GL(2)'s and quantum SL(2)'s. These are quantum matrix groups and the associated coordinate rings are noncommutative Hopf algebras.
April 9, 2008	Speaker: Tom Howard Title: A Flavor for Derived Categories Abstract: Chain complexes are widely studied throughout algebra and in applications such as algebraic topology. Like peas in a pod, we are often only interested in plucking out the delicious homology groups from the complex, discarding (or stir frying) the remains. This suggests forming a new category by identifying two complexes whose homology matches. The resulting category is called the derived category. To get a full introduction to derived categories requires lengthy discussion of homological algebra, projective resolutions, mapping cones, triangulated categories, and localization of categories. So instead I'll be a bit sketchy on these details, and try to convey some intuition into how derived categories behave and why they are useful.
April 2, 2008	Speaker: Chris Nowlin Title: Universality and Duality Abstract: This will be an informal look at concepts that have come up in my research into quantum groups. We will look at a universal property intendedto capture the notion of dualizing how an algebra acts on a module, discussing the basics of the concepts involved and seeing how this ties into the study of quantum groups. The aim of the talk will be for the audience to understand everything.
March 6, 2008	Speaker: Tom Howard Title: Endomorphism Rings and Direct Sum Decompositions of Modules Abstract: Given any module M, there a one-to-one correspondence between the direct summands of M and the idempotents in the endomorphism ring End(M). We will explore this correspondence, and discuss how endomorphism rings play a vital role in the decomposition theory of modules. In particular, we will discuss when a module may be written as a direct sum of indecomposable modules, and moreover when such a decomposition is essentially unique.
February 28, 2008	Speaker: Keith Thompson Title: Symmetric Polynomials Abstract: A symmetric polynomial in $n$ variables is one which is unchanged by any permutation of the variables. We will examine a proof of the fundamental theorem of symmetric polynomials, which states that every symmetric polynomial is a (standard) polynomial in the elementary symmetric polynomials, giving a ring isomorphism between the ring of symmetric polynomials and the standard polynomial ring. Time permitting we will cover the fundamental theorem of rational functions as well, and look at other bases for the ring of symmetric polynomials such as the complete homogeneous symmetric polynomials and, in the case where the coefficients are from a field of characteristic 0, the power sum symmetric polynomials.
February 21, 2008	Speaker: cancelled Title: Abstract: The seminar is cancelled this week so that we may see Bill Jacob speak at the Graduate Student Colloquium.
February 14, 2008	Speaker: Jesse Liptrap Title: Translation Invariant Symmetric Polynomials Abstract: Translation invariant (anti)symmetric complex polynomials in $n$ variables correspond to $n$-electron wave functions for the fractional quantum Hall effect. Therefore we are interested in the structure of such polynomials. We establish an isomorphism between the algebra of translation invariant symmetric polynomials in $n$ variables and the full polynomial ring in $n-1$ variables, over any field of characteristic $0$. Antisymmetric polynomials don't need special treatment, as they are merely symmetric polynomials multiplied by the Vandermonde determinant $\prod_{i < j}(z_i-z_j)$.
February 7, 2008	Speaker: Chris Nowlin Title: Supertropical Algebras Abstract: The goal of this talk is to be very elementary and accessible. Hell, the goal of this talk is to redefine "accessible'' for a new generation. The object of focus will be supertropical semirings. Specifically, it will be on constructing the motivating example of supertropical semirings. We will begin with an ordered abelian group G (that's an example of the extent of vocabulary needed to follow this talk; and even then, we'll mainly talk about adding real numbers), with its operation written multiplicatively, and define addition on G by x+y=max{x,y}. Adding in an element called "negative infinity" will give this a zero element and thus the structure of a semiring (Don't worry; nobody else knows what a semiring is either). We will then observe some things about this structure that are not... ideal. To deal with this, we will throw in another copy of the group, whose elements will be called ghosts (If you get scared and need somebody to hold you at this point, nobody will judge you). Once we're clear on this new algebraic object and why it's cool, we'll specialise to the group of real numbers under addition (continuing, of course to write the operation multiplicatively) and see what polynomials look like. For example, (x+7)(x+3)=x^2+7x+10. Time permitting, we'll see analogues of some major facts about polynomials, like the Euclidean Algorithm, the Fundamental Theorem of Algebra, or Hilbert's Nullstellensatz.
January 31, 2008	Speaker: Garrett Johnson Title: Simple Lie Algebras Abstract: Here, I will show how the simple Lie algebras are classified over an algebraically closed field of characteristic zero. Dynkin diagrams, root systems, etc.. I will assume no previous knowledge of the subject.
December 12, 2007	Speaker: Garrett Johnson Title: r-matrices Associated to the Maximal Parabolic Subalgebras of sl(n) Abstract: The non-skew r-matrices on sl(n) have been classified by combinatorial objects on the Dynkin graph called Belavin-Drinfel'd triples. Classifying skew r-matrices is much more difficult and amounts to understanding all subalgebras of a special type, called carriers, which have a homological description. The maximal parabolic subalgebras are obtained by removing exactly one vertex from the Dynkin diagram. Of particular interest is the fact that the only maximal parabolic subalgebras of sl(n) which are carriers for skew r-matrices are those when the i-th node is removed for i relatively prime to n. In this case, the r-matrix is unique up to an equivalence relation which I will describe. Furthermore, the only maximal Belavin-Drinfel'd triples are those for which the i-th node is removed from either subset of the vertices for i relatively prime to n. Again, the non-skew r-matrix obtained in this manner is unique up to equivalence.
December 5, 2007	Speaker: Tom Howard Title: Global and Projective Complexity for Finite Dimensional Algebras II Abstract: This is a continuation of last week's talk.
November 28, 2007	Speaker: Tom Howard Title: Global and Projective Complexity for Finite Dimensional Algebras I Abstract: Every module X over a finite dimensional algebra has a minimal projective resolution, the length of which is called the projective dimension of X. In the case that the projective dimension is infinite,one might hope that the dimensions of the terms in the resolution do not grow too quickly. The asymptotic growth of the dimensions of the projective modules in the minimal projective resolution is the projective complexity of X. I'll discuss a few results regarding projective complexity that are analogous to elementary facts about projective dimension.
November 14, 2007	Speaker: Charlie Beil Title: Dimension Theory in Commutative Algebra Abstract: I will talk about dimension theory in commutative algebra, such as the relations among Krull dimension, embedding dimension, regularity, depth, Cohen-Macaulayness, projective dimension, and global dimension, with results such as the Auslander-Buchsbaum formula. I will say what all these words mean and will give lots of examples. I will even draw some nice pictures.
November 7, 2007	Speaker: Chris Nowlin Title: An Application of Gröbner Bases Abstract: We will review the definition of a Gröbner basis with some examples. Then we will look at how these can be useful in showing a certain ideal in the coordinate ring of matrices is prime. In particular, Gröbner bases can help us determine an ideal is radical and help pin down the intersection of ideals. The most important thing to note about a Gröbner Basis is the following: it is not a basis for anything!! As an added incentive to come, there will be Coke served; exactly one can, which will be served to me.
October 31, 2007	Speaker: Charlie Beil Title: Explicit Computation of Syzygies using Gröbner Bases Abstract: Given a module M, the first syzygy module of M is generated by the relations of M, and in general the ith syzygy module of M is generated by the relations in its (i-1)th syzygy module. In simpler terms, you consider the relations of the relations of the relations of the relations... of M. This information encodes many important properties of rings used in algebraic geometry. I will show how to explicitly compute syzygies for polynomial rings over fields using Gröbner bases. The talk will involve no talking, chalk, or weaponry, and should be accessible to anyone with 14 fingers.
October 24, 2007	Speaker: Garrett Johnson Title: r-Matrices Abstract: This will be a short introduction to the subject. First, I will have to say what an r-matrix is. After that, I will spend some time motivating the idea and showing how r-matrices occur naturally. By the end of the talk, I hope prove some of the first fundamental results concerning r-matrices on the simple Lie algebras.
October 17, 2007	Speaker: Tom Howard Title: Functor Categories in Representation Theory Abstract: Given two categories, C and D, the collection of functors from C to D, denoted Fun(C,D), is itself a category with natural transformations as morphisms. Many types of representations found in algebra may be realized as functors, including group actions, linear group representations, modules, representations of quivers, and sheaves. Given a ring R, one can consider the category of (additive) functors from R-Mod (left R-modules) to Ab (abelian groups). The structure of this category is strikingly similar to that of a module category, and I will characterize the projective, finitely generated, simple, and injective objects in the category. Many important concepts which are relatively cumbersome to define explicitly in R-Mod may be phrased quite elegantly and usefully in terms of the functor category above. No prior knowledge of functor categories or representation theory will be assumed.

Date	2006 - 2007 Schedule
August 8, 2007	Speaker: Tom Howard Title: Preprojective Modules over Artin Algebras Abstract: This talk is based on the theory of preprojective modules over Artin algebras developed by Auslander and Smalø in a paper during the late 70s. The idea is to construct a special partition the class of indecomposable modules called a preprojective partition. Auslander and Smalø show that a subcategory (full, and closed under summands) of the category of finitely generated modules over your algebra has a preprojective partition if it has a property known as covariant finiteness. I'll discuss this, and possibly relate it to Auslander and Reiten's theory of almost split sequences as time permits.
June 7, 2007	Speaker: Tom Howard Title: The Hidden Subgroup Problem and Quantum Computation Using Group Representations Abstract: Many important problems, such as factoring numbers and finding the shortest vector in a lattice, are special cases of a more general problem known as the hidden subgroup problem, which goes as follows. You have a finite group G and a function f: G -> S, where S is some set that won't concern us. Suppose further that there is some subgroup H such that f is constant on cosets of H, but distinct on distinct cosets. H is called the hidden subgroup, and your task is to use f to determine H. While no efficient algorithm is known in general, even for a quantum computer, I will discuss an efficient algorithm that computes the normal core of H, i.e. the largest subgroup of H normal in G. In particular this solves the case where G is abelian, which is sufficient for factoring numbers and yields Shor's algorithm.
May 31, 2007	Speaker: John Levitt Title: A Gentle Survey of the Minimal Model Program Abstract: The MMP is an algorithm proposed by Mori in the late 70s to classify algebraic varieties up to birational equivalence, for which he won a fields medal. Recent progress by Cascini and McKernan (and collaborators) in our own department has nearly completed the proof that this classification method works in characteristic 0, with hope for characteristic p. Keeping in mind that this is an algebra seminar, I plan on illustrating some of the tools used in birational geometry, particularly with regard to algebraic curves and surfaces. Some familiarity with algebraic geometry would be nice, and a knowledge of sheaves even better, but I will attempt to avoid too much technical detail.
May 24, 2007	Speaker: Charlie Beil Title: How Lie Theory has been used to Predict the Existence of Particles Abstract: Particle physics is the joining of quantum mechanics with representation theory. A particle itself is an irreducible representation of the symmetries of space-time. I will explain how certain particles have been predicted to exist, and then later found, using representations of certain Lie groups such as U(1)xSU(2) and SU(3).
May 17, 2007	Speaker: Chris Nowlin Title: Relations in Quantized Coordinate Rings Abstract: We will examine relations in Oq(SOn), a deformation of the coordinate ring of SOn, and a "quantum group". As far as I can tell, a deformation is when somebody takes a classical object and throws in q's to the relations in random places. Except they're maybe not so random. But before one can understand the patterns showing up, one should see where these relations come from. We will look in more depth at Oq(M2) because it's simpler,people have a better handle on it, and it seems to form a building block for the others. We will attempt to look at why this algebra was defined the way it was and see that 3 different approaches yield the same relations. Time restraints will force many of the details to be left out and processes to only be outlined. It will be left to the audience to attempt to deduce what details are left out due to time issues and what are due to the speaker's ignorance. This talk should be accessible to anybody who knows what an algebra is and what a matrix is. And if you don't know what an algebra is, it's a vector space with a multiplication defined on the vectors. There. Now you can come. The tensor product will come up too, but you don't need to know anything about it. You just have to not be scared or angry if I draw a circle around x's. As an added incentive, I will be drinking a Coke.
May 3, 2007	Speaker: Garrett Johnson Title: Poisson Geometry Abstract: I will define what a Poisson manifold is and I will give some examples that appear in classical dynamics. A specific class of Poisson manifolds, called Poisson-Lie groups are important objects used in the study of quantum groups.
April 26, 2007	Speaker: Tom Howard Title: Additive and Abelian Categories Abstract: In attempting to generalize the techniques of homological algebra to arbitrary categories, one is naturally led to abelian categories. Roughly speaking, they are categories that behave similarly enough to the category of abelian groups. I will define abelian categories in stages, and discuss the consequences and motivations of the new conditions as I added them. By the end of the talk I hope to prove some basic homological facts for arbitrary abelian categories, but the emphasis will be on the development as an example of how one can work in categories that aren't concrete.
Feb 15, 2007	Speaker: Tom Howard Title: Examples of Adjoint Functors Abstract: A pair of functors is called adjoint if they satisfy a certain natural relationship that is superficially similar to that of adjoint operators. While you may not have known what an adjoint pair is, you no doubt come across them frequently. I plan to describe what adjoint functors are, and state some of the most useful facts about them, relating to continuity and exactness. I'll spare the audience the abstractly nonsensical proofs and instead focus on giving lots of accessible examples, primarily from algebra and topology.
Feb 1, 2007	Speaker: Ryan Ottman Title: Reflection and Coxeter Groups Abstract: I will talk about reflection groups and illustrate with some simple examples. Then I will show that these reflections groups are also coxeter groups and do some examples finding the coxeter graph and showing the geometric side of things.
Nov 28, 2006	Speaker: Garrett Johnson Title: Coboundary Lie Bialgebras and the Classical Yang-Baxter Equation II Abstract: We will define a Lie bialgebra structure on sl(2,C), then we will see how it is coboundary and determine under what conditions an element r defines a Lie cobracket.
Nov 14, 2006	Speaker: Garrett Johnson Title: Coboundary Lie Bialgebras and the Classical Yang-Baxter Equation I Abstract: Lie Bialgebras are tangent spaces at the identity of Poisson-Lie groups. I will define these objects and discuss the conditions needed to define a coboundary structure. We will see how the Classical Yang-Baxter Equation arises.
Nov 7, 2006	Speaker: Chris Nowlin Title: An Introduction to Hopf Algebras Abstract: A Hopf algebra is a special type of bialgebra, which is a special type of algebra, which is a special type of vector space, which is a special type of abelian group, which is a special type of monoid, which is a special type of magma, which is a special type of set. We will learn what (some of) these words mean and why they are cool. If time permits, we will formally define the word "cool". We will start with a vector space, and then see how we would define an algebra, and not in any simple you can multiply the vectors type of way, but in a category-theoretical way featuring diagrams. We will see why these fancy commutative diagrams match our intuition and proceed to define a bialgebra and finally a Hopf algebra, with motivating examples along the way to see precisely why these definitions are sensible and useful. As an added incentive, I will be drinking a nice, refreshing Coca-Cola.
Oct 31, 2006	Speaker: Garrett Johnson Title: Extensions of Groups and H^2 Abstract: We are able to classify all extensions of a group G with abelian kernel A up to an equivalence relation by looking at the second cohomology group H^2(G,A). I will define group cohomology and describe the one-to-one correspondence between elements of H^2 and equivalency classes of extensions.

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Last Modified on 3/May/2008