Temporary Time: Monday 2:10-3pm
Location: SH 4519
Interested in speaking in the algebra seminar? Contact Julia Galstad, galstad (at) math (dot) ucsb (dot) edu.
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Date |
2010 - 2011 Schedule |
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March ?, 2011
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Speaker: Tomas Kababbe? Abstract: tbd |
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February 24, 2011
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Speaker: Tom Howard Abstract: A common theme in algebra is understanding general objects by factoring them into irreducible ones, i.e. integers into primes or polynomials into linear factors. This same approach may be taken for understanding module homomorphisms. In our search for irreducible homomorphisms we will discover the rich structure of Auslander-Reiten theory, including almost split sequences and AR quivers. Much useful information can be read directly from these AR quivers. |
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February 17, 2011
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Speaker: Drew Jaramillo Abstract: I plan to give an elementary introduction to some of the basic structures needed to study Quantum groups. We will start by defining coalgebras, bialgebras, and Hopf Algebras and use these to investigate the "quantized" versions of familiar mathematical objects. |
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February 10, 2011
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Speaker: Julia Galstad Abstract: Projective modules are a nice generalization of free modules. Modules with finite projective dimension are also nice because we can approximate them with projective modules. The next step is asking whether or not the category of finitely generated modules over a certain algebra can be approximated by those of finite projective dimension. In this talk, I will define these different versions of approximations and describe some constructions that are possible once you have an algebra that has this nice approximation property. |
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February 3, 2011
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Speaker: Stepan Paul Abstract: How large an integer d is required to ensure that there exists a degree-d polynomial curve in C^2 which passes through r > 9 general points with given multiplicities? The Nagata Conjecture predicts a sharp lower bound on d in terms of r and the sum of the multiplicities. The conjecture has its roots in algebraic geometry, but here I will present a purely linear algebraic approach. |
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January 27, 2011 |
Speaker: Arielle Leitner Abstract: In this talk, we will develop some basic quadratic form theory. We will use these techniques to prove the Cassels-Pfister theorem, which says that if $p(x) \in F(x)$ is a sum of $n$ squares, then $p(x)$ is already a sum of $n$ squares in $F[x]$. No background knowledge is required.
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January 20, 2011 |
Speaker: Stepan Paul Abstract: Toric varieties are (partial) compactifications of torus groups T = (C*)^n onto which the torus action can be extended. Familiar examples include C^n, and CP^n. They provide a broad yet accessible arsenal of examples for algebraic geometers to work with. I will give some basic definitions, explain the connection to cones in the integer lattice Z^n,and give some nice properties and examples. |
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January 13, 2011 |
Speaker: Julia Galstad Abstract: This talk will be a simpler version of the talk I gave at Pomona College in December. I will start out with some background material for understanding finite dimensional algebras with a focus on examples. I will discuss results of Birge as well as Tan and Koenig for some specific kinds of algebras, focusing on diagrams that look cool, as well as share a conjecture. |
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Fall Quarter, 2010 |
Speakers: Jon Cass, Julia Galstad, Tom Howard,
Drew Jaramillo Abstracts: coming soon. |
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Date |
2009 - 2010 Schedule |
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May 24, 2010 |
Speaker:
Chris Nowlin Abstract:
In geometric group theory, Coxeter groups are groups generated
by reflections of euclidean space. Once some generating
reflections are chosen, a natural notion of length emerges,
which leads to an ordering called the Bruhat ordering. We will
consider the poset structure of Bruhat order intervals.
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May 17, 2010 |
Speaker:
Tom Howard Abstract: Homological algebra, algebraic geometry, and algebraic topology abound with additive categories. Among the most familiar and most well-behaved are the abelian categories. While the class of abelian categories is quite robust -- closed under numerous useful constructions -- any intrepid algebraist will soon fine themselves interested in a nonabelian additive category. We will encounter some of these gems, and introduce two examples of structure to grasp for in desperation: those of exact categories and triangulated categories.
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May 10, 2010 |
Speaker:
Julia Galstad Abstract: Spectral sequences are a useful tool for algebraic topologists, algebraic geometers, and many others. They are used to compute homology groups by successive approximation. This talk will be an introduction to the topic. Motivation and examples will be given mainly from a topological point of view.
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May 3, 2010 |
Speaker:
Garrett Johnson Abstract: RSA (named after Rivest, Shamir and Adleman) is an algorithm used for sending secret messages along public airwaves. The mathematics of the algorithm uses theorems about the prime numbers. The main strength of RSA lies in the fact that given two prime numbers p and q, we can very easily multiply them together to get a number n (or, if p and q are very large, get a computer to do it for us). However, if we are given a large number n that is known to be a product of two primes p and q, it is in general very difficult to factor n into the product pq. In the RSA algorithm, this translates into the fact that we can only decrypt an encrypted message if we know what the two primes p and q are; the person receiving a secret message should be the only one having this knowledge.
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April 28, 2010 |
Speaker:
Chris Nowlin Abstract: This talk will summarize some recent results and conjectures I hope to throw into my dissertation at the last minute. The topic will be a universal way of extending partially ordered sets which satisfy certain properties. We will then see that Bruhat order intervals in Coxeter groups satisfy these properties and that the torus-invariant prime spectra of certain noncommutative polynomial rings might also satisfy these properties. It should be accessible to everybody and interesting to nobody.
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April 21, 2010 |
Speaker:
Garrett Johnson Abstract: An extension $R\subseteq S$ of rings is called quadratic if S is a free left R-module of rank two (where R acts on S via multiplication). Using a slightly modified version of the Hilbert Basis Theorem, we'll give a proof that under mild restrictions, the amalgamated product of quadratic extensions S_1 and S_2 over a noetherian base ring R is also noetherian. Secondly, we'll show that if R is a division ring, then the amalgamated product S_1*S_2 is a principal ideal ring. These results can be applied to examples such as skew group rings of the infinite dihedral group or the affine and double affine Hecke algebras of type A_1.
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April 14, 2010 |
Speaker:
Tom Howard Abstract: Let Q be a finite directed graph. Fix a vertex v, and let p_n be the number of paths of length n starting at the vertex v. I will show that, asymptotically, p_n grows as b^n * n^r for some b,r > 0. Along the way, I'll describe how to compute b and r by hand, and do several examples. I will briefly describe how this combinatorial procedure may be used to compute the complexity of certain modules (complexity will be defined).
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January 27, 2010 |
Speaker:
Keith Thompson Abstract: We'll continue with complexes of groups - what they are, how they are different from orbifolds, and what we can do with them. We'll begin with the definition and the construction of the fundamental group. We'll then focus on simple examples that illustrate the role of twisting elements in determining the local structure and existence of a universal cover.
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January 20, 2010 |
Speaker:
Ellie Grano Abstract: I will introduce the Temperley-Lieb (planar) algebra. I will give one or two examples of other planar algebras via generators and relations and some consequences.
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December 2, 2009 |
Speaker:
Chris Nowlin Abstract:
In considering the spectrum of prime ideals of certain algebras,
the Stratification Theorem of Goodearl-Letzer tells us the
flavor of the entire spectrum is captured by a particular finite
poset. Even more amazingly, we observe these posets are
isomorphic to the posets of subwords of words in Coxeter groups.
We will begin with a few examples to illustrate this phenomenon.
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November 18, 2009 |
Speaker:
Ryan Ottman Abstract: I'll explain what it means for a Coxeter group to have hyperbolic signature, then I will describe some results I have obtained.
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November 11, 2009 |
Speaker:
Keith Thompson Abstract:
One of the most productive ways to study the structure of a
group is through its geometric actions on polyhedral complexes.
Small categories without loops (scwols) provide natural ways of
encoding the structure of the complex on which the group acts as
well as that of the compact quotient. A complex of groups over a
scwol can encode all of the information necessary to reconstruct
both the group which produced the quotient and the complex on
which it acted. There will be lots of examples, and if there's
time we'll get to examples involving twisting elements. Also,
there may be candy.
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November 4, 2009 |
Speaker:
Brie Finegold Abstract: In the 1970's when Bass-Serre theory (graphs of groups) was published, it became popular to use topological methods to find presentations of 2 dimensional special linear groups, SL(2,R). Richard Swan used the fact that SL(2,C) is isomorphic to the isometry group of hyperbolic space to find presentations of SL(2,R) where R is a ring of integers in an imaginary quadratic number field. I'll give a quick overview of the different methods used to find group presentations of SL(2,R), SL(3,R), and how these relate to/differ from my method. I'll also give some examples of the presentations derived from this.
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October 28, 2009 |
Speaker:
Julia Galstad Abstract: After giving a brief history of design theory, along with some fun examples of designs, we'll focus on designs that appear in codes. The discussion will include the Assmus-Mattson Theorem and some of its generalizations.
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October 14, 2009 |
Speaker:
Jon Cass Abstract: An elliptic curve E is the locus of solutions to a degree 3 equation. Many interesting results are obtained by looking at the structure of the solutions whose coordinates are in various fields. We denote the solutions to E with coordinates in a ring F by E(F). We can introduce a group structure on E(F), and then examine what kinds of groups we get when F is one of a number of different fields. In this talk we will discuss the structure of E(C), E(R), E(Q), and E(Z_p). The main theorems to be proven are that E(C) is isomorphic to a torus and that E(Q) is a finitely generated group.
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October 7, 2009 |
Speaker:
Tom Howard Abstract: Given two categories, C and D, the functor category Fun^o (C,D) has as objects the contravariant functors from C to D and as morphisms the natural transformations. Yoneda's Lemma describes how any category C may be fully embedded into the functor category Fun^o (C, Set), with an additive version obtained by replacing Set with Ab. The functor category Fun^o (C, Ab) has many very nice properties. In particular, it is abelian so one can use homological algebra. After discussing Yoneda's Lemma, we'll use it to show that in case C is a module category, the finitely generated projective objects are precisely the Hom functors. If time permits, I will explain where so-called "contravariantly finite" subcategories got their name.
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Date |
2008 - 2009 Schedule |
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September 2, 2009 |
Speaker:
Garrett Johnson Abstract: We will look at the example of Oq(M_2), the 2x2 quantum matrices. The approach will be different than the approach Chris took last week. For each element in a Weyl group, one can associate to it a quantized nilpotent Lie algebra. I will show how Oq(M_2) can be realized as one of these creatures. So we should be able to determine a lot just from the particular group element we used, and that's what we'll do here to classify the torus-invariant prime ideals.
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August 26, 2009 |
Speaker:
Chris Nowlin Abstract: We will use the example of the Ring of Quantum 2 by 2 Matrices to illustrate some techniques for classifying the prime ideals of a quantized coordinate ring invariant under the action of a torus. This is a simple example in which these techniques will seem a sledgehammer, but they should be illustrated well enough. We will use a combination of factoring, localizing and a cool derivation-deleting homomorphism to reduce the problem to finding prime ideals in simpler quantized coordinate rings.
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June 3, 2009 |
Speaker:
Julia Galstad Abstract: Under certain reasonable hypothesis, the centralizer of an element of positive degree in a pseudo-differential operator ring is commutative. The original argument is of Schur from 1904. The talk will be based on a paper by K. R. Goodearl on the subject.
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May 27, 2009 |
Speaker:
Chris Nowlin Abstract:
In examining n-dimensional Euclidean space, we consider its
coordinate ring, which consists of polynomials in n variables
where each variable is thought of as coordinate functions from
Euclidean Space to the base field.
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May 20, 2009 |
Speaker:
Brian Sittinger (CSU Channel Islands) Abstract:
As many of you may have at least heard about the Prime Number
Theorem, probability (density) questions about the integers form
a branch of number theory. One can ask what is the probability of
randomly picking a square-free integer or picking a pair of
integers that is relatively prime. Stunningly, the answer to both
of these questions is the unfathomable 6/(π2). If
you have studied series, you may recognize this number as the
reciprocal of ζ(2)=1+1/(22)+1/(32)+....
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May 13, 2009 |
Speaker:
Tom Howard Abstract: The complexity of a module is an extension of projective dimension, measuring the asymptotic growth of a minimal projective resolution. I will use ideas from homological algebra to explore basic properties of complexity and give conditions on when the growths are reasonably well-behaved. This talk should be accessible to anyone currently in Birge's Math 236B or familiar with projective resolutions, long exact sequences, and the Ext functor.
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May 6, 2009 |
Speaker:
Chris Nowlin Abstract: This talk will be a report on my research, which lately has been trying to pin down prime ideals in a particular example of a Quantized Coordinate Ring I've been examining. The talk will be given in such a way that most people should understand what is being said, but no motivation for why anybody would care should be expected. Any motivation or application given in the talk will likely be fabricated.
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April 29, 2009 |
Speaker:
Garrett Johnson Abstract: One way of constructing a quantum group is via specific linear operators called R-matrices. I will give some examples of this construction. When studying the representation theory of the double affine Hecke algebras, one notices some operators that also happen to be R-matrices. This gives another interpretation of these quantum groups.
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February 25, 2009 |
Speaker:
Tom Howard Abstract: I'll finish the description of the module G-varieties I began last week. The orbit spaces induce a topology on the set of isomorphism classes of modules. A property of modules is said to be generic if, given any irreducible subset of modules, there is either a dense open subset of modules with the property or without the property. I'll introduce generically continuous functions and show how they can be used to find new generic properties.
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February 18, 2009 |
Speaker:
Tom Howard Abstract: When studying modules, over a finite dimensional algebra in our case, one often comes across families of modules which differ by just a few scalars, and hence can studied all at once. This approach can be put on a rigorous setting by introducing a variety parametrizing the modules. This week, I'll compare and contrast three candidates for such a variety.
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February 11, 2009 |
Speaker:
Chris Nowlin Abstract:
Yes it can. We will look at a particular example of a
noncommutative polynomial ring, one that is of interest to nobody
save myself. We will notice pretty easily that the variables
don't commute, but we will see that this ring is nonetheless
"nice". This example will be a springboard to a (rather
one-sided) discussion of what is meant by the word "nice"
in this context. For those who feel an abstract should include
technical words, nice noncommutative polynomial rings as I mean
them can also be referred to as normally separated torsion free
CGL extensions.
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December 4, 2008 |
Speaker:
Joules Nahas Abstract: We use techniques from harmonic analysis on the Heisenberg group to relax a technical condition on the sum of squares decomposition for the Weyl Algebra.
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November 20, 2008 |
Speaker:
Julia Galstad Abstract: Analogies are known to exist between Kleinian codes (codes over the Kleinian Four Group), binary codes, lattices and vertex operator algebras [see Hoehn's Self-dual Codes over the Kleinian Four Group]. A new and natural fifth step in this sequence is defined and explored. L-codes are also codes over the Kleinian four group, but live in a different quadratic space than Kleinian codes, hence the new name. Because of the relationship between L-codes and vertex operator superalgebras, the classification of extremal even self-dual L-codes is of special interest. I will give some theorems about L-codes, then I will present some remaining open questions.
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November 6, 2008 |
Speaker:
Tom Howard Abstract: The holy grail of representation theory is, given an algebra A, completely understand the category A-Mod of representations. The usual strategy for classifying the objects of A-Mod is to restrict attention to the indecomposable modules, but the approach to classifying the morphisms in A-Mod is less clear cut. I will discuss one approach through the classification of irreducible morphisms. The main tools for computing irreducible morphisms are the almost split sequences, also called Auslander-Reiten sequences. Finally, the data of the indecomposable modules and irreducible morphisms can be depicted graphically in terms of the Auslander-Reiten quiver of the category A-Mod.
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October 30, 2008 |
Speaker:
Chris Nowlin Abstract:
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October 22, 2008 |
Speaker:
Garrett Johnson Abstract: This week I'll actually get to the research I'm currently working on. I'll have those funny pictures up on the board again and I'll give more examples. We'll look at some examples of quantizing (whatever that means?) classical r-matrices for some small quantum groups like SL(2) and depending on how ambitious we get, SL(n). You can expect more equations too!
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October 15, 2008 |
Speaker:
Garrett Johnson Abstract: I probably won't get to what a quantum group is, but we'll definitely have time to go over the "pre-quantized" quantum groups, also know as Poisson-Lie groups (or Poisson algebraic groups). These are groups with the added structure of being a Poisson manifold (or Poisson variety). I'll explain more in my talk. For the groups I'm considering, the geometric structure has a nice algebraic interpretation, called a classical r-matrix. A classification of r-matrices, and hence Poisson-Lie groups, was achieved in the early 1980's by Belavin and Drinfeld. If time permits I'll go over what the classification theorem says and then try to motivate quantum R-matrices, but this will probably be a task better left to Chris.
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October 8, 2008 |
Speaker:
Tom Howard Abstract:
In studying differential and algebraic geometry, I was left with
two nagging questions:
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Date |
2007 - 2008 Schedule |
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May 28, 2008 |
Speaker:
Brian Sittinger (CSU Channel Islands) Abstract:
"Integration is not as straightforward as differentiation;
there are no rules that absolutely guarantee obtaining an
indefinite integral of a function."- James Stewart,Calculus
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May 21, 2008 |
Speaker:
Ed Rehkopf Abstract: This is a continuation of last week's talk.
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May 14, 2008 |
Speaker:
Ed Rehkopf 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.
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May 7, 2008 |
Speaker:
Chris Nowlin 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.
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April 30, 2008 |
Speaker:
Garrett Johnson 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.
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April 9, 2008 |
Speaker:
Tom Howard 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.
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April 2, 2008 |
Speaker:
Chris Nowlin 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.
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March 6, 2008 |
Speaker:
Tom Howard 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.
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February 28, 2008 |
Speaker:
Keith Thompson 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.
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February 21, 2008 |
Speaker:
cancelled Abstract: The seminar is cancelled this week so that we may see Bill Jacob speak at the Graduate Student Colloquium.
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February 14, 2008 |
Speaker:
Jesse Liptrap 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 $\prod_{i < j}(z_i-z_j)$.
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February 7, 2008 |
Speaker:
Chris Nowlin 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.
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January 31, 2008 |
Speaker:
Garrett Johnson 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.
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December 12, 2007 |
Speaker:
Garrett Johnson 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.
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December 5, 2007 |
Speaker:
Tom Howard Abstract: This is a continuation of last week's talk.
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November 28, 2007 |
Speaker:
Tom Howard 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.
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November 14, 2007 |
Speaker:
Charlie Beil 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.
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November 7, 2007 |
Speaker:
Chris Nowlin 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.
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October 31, 2007 |
Speaker:
Charlie Beil 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.
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October 24, 2007 |
Speaker:
Garrett Johnson 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.
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October 17, 2007 |
Speaker:
Tom Howard 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.
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Date |
2006 - 2007 Schedule |
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August 8, 2007 |
Speaker:
Tom Howard 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.
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June 7, 2007 |
Speaker:
Tom Howard 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.
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May 31, 2007 |
Speaker:
John Levitt 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.
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May 24, 2007 |
Speaker:
Charlie Beil 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).
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May 17, 2007 |
Speaker:
Chris Nowlin 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.
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May 3, 2007 |
Speaker:
Garrett Johnson 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.
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April 26, 2007 |
Speaker:
Tom Howard 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.
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Feb 15, 2007 |
Speaker:
Tom Howard 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.
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Feb 1, 2007 |
Speaker:
Ryan Ottman 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.
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Nov 28, 2006 |
Speaker:
Garrett Johnson 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.
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Nov 14, 2006 |
Speaker:
Garrett Johnson 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.
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Nov 7, 2006 |
Speaker:
Chris Nowlin 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".
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Oct 31, 2006 |
Speaker:
Garrett Johnson 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 26/May/2010