Download Paper (version of 2011-06-09)

Abstract:
In our paper “Non-commutative desingularization of determinantal varieties, I” we constructed and studied non-commutative resolutions of determinantal varieties defined by maximal minors. At the end of the introduction we asserted that the results could be generalized to determinantal varieties defined by non-maximal minors, at least in characteristic zero. In this paper we prove the existence of non-commutative resolutions in the general case in a manner which is still characteristic free. The explicit description of the resolution by generators and relations is deferred to a later paper. As an application of our results we prove that there is a fully faithful embedding between the bounded derived categories of the two canonical (commutative) resolutions of a determinantal variety, confirming a well-known conjecture of Bondal and Orlov in this special case.

Notes:

Check back soon for a major revision to this preprint.

BibTeX code:

Coming eventually.

Download Paper (completely revised version of 2013-03-27)

Abstract:
In this paper we consider Grassmannians in arbitrary characteristic. Generalizing Kapranov’s well-known characteristic-zero results we construct dual exceptional collections on them (which are however not strong) as well as a tilting bundle. We show that this tilting bundle has a quasi-hereditary endomorphism ring and we identify the standard, costandard, projective and simple modules of the latter.

Notes:
This version is more up-to-date than the arXiv copy: it both contains the Acknowledgments that an earlier draft lacked, and has the embarrassing typo fixed. This material will not be published on its own; it will be included in Non-commutative desingularization of determinantal varieties, II. We thought it was interesting enough to post on the arXiv by itself, particularly since it seemed to us that as soon as anyone knew the statement of the result, they would probably be able to write down the proof.

This is a complete revision of the original note, now four times as long with a lot of new material. Rather than folding it into part 2 of the determinantal varieties paper, we now plan to publish it on its own.

Joint work with Roger Wiegand.

Download (version of 2012-06-22)

Abstract:
The Brauer-Thrall Conjectures, now theorems, were originally stated for finitely generated modules over a finite-dimensional k-algebra. They say, roughly speaking, that infinite representation type implies the existence of lots of indecomposable modules of arbitrarily large k-dimension. These conjectures have natural interpretations in the context of maximal Cohen-Macaulay modules over Cohen-Macaulay local rings. This is a survey of progress on these transplanted conjectures.

Notes:

Much of this material appears in our book, but here it is collected in one place, and explained a bit more clearly. The paper also lists some open questions.

Bibtex code:

 coming soon

The course is over! Final grades have been submitted to MySlice. Thanks to everyone for a great semester, and enjoy your break.

Basic Info

Instructor

Prof. Graham Leuschke
email: gjleusch@math.syr.edu
AIM: leuschkeg
Google: leuschke@gmail

Time & Place

TR 9:30–10:50 in Carn 300

Textbook

Carl D. Meyer, Matrix Analysis and Applied Linear Algebra

Office Hours

MW 1:30–4:30; by appointment; and any time my door is open (317G Carnegie)

Important documents

syllabus

Announcements

Here is last year’s final exam. As always, it should be used with caution: we did not cover exactly the same material this year. Also, this year’s final will have a large section consisting of definitions and formulas, like Exam II.

Exam II was in class on Thursday 29 November. It covered up through orthogonal projections and reflections, that is, Sections 5.1–5.6, plus the PageRank material. Here is last fall’s Exam II for your reference. Note that we did not cover exactly the same material this year. Also I announced in class that about half the points on this exam will be “theoretical” questions: definitions and formulas, rather than computations.

Exam I was in class on Tuesday 9 October. It covered up through least-squares, that is, all of the material we covered from Chapters 1–4. Here is last fall’s Exam I for your reference. Note that we did not cover exactly the same material before the first exam this semester as last year.

It has been said that ‘the human mind has never invented a labor-saving machine equal to algebra.’ If this be true, it is but natural and proper that an age like our own, characterized by the multiplication of labor-saving machinery, should be distinguished by the unexampled development of this most refined and most beautiful of machines.

Josiah Willard Gibbs, 1887

Problem Sets

• HW #1, due Thursday 6 Sept
• HW #2, due Thursday 13 Sept. Here is a solution to problem #2, and here are the calculations in 3-digit floating-point arithmetic for problem #4 (a.k.a. #1.1.5).
• HW #3, due Thursday 20 Sept (corrected)
• HW #4, due Thursday 27 Sept
• HW #5, due Thursday 4 Oct
• HW #6, due Thursday 18 Oct
• HW #7, due Thursday 25 Oct
• HW #8, due Thursday 8 Nov
• HW #9, due Thursday 15 Nov (OMIT problem 1(c))

Links

• interactive data-fitting (least-squares/regression lines). Place a bunch of points more-or-less on a line, watch the line snap to them. Then move one waaaaay off the line, and see how robust the regression line is.
• Mathematics at Google: “There is a wide variety of Mathematics used at Google. For example Linear Algebra in the PageRank algorithm, used to rank web pages in search results. Or Game Theory, used in ad auctions, or Graph Theory in Google Maps. At Google there are literally dozens of products which use interesting Mathematics. These are not just research prototypes, but real Google products; in which Mathematics play a crucial role. In this presentation, I introduce several applications of Mathematics at Google.”
• an online linear algebra toolkit
• an application where rank-one updates and their inverses arise
• The Anatomy of a Large-Scale Hypertextual Web Search Engine [PDF] (Sergey Brin and Larry Page’s original paper describing the PageRank algorithm. Surprisingly readable.
• The Use of the Linear Algebra by Web Search Engines [PDF] (Again, surprisingly readable. Has sections on PageRank and Jon Kleinberg’s HITS ["hubs and authorities"] algorithm. Co-written by our textbook’s author.)
• The second-order linear differential equation we studied on 4 September is essentially the same thing as a Sturm-Liouville equation.  Examples include the Bessel equation and the Legendre equation.
• What Every Computer Scientist Should Know About Floating-Point Arithmetic (a slightly different approach than ours)
• The Nine Chapters on the Mathematical Art (wikipedia)
• A Brief History of Linear Algebra and Matrix Theory from Marie Vitulli at Oregon
• James Joseph Sylvester (1814-1897) first coined the term ”matrix”: “a rectangular array of terms out of which different systems of determinants may be engendered, as from the womb of a common parent.”  (”Matrix” is the Latin word for ”womb.”  Sylvester was  also responsible for the mathematical terms ”syzygy”, ”meicatecticizant,” and ”tamisage.”  His articles are still great fun to read.) Here’s some more on Sylvester.
• Carl Friedrich Gauss (Wikipedia) was the greatest mathematician of all time (OF ALL TIME), but he did not invent Gaussian elimination.
• history and motivation of determinants (nothing about why you should *avoid* them, but we’ll see that later)
• A complete explanation of the operation counts in Gaussian elimination.  Also Gauss-Jordan and finding the inverse.
• World Freehand Circle Drawing Champion because why not