Autumn 2000
Math 697R: Applicable Algebraic Geometry

Instructor: Frank Sottile
Office: LGRT 1244
Telephone: 545 - 6010
Home: 256 - 1223
Cell: 695 - 1574
email: sottile@math.umass.edu
WWW: http://www.math.tamu.edu/~sottile
two lines meeting 4 given lines in R^3
Office Hours: Tu, Th 11:30 --- 1:00 PM     and by appointment.
Lectures: Tu, Th 1:00 - 2:15 PM, LGRT 111.
Prerequisites: Some familiarity with abstract algebra, particularly linear algebra and rings of polynomials.
No familiarity with algebraic geometry is expected.
Text: Draft of Applicable Algebraic Geometry, by Sottile, J. Rosenthal, and A. Wang. Relevant chapters will be distributed during the course, and student input on the material is encouraged.
Notes from the course
Other books that are available
Overview
Course Particulars The course requirement is participation, as defined by class attendance and two problem sessions/assignments.
Make up class Wednesday, 25 October 2000 Early evening, place TBA
Rough Schedule

Notes from the course

Outline of 7 September.
Outline of 19 September.
TeX file of part of lecture on deadbeat control on 19 September. (20-09-00).
Maple file of computation in class Output of that computation.
TeX file concerning observability from lecture on deadbeat control on 19 September. (21-09-00).
TeX file with matrix manipulations of characteristic equation from 21 September. (21-09-00).
Outline of 21 September.
Outline of 26 September.
TeX file describing the algebra of polynomial matrices from 26 & 28 September. (17-10-00).
Outline of 28 September.
Outline of 3 October.
Outline of 5 October.
Outline of 10 October.
Outline of 12 October.
Outline of 17 October.
Outline of 19 October.
Outline of 24 October.
Outline of 25 October.
Outline of 26 October.
Outline of 31 October.
Outline of 2 November.
Outline of 7 November.
Outline of 8 November.
Outline of 9 November.
Outline of 14 November.
Outline of 16 November.
Outline of 28 November.

Overview

Algebraic geometry, which is the study of solutions to systems of polynomial equations, is important for its potential applications---polynomial equations are ubiquitous in mathematics and the applied sciences. Applications often demand explicit, real-number answers and tools for this have been developed in recent years from within algebraic geometry.

This course is concerned with complementary developments---problems from the applied sciences whose understanding uses algebraic geometry in an essential way. These include problems of pole placement and stabilization (linear systems theory) as well as matrix completion and eigenvalue inequalities (matrix theory). These are intended to illustrate, rather than delimit, the range of problems for which an algebraic-geometric perspective is useful.

This course is intended to provide a motivated and very concrete introduction to algebraic geometry---in particular, no knowledge of algebraic geometry is expected. Particular attention will be paid to the real numbers and to special spaces which have been important in applications.

Rough Schedule

Some mathematical control theory
            State-space realization
            Deadbeat Control
            Pole placement problem
            McMillan degree
            transfer functions
            Coprime factorization of matrices
Introduction to Affine Algebraic Geometry
            The algebraic-geometric dictionary
            Unique factorization for varieties
            Regular and rational functions
            Smooth and singular points
Some real and effective algebraic geometry
            Real algebraic geometry
            Gröbner bases
            Solving systems of polynomial equations
Projective algebraic geometry
            Subvarieties of projective space
            Maps and examples of projective varieties
            Central projection
Schubert Calculus
            Grassmann varieties
            Schubert decompositions
            Enumerative geometry
            Curves in Grassmann Varieties
Applications to Control Theory
            Hermann-Martin curve
            Constant linear feedback
                  pole placement; Theorem of Brockett and Byrnes
                  Real pole placement; Wang's Theorem
            Dynamic compensators
                  Degree limits
                  Number of compensators in the critical dimension


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Last modified: 31 October 2000 by Frank Sottile