Department of Mathematics

The Florida State University

This Week in Mathematics

29 March - 2 April 1999

Monday: 29 March 1999

Graduate Student Seminar, 1:30 p.m., 204B Love Building

moved this week to Wednesday, 31 March, 1:30pm

Special Colloquium Coffee, 3:00 p.m., 204 Love Building
Special Colloquium, 3:35 p.m., 101 Love Building

Thorsten Woermann, Florida State University

Real Roots and Sums of Squares of Polynomials

Tuesday: 30 March 1999

Algebraic Geometry Seminar, 2:00 p.m., 102 Love Building

Angela Vierling, Florida State University

Resolving Singularities of Curves

No Applied Topology Seminar, 3:35 p.m., 104 Love Building

Wednesday: 31 March 1999

Graduate Student Seminar, 1:30 p.m., 204B Love Building

Ben Fusaro, Florida State University

Finite Math and Logic

(Real) Analysis Seminar, 2:30 p.m., 201 Love Building

Reading through “Oscillatory Integrals with Polynomial Phases” by Phong and Stein

Complex/Symbolic Seminar, 3:35 p.m., 102 Love Building

A. H. M. Levelt, University of Nijmegen

Formal Solutions of Linear Differential Equations

Linear differential equations with analytic (e.g. meromorphic or rational function) coefficients are a classical and much studied subject showing special phenomena and characteristics. As an example look at the differntial equation x^2 y"+(3x-1)y'+y=0. This equation has y=\sum_{n=0}^{\infty} n! x^n as a 'solution'. Even if analytic solutions are the ultimate goal, a good understanding of formal solutions is indispensible. Since more than a century papers have been devoted to this subject. A complete classification has been given by W. Balser, W.B. Jurkat and D.A. Lutz (A General Theory of Invariants for Meromorphic Differential Equations; Part I, Formal Invariants, Funkcialaj Evacioj, 22 (1979), 197-221).
          An easy proof is the main objective of the talk. A simple proof of the (rational) Jordan decomposition of square matrices can be turned into one of the BJL classification by a minor adaptation (`endomorphisms of certain simple modules'). In fact a more intrinsic set of invariants will be given and their relation will be explained to fundamental systems of (formal) solutions, certain differential Galois groups, characteristic classes, exponential parts and generalized exponents.

Refreshments, 3:00 p.m., SCRI Seminar Room - 499 DSL
Interdisciplinary Lectures in Science & Computing, 3:30 p.m., 499 DSL

J. Tinsley Oden, University of Texas

A Posteriori Error Estimates and Goal-Oriented Adaptivity -- Application to Heterogeneous Media

The cornerstone of adaptive modeling and meshing is the availability of reliable a posteriori estimates of error. This error can be the result of a poor mathematical model of the physical event or of error in the numerical approximations. In this lecture, procedures for obtaining upper and lower bounds of modeling error in the analysis of the heterogeneous materials are presented. Similar procedures can be used to estimate numerical error in the numerical solutions of partial differential equations. Numerical results are described in which estimates of modeling and approximation error are used to drive goal-oriented adaptive methods.

Thursday: 1 April 1999

Algebra Seminar, 2:00 p.m., 104 Love Building

Angela Vierling, Florida State University

Resolving Singularities of Curves

Topology Tea Time, 3:00 p.m., 204 Love Building

Topology Seminar, 3:35 p.m., 104 Love Building

Ulrich Koschorke, Universitat-Gesamthochschule Siegen

Milnor's mu-Invariant and Higher Dimensional Link Homotopy

Special Colloquium Coffee, 3:00 p.m., 204 Love Building
Special Colloquium, 3:35 p.m., 201 Love Building

Agnes Szanto, MSRI, Berkeley

Solution of Degenerate Polynomial Systems of the Complex Field

The main emphasis of the talk is to find the set of complex roots of systems polynomial equations in degenerate cases, i.e. when the codimension of the algebraic set and the number of defining equations are not equal. First, I investigate a representation for algebraic sets originally proposed by Michael Kalkbrener. This representation gives an alternative to Groebner bases for many applications, and is suitable for deriving improved time and space complexity bounds. Although the computation of Kalkbrener's representation appeared to be more efficient in practice than Groebner bases techniques, but it was not known if there exists sub-exponential worst case complexity bound.
     I will describe new algorithms both for the computation of Kalkbrener's representation and for conducting geometric operations on algebraic sets using Kalkbrener's representation. To compute Kalkbrener's representation, I describe methods based on the sparse elimination theory of Gelfand, Kapranov and Zelevinsky. These methods exploit the sparseness of the input polynomials and in the dense representation of the polynomials I could prove sub-exponential complexity bounds. Moreover, using Kalkbrener's representation, I present efficient parallel algorithms for geometric operations on algebraic sets, generalizing the zero dimensional "lazy factorization" algorithms of Teitelbaum to higher dimensions.
     The other issue addressed in the talk is how finite precision methods can be combined with symbolic techniques to solve systems of polynomial equations. I present a new predictor-corrector method for solving over-constrained systems of polynomial equations. The algorithm combines the homotopy method developed by Shub and Smale to analyze the complexity of Bezout's theorem and the theory of general resultants developed by Gelfand, Kapranov and Zelevinsky.

Friday: 2 April 1999

Colloquium Coffee, 3:00 p.m., 204 Love Building
Colloquium, 3:30 p.m., 101 Love Building

A. H. M. Levelt, University of Nijmegen

The Cycloheptane Molecule: A Challenge to Computer Algebra

The picture on the left represents cyclohexane, a ring-shaped molecule formed by 6 carbon atoms with single bonds (not to confuse with benzene!) where the remaining 12 bonds are occupied by hydrogen atoms.
     The picture on the right shows the cycloheptane molecule. It has 7 carbon atoms instead of 6.
     In our model of the molecules the distances of successive carbon atoms are fixed and equal; the angle between the bonds is the characteristic one of charbon (about $109^\circ$). A macroscoping realization of these models shows that cyclohexane has two {\em conformations} (as the chemists say): a rigid one and another one with one degree of freedom. Cycloheptane also has two conformations, both of them having one degree of freedom. They cannot be transformed into each other by continuous moves.
     The question arizes whether this phenomenon can be understood in mathematical terms. In his Ph.D. thesis L.J. Oosterhoff (Leiden, 1949) showed by an easy computation that the configuration space of cyclohexane consists of a fourth degree plane curve (genus 1) and an isolated point.
     Cycloheptane turns out to be much harder. The configuration space $\cal C$ is an algebraic curve in 7-space, the solution space of a complicated set of polynomial equations. The projections of $\cal C$ on the 2-dimensional coordinate spaces are such that a computer seems indispensible (algebraic curves of degree 32, about 250 terms, 30 digit coefficients). Nevertheless, one can compute points (floating point, algebraic) on $\cal C$ sufficiently efficiently to get a graphic representation of $\cal C$ where the two components are clearly visible.
     Some questions have been answered, many remain. E.g. what is the physical (resp. chemical) meaning our results? It is a work in progress.

No Scientific Computing Seminar, 4:30 p.m., 200 Love Building

1st Annual Mathematics Honors Day, 3:30 p.m., 102 Love Building
Goodner TA Awards followed by Pi Mu Epsilon Induction Reception Following in 204 Love Building

Donald Foss, Florida State University

Measuring Mentation: A Look at Listening to Language

Your eyes move when you read, and we can learn some things about what is happening during reading by measuring those movements. But your ears don't move when you listen; you just sit there. How, then, can we measure what is happening during auditory language comprehension, the most natural way we comprehend? Recent advances in cognitive neuroscience permit us noninvasively to observe some brain activity during listening comprehension, but the temporal resolving power of those techniques is inadequate--in other words, comprehension is fast and the brain imaging techniques are (so far) just too slow. In this talk I will discuss these problems and describe measurement techniques, some of which were devised in my laboratory, that can get at auditory comprehension while it is happening--some ways of measuring mentation.

Seminars and colloquia at "that other" university [a.k.a. the University of Florida]

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