# Objectives of the Cartan-Lie Group

For more on Elie Cartan

For more on Sophus Lie

This is a small working group which started meeting in the Fall of 1995. The goal is to discuss research problems in computer science involving geometry. In the Fall semester 95, we tried to see how the theory of Lie groups could be applied to certain problems in computer vision and motion planning. This led us to study differential forms. This study was continued in the Spring, with fibre bundles and connections. Other topics were considered later on.

This semester (Spring 2000), we are planning to meet every Friday, 4:03-4:33pm, in Moore 554

I will need volonteer speakers. As in the past, we will also hear people talk about their current pet problems.

## Next Meeting:

Friday April 21, 3:03-4:33pm
Moore 554

## Topic:

Structure of Linear Shift-invariant Operators on Sequences

## Speaker:

Paul Hughett
University of Pennsylvania, Neuropsychiatry Section

Abstract: The theory of digital signal processing is based on three fundamental results about the structure of a linear shift-invariant (LSI) operator on sequences. (1) The action of any LSI operator can be represented as convolution by its impulse response. (2) Every complex exponential sequence is an eigenfunction for every LSI operator. (3) The frequency response of an LSI operator is the Z-transform of its impulse response. Recent work has shown, however, that the traditional proofs of these facts are not valid if non-causal operators or two-sided sequences are admitted and that it is possible to construct LSI operators not representable by their impulse responses and whose frequency responses are unrelated to their impulse responses. (On the other hand, the exponential eigenfunction property can be shown rigorously, even though the traditional proof is invalid.) In this talk, I will exhibit counterexamples to the traditional structure theorems, explain conditions under which the traditional theorems (or close analogs) remain valid, and discuss the structure of general LSI operators on two-sided sequences.

## Old Problems

1. Find an efficient method for defining triangular surface splines meeting with C^k-continuity. Find some kind of de Boor algorithm (contributed by Jean Gallier).

2. Find an efficient method for determining the intersection of a quadric (say a sphere) with a parametric surface of bidegree <2, 2> (contributed by Deepak Tolani).