Kevin P. Knudson

Generating Discrete Morse Functions from Point Data

If K is a finite simplicial complex and h is an injective map from the vertices of K to ℝ, we show how to extend h to a discrete Morse function in the sense of Forman [Forman 02] in a reasonably efficient manner so that the resulting discrete Morse function mirrors the large-scale behavior of h. A concrete algorithm is given for the case where K is a subcomplex of ℝ3.

NONLINEAR STATISTICS OF HUMAN SPEECH DATA

We study the structure of point clouds obtained as time delay embeddings of human speech signals by approximating the data sets with certain simplicial complexes and analyzing their persistent homology. Results for several different sounds are presented in embedding dimensions 3 and 4. The first Betti number allows a coarse classification of sounds into three groups: vowels, nasals and noise.

The homology of special linear groups over polynomial rings

Abstract We study the homology of SL n ( F [ t, t −1 ]) by examining the action of the group on a suitable simplicial complex. The E 1 -term of the resulting spectral sequence is computed and the differential, d 1 , is calculated in some special cases to yield information about the low-dimensional homology groups of SL n ( F [ t,t −1 ]). In particular, we show that if F is an infinite field, then H 2 ( SL n ( F [ t, t −1 ]), ℤ) = K 2 ( F [ t, t −1 ]) for n ≥ 3. We also prove an unstable analogue of homotopy invariance in algebraic K -theory; namely, if F is an infinite field, then the natural map SL n ( F ) → SL n ( F [ t ]) induces an isomorphism on integral homology for all n ≥ 2.

Birth and death in discrete Morse theory

Suppose

Relative Completions and the Cohomology of Linear Groups Over Local Rings

M

The Friedlander-Milnor Conjecture

is a finite simplicial complex and that for

Unstable homotopy invariance and the homology of SL2(Z[t])

0=t_0,t_1,...,t_r=1

Relative Completions of Linear Groups Over Coordinate Rings of Curves

we have a discrete Morse function

Low-dimensional Results

F_{t_i}:M\to \zr

Rank One Groups

. In this paper, we study the births and deaths of critical cells for the functions