2 research outputs found
Tverberg's theorem and graph coloring
The topological Tverberg theorem has been generalized in several directions
by setting extra restrictions on the Tverberg partitions.
Restricted Tverberg partitions, defined by the idea that certain points
cannot be in the same part, are encoded with graphs. When two points are
adjacent in the graph, they are not in the same part. If the restrictions are
too harsh, then the topological Tverberg theorem fails. The colored Tverberg
theorem corresponds to graphs constructed as disjoint unions of small complete
graphs. Hell studied the case of paths and cycles.
In graph theory these partitions are usually viewed as graph colorings. As
explored by Aharoni, Haxell, Meshulam and others there are fundamental
connections between several notions of graph colorings and topological
combinatorics.
For ordinary graph colorings it is enough to require that the number of
colors q satisfy q>Delta, where Delta is the maximal degree of the graph. It
was proven by the first author using equivariant topology that if q>\Delta^2
then the topological Tverberg theorem still works. It is conjectured that
q>K\Delta is also enough for some constant K, and in this paper we prove a
fixed-parameter version of that conjecture.
The required topological connectivity results are proven with shellability,
which also strengthens some previous partial results where the topological
connectivity was proven with the nerve lemma.Comment: To appear in Discrete and Computational Geometry, 13 pages, 1 figure.
Updated languag
Towards an integrated understanding of neural networks
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018.Cataloged from PDF version of thesis.Includes bibliographical references (pages 123-136).Neural networks underpin both biological intelligence and modern Al systems, yet there is relatively little theory for how the observed behavior of these networks arises. Even the connectivity of neurons within the brain remains largely unknown, and popular deep learning algorithms lack theoretical justification or reliability guarantees. This thesis aims towards a more rigorous understanding of neural networks. We characterize and, where possible, prove essential properties of neural algorithms: expressivity, learning, and robustness. We show how observed emergent behavior can arise from network dynamics, and we develop algorithms for learning more about the network structure of the brain.by David Rolnick.Ph. D