Nanoscale Pattern Formation at
Collaborator: Mark Bradley, CSU Department of Physics
Bombarding a solid surface with a broad ion beam can lead to the spontaneous formation of a
remarkable variety of self-assembled nanoscale patterns, including periodic height modulations or
"ripples," mounds arranged in hexagonal arrays of astonishing regularity, and arrays of sharply
pointed conical protrusions. The emergence of these patterns is fascinating in its own right,
but active experimental investigations are aimed at developing ion bombardment as an important tool
in the fabrication of nanostructures for a wide range of applications. Our team has developed
a theory that explains the genesis of patterns that form spontaneously when a binary solid is
subjected to ion bombardment. We are investigating defects in these patterns, with the aim of
suggesting techniques whereby defects may be minimized.
Collaborators: Stephen Thompson, CSU Department of Chemistry; Jaime Shinn, Department of
We are seeking a universal mathematical description of experiments involving diffusion and
nucleation and growth processes that are being performed in The Laboratory for Mathematics in the
Sciences. This includes gas-phase periodic precipitation phenomena and patterns produced
by polymer-gas interactions.
Phyllotaxis and Growth in Biological
Collaborators: Alan Newell, University of Arizona; Todd Cooke, University of Maryland-College
Through mathematical models for the formation of phyllotactic patterns on plants from
biochemical and biomechanical mechanisms, we suggest ways to understand both universal aspects of
phyllotactic patterns as well as how interacting mechanisms can cooperate or compete to produce the
array of phyllotactic lattices and polygonal planforms seen in nature.
Supported by NSF Grant DMS-000000
Invasive Species Spread
Collaborators: Christopher Strickland, Department of Mathematics; Gerhard Dangelmayr,
Department of Mathematics; Sunil Kumar, Natural Resource Ecology Lab; Tom Stohlgren, Natural
Resource Ecology Lab
Many scientists now recognize invasive species as the number one environmental threat of the
21st Century. Invasive species pose threats to global ecosystems, including processes, functions,
and the life they sustain. The invasion of non-native plants, animals and pathogens has escalated
dramatically over the last few decades with the increase of trade, transportation and other
elements of globalization. Impacts include loss of native species and habitat, economic
suppression, reduced food and water security, and direct threats to human health.
Our focus species is cheatgrass, which is one of the worst invaders in the western United
States. Cheatgrass has also invaded many parts of the Rocky Mountain National Park (RMNP) and
is spreading further into the park. Our team has developed a prototype model, using as a basis data
available for the invasive cheatgrass (Bromus tectorum) in RMNP. We are working to
incorporate more available data into the model, and we are in the beginning stages of mathematical
analysis of the model to extract information useful for managers at RMNP.
Orthogonal Coordinate Systems
Collaborator: Barbara Shipman, Department of Mathematics, University of Texas--Arlington
We investigate a Lie-algebraic framework for the Gauss-Mainardi-Codazzi equations governing
coordinate systems that are orthogonal with respect to various metrices.