had a spirit of international cooperation. As a result, whenever one center developed an
idea that resulted in a major improvement in the operational forecast skill, the idea was
discussed in scientific meetings and workshops, and different centers tested variations
of the same idea in their own center. There is no doubt that this friendly international
competition has resulted in much faster progress than what would have taken place if
the method of improvement had been kept secret.
4.2. Specific recommendations
In addition to the relatively high-level recommendations of the previous section, a
number of potentially promising avenues for specific areas of the mathematical sciences
also emerged from group discussions. We describe these areas in bullet point format
below. These recommendations are not intended to be an exhaustive list of potentially
promising areas of research.
- Mathematical formulation of sustainable, composable, multiscale dynamical
systems acting on discrete, non-smooth real world data; multi-scale
characterization of features emerging in the data; - New theoretical frameworks for the dynamics of hybrid HES models
consisting of different types of models for human versus environmental
submodels; - Methods to characterize dynamic topologies relevant to analysis of social and
biological networks; - An expanded mathematical toolkit to explore the impact of coupling in
dynamic, complex, adaptive HESs and to characterize the interaction of
shifting rates of dynamics in coupled systems; - Further development of complex dynamical systems theory as it relates to
HES; - Further developing of coding and information theory, time series analysis,
and projection geometry as it relates to HES; and - Theory of ensemble models, such as development of theoretically justified
simulation, and machine learning and pattern recognition with neural
networks and genetic algorithms.
These areas can obviously involve a broad segment of the mathematics
community, including the computational harmonic analysis community, the dynamical
systems community, the computational topology community, the distributed systems
community in computer sciences, the algebraic and analytic geometry community, the
topological quantum computing community, the stochastic PDE community, the
stochastic processes community, and the information theory community.
References
- Anand M, Gonzalez A, Guichard F, Kolasa J, Parrott L (2010). Ecological systems
as complex systems: Challenges for an emerging science. Diversity, 2 : 395-410. - Beddoe R, Costanza R, Farley J, et al. (2009). Overcoming systemic roadblocks to
sustainability: The evolutionary redesign of worldviews, institutions and
technologies. PNAS 106(8): 2483-2489.