These are notes for a very rapid introduction to the basics of exterior differential systems and their connection with what is now known as Lie theory, together with some typical and not-so-typical applications to illustrate their use.

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With the ever-increasing global demand for high quality rice in both local production regions and with Western consumers, we have a strong desire to understand better the importance of the different traits that make up the quality of the rice grain and obtain a full picture of rice quality demographics. Rice is by no means a ‘one size fits all’ crop. Regional preferences are not only striking, they drive the market and hence are of major economic importance in any rice breeding / improvement strategy. In this analysis, we have engaged local experts across the world to perform a full assessment of all the major rice quality trait characteristics and importantly, to determine how these are combined in the most preferred varieties for each of their regions. Physical as well as biochemical characteristics have been monitored and this has resulted in the identification of no less than 18 quality trait combinations. This complexity immediately reveals the extent of the specificity of consumer preference. Nevertheless, further assessment of these combinations at the variety level reveals that several groups still comprise varieties which consumers can readily identify as being different. This emphasises the shortcomings in the current tools we have available to assess rice quality and raises the issue of how we might correct for this in the future. Only with additional tools and research will we be able to define directed strategies for rice breeding which are able to combine important agronomic features with the demands of local consumers for specific quality attributes and hence, design new, improved crop varieties which will be awarded success in the global market.