Mathematical problem solving:
What do you need? Patience, curiosity, optimism, and techniques, to name a few. You gain some of these skills by solving problems that are quite doable, but even more by trying to solve problems, that are actually a bit too hard for you.
Solving a problem different to anything you have seen before is like a first ascent of a mountain - a genuine exploration, with greater risks, challenges and satisfaction. Taking wrong turns and climbing backwards to try another route. And again. And again...
Eventually you may give up solving it on your own, but you can still convert your efforts into valuable experience. Read the solutions. Does that mean you actually understand them? Usually not. Try to solve it on your own again a week later... Read, wait, solve, repeat until you can.
George Polya's classic book "How to solve it"
Terence Tao's blog on "Solving mathematical problems"
Mathematics and the pandemic:
With the start of the Covid-19 pandemic, mathematics has entered media and politics. It was unusual for Germany to have a chancellor with a PhD in quantum chemistry, but even more unexpected Angela Merkel would go on explaining the definition and role of R0 in interviews with journalists about the rationale for the lockdown. Christophe Fraser's team at the Big Data Institute has been developing a system of digital contact tracing in conjunction with testing to control R0 while opening up the country.
Simulations of pandemic and effect of countermeasures (3Blue1Brown)
What types of thinking underly problem solving? There is logical thinking as in the Green-eyed Prisoners Puzzles. There is combinatorial thinking in the this Passcode riddle. A few examples for the so-called out-of-the-box thinking are discussed in this video about the Psychology of Problem-Solving. And there are the enemies of thinking, as illustrated for example in the video How to stay calm under pressure.
If you could not get hold of yarn for the mathematical knitting projects linked to in the last package, here is your second chance to use your hands for doing and understanding mathematics. All you need is paper and a flat hard horizontal surface (e.g. a table)!
Data-driven research in ecology
Mathematicians, statisticians and data scientists are increasingly needed for quantitative research in ecology. Conservation and evolution of species and ecosystems under changing environments are crucial questions in today's world. For decision making industrial countries, trade-offs have to be found between ecological and economical objectives, and they need to be based on both theory and empirical observations. The availability of data is impressive due to shared data from scientific, repositories of volunteers observations (aka citizen scientists) and the increasing use of remote sensing technologies in the field.
You can start exploring such yourself using publicly accessible datasets. Here are some repositories:
- The UK Butterfly Monitoring scheme (UKBMS) provides annual data an the population status of butterflies derived from a wide-scale programme of site-based monitoring and sampling in randomly selected 1km squares. The scheme has monitored changes in the abundance of butterflies throughout the United Kingdom since 1976. Forty years later, trends in butterfly populations were compiled from a network of over 4,000 locations across all years, with nearly 2,500 sample locations monitored in 2015. The UKBMS is based on a well-established and enjoyable recording method listed above and has produced important insights into almost all aspects of butterfly ecology. The UKBMS data repository is managed by the UK Centre of Ecology and Hydrology (CEH) under their Find data tab. Simply type UKBMS into the search box.
- The UK Centre of Ecology and Hydrology (CEH) is an independent, not-for-profit research institute. The 500 scientists provide the data and insights that researchers, governments and businesses need to create a productive, resilient and healthy environment. To facilitate quantitative research they provide a rich data sets collection covering many animal and plant species as well as other environmental data, usually ready for free download.
- Movebank is a free online platform that helps researchers manage, share, analyze and archive animal movement data. You can choose from whales, vultures, bats and many more species. Movebank is hosted by the Max Planck Institute of Animal Behavior in coordination with the North Carolina Museum of Natural Sciences, the Ohio State University and the University of Konstanz. More details can be found in their About tab. Under their Data tab many data sets along with documentation can be accessed immediately and the drop down filter at the top helps finding them. (Some require an application for permission, but the freely accessible ones are enough to keep you busy for years.) The download tab in the Studies window just asks you to agree to the Movement conditions and then gives you a choice of downloading formats. CSV means comma separated values and can be read by most programming languages. If the columns are not self explanatory, you can request more information and also see the Help Tab and Tools Tab provide further information.
Before such data can be analysed, you need to do a lot of detective work. What are the meanings of the columns and rows in the data files? Try to find information on the site where you got the data from and search for suitable website for more context about the domain to get ideas of how the data should be interpreted. Are the data complete or are entries missing? Are they missing at random or are there patterns (e.g. related to certain times or locations)? Once you have understood what the data entries actually mean, you can think of good visualisations and summaries of the information contained in the raw data sets. Why not acquiring some basic R skills using the links above and trying your hand on some of the datasets? See Package 1 for resources on how to learn R. (If you know other languages, e.g. Python, you may use this as well.)