“Donaldson has opened up an entirely new area; unexpected and mysterious phenomena about the geometry of 4-dimensions have been discovered. Moreover, the methods are new and extremely subtle, using difficult nonlinear partial differential equations. On the other hand, this theory is firmly in the mainstream of mathematics, having intimate links with the past, incorporating ideas from theoretical physics, and tying in beautifully with algebraic geometry. It is remarkable and encouraging that such a young mathematician can understand and harness such a wide range of ideas and techniques in so short a time and put them to such brilliant use. It is an indication that mathematics has not lost its unity, or its vitality.” – Sir Michael Atiyah
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In my previous post, I discussed how to adapt a problem that you have found in order to make the problem groupworthy. One of the important things to consider when adapting real-world problems is to avoid giving step-by-step instructions and formulas to students. Instead, a teacher should maximize the opportunities for groups to make their own decisions about problems. In other words, in order to have challenging and productive group discussions, there must be an element of uncertainty so that students engage with the problem and with each other.
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“Have you ever thought about how strange it is that we think about infinity every day, but most people think about it only on the rarest of occasions, if ever?” This is the text message I recently sent two of my close friends, who also happen to be mathematicians in my department. I was deep in the midst of studying for preliminary exams, trying to prove Riemann’s Theorem on removable singularities, when I started to think – really think – about infinity.
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