Philosophical Statement:  There is obviously a connection between education and doing research (mathematical or otherwise), but making that connection precise is a lot harder than it should seem to be. One of the difficulties lies in acquiring a good intuition for all of the background knowledge and tools that go into any modern scientific research project. Without this intuition, it is difficult to ask new, reasonable questions, answer these questions, and bridge the gap between disciplines. Intuition, albeit hard to rigorously define, is a compression (with loss) of more detailed information. Furthermore, it is an alterable thing, that is formed via the filtering for and synthesis of certain poignant ideas. Probably, it is actually a story (even in math) we tell ourselves, that we believe captures the essential plot points. With intuition, details can be temporarily swept aside. Of course, with time, details may reemerge as poignant and alter this intuition. Certain elements of the plot become more essential in certain contexts, but despite its need for alternation, the thing that makes intuition essential, is that it is testable against the truth. When required, the details, written and stored in a vast number of textbooks and papers, can be brought back into focus to reinform our intuition. This is all to argue, that the building of intuition about a larger number of ideas communally is more primary to research than is currently acknowledge. Instead of strengthening the broader scientific communities understanding of older ideas, the focus is and incentives for: new and more, which can lead to niche and esoteric. New and more can of course be reasonable goals, and they are also embedded in the dominant paradigm for what makes a good society; but I think more social mechanisms, reward systems, and celebrations of older ideas could do some good in compressing the vast amounts of new information that have been produced in the explosion of scientific ideas over the past century. Hopefully then, more people can have greater insight into the tools and knowledge that may generally be taken for granted or hidden in a black box.

Overview: My statement above is more or less only a justification for having a section of this website devoted to showing off ideas I find interesting.  In no way do I hope to achieve anything like I argue for above, but let it be like a star that we might dream of getting to one day. Unlike a paper or textbook, the goal here will be to stick to small examples that I think tell an important story. Of course, the story will not capture all of the details (and will often require background knowledge), but I hope to provide what I think are like informative vignettes . This site is new, so please excuse the small number of  such vignettes, more shall come with time...