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jgill
Boulder climber
Colorado
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Nov 24, 2014 - 10:37pm PT
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. . . maybe one of our house mathematicians could come up with some examples of the "practical utility" of some of these ideas, and perhaps the more general beauty of the ideas themselves... (Ed)
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Good question
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yanqui
climber
Balcarce, Argentina
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Nov 25, 2014 - 05:52am PT
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I suppose "practical" mainly refers to the way Grothendieck's work has turned out to be useful to the work that other mathematicians are doing. This may sound like too much of a closed system, but anyone who has ever tried to do serious research in mathematics knows how difficult it is to develop new mathematics that, in turn, has a wide range of applications to what other mathematicians are working on, especially if we're talking about a lot of mathematicians in diverse areas (some relation to what other mathematicians are working on is an essential ingredient of publishable research).
I have actually used some of Grothendeick's work on topological vector spaces in published research (although I learned this stuff in a book by Schaefer) and have studied another interesting application of these constructions due to another mathematician. These applications have a lot to do with the fact that the space of functions in question is the space of analytic functions and the natural topology here is not defined by a norm (at least this should mean something to Dr. Gill and Dr. Gold). Algebraic geometry also has powerful applications to my field of choice, but the legacy comes more from Serre (by way of Beilinson and Bernstein) than from Grothendieck. However, Grothendieck's philosophy to build the big machine and let the problems fall out simply, has, to some extent, worked in my field.
I suppose with time and patience I could explain my field, in a way that makes sense, to someone who has had a strong year-long course in Freshman Calculus and a good linear algebra course. A course in advanced calculus would make it even easier. Then maybe I could expalin how Grothendieck's constructions really do apply in a beautiful way to solve some problems in my field. Who knows, maybe I'll try this later.
Rather than do that, I would like to say something about mathematical research, from my point of view, that might make sense to anyone. The area of mathematics that most interests me is known as "Representation Theory". To some extent, I like this area of mathematics because I like it, the same way a climber may like bouldering, or long trad routes, or difficult sport routes. I can give "reasons" I like: it uses different methods: algebra, analysis and geometry are all important tools to understand representation theory; it has a long history (more than 100 years) and a rich development in the mainstream of mathematics; plus I've developed some skills working in this field and it's natural for people to pursue activities they have some capacity for. The field has had some interesting applications to physics and even more practical applications, like imaging technologies, but the problems I think about seem "natural" to me from a mathematical standpoint (and hopefully to other mathematicians as well) and I don't worry about whether or not the solutions of these problems has any applications to outside fields (although it would be great if they do). Representation theory is like a vast landscape, out there somewhere, that you must focus on intensely in order to try and see. The landscape is so vast, that it seems impossible to take in the complete view, so we often focus on little pieces of the land, to see these parts more clearly. From these little pieces we try to build a map, but the map is always incomplete. There are many maps left over by other explorers, and it's important to use those, to try to understand what these maps show. However, if you ever hope to learn anything about the lay of the land, you must set out on your own, sometimes bumping around in the dark, constructing your own version of the map, as you go.
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Crump
Social climber
Lakewood, CO
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Nov 25, 2014 - 09:27am PT
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In memory of my Mentor, Dr Frederic Albert Matsen, and sung to Ringo's snappy tune...
We all live in a finite Hilbert Space
A finite Hilbert Space
A finite Hilbert Space
We all live in a finite Hilbert Space
...
Cheers to the adventures of the mind.
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