Timeline for Computing homotopies
Current License: CC BY-SA 2.5
7 events
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| Jan 10, 2011 at 3:17 | comment | added | Bill Thurston | @Sergey: I should have mentioned (as you must know but others may not) that finite topologies are equivalent to posets; geometric realizations can in fact be thought of as countable posets (like the decimal system). The main point about geometry is that humans have powerful special purpose geometric circuitry, but weak combinatorial circuitry, especially with symbolic data. The situation for computers is reversed; even GPUs are good for general purpose arithmetic and combinatorial computation. | |
| Jan 10, 2011 at 3:09 | comment | added | Bill Thurston | @Sergey: I do appreciate simple combinatorial descriptions of homotopies, along with the geometric descriptions. Long ago in college I wrote a senior thesis developing the homotopy theory of finite topologies, and its relationship to cell complexes etc., without being aware of any literature about it. Combinatorial descriptions can be simple and elegant, but they're just different and seem to use different parts of our brain than the geometric descriptions. Translation back and forth is not easy or automatic, although it can be learned. Certainly it's very important for machine computation. | |
| Jan 10, 2011 at 2:06 | comment | added | Sergey Melikhov | In fact, I believe the problem is really in that our basic (algebraic and geometric) topology is still missing true combinatorial foundations. By this I mean dealing with simplicial/cubical complexes, or posets, or simplicial sets in a purely combinatorial way without ever resorting to "obvious" homotopies or homeomorphisms on geometric realizations. At first glance this insane idea must seem an overkill (or even making things still worse), if feasible at all. Anyhow, I happen to be working on this, and not out of pedantism but in pursuit of applications that are out of reach otherwise... | |
| Jan 10, 2011 at 1:35 | comment | added | Sergey Melikhov | I used to think like that - clear and neat geometric constructions sometimes "happen to" have awkward symbolic descriptions. Certainly, whenever I encounter a clumsy symbolic construction in some paper it immediately becomes a challenge for me to recreate it geometrically without the pain of reading the symbols. This said, there's also a deeper look on this issue. If I cannot write up my homotopy in a simple way while geometrically it looks simple, this means I haven't developed an adequate language; and that might prevent me from being able to see how to solve a harder problem. | |
| Jan 6, 2011 at 11:18 | vote | accept | Harry Gindi | ||
| Feb 5, 2011 at 0:05 | |||||
| Jan 4, 2011 at 19:44 | history | edited | Bill Thurston | CC BY-SA 2.5 |
added 101 characters in body
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| Jan 4, 2011 at 16:15 | history | answered | Bill Thurston | CC BY-SA 2.5 |