James Tauber's Blog 2004/11/29
Film Project Update: Mailing the DVD
With a climax worthy of a film, I got the DVD of Alibi Phone Network sent off to Tom for duplication and festival submission.
I had arranged to visit a friend in the afternoon and my original plan was to spend the morning doing the DVD burning, mail it off and then go visit the friend. However, the burning took longer than I planned and so I decided I'd go to the post office after I'd paid the visit.
Somewhere between when I left to go to the friend's house and when I got back home, I misplaced my wallet so I had no money to pay for the shipping. I rang my mum (who lives ten minutes away) and asked if I could come over and borrow some money. (Oh how many times in my thirty-one years my mum has come to my rescue!)
I got the money, rushed to the post office just before closing time and, as I put the parcel on the counter, the lady said "can I see some ID?". I'd forgotten that international shipping requires ID. And my drivers licence was...you guessed it...in my wallet.
So I raced back home, found my passport, raced back to the post office and got the DVDs sent off within minutes of closing.
by jtauber : Created on Nov. 29, 2004 : Last modified Feb. 8, 2005 : Categories filmmaking alibi_phone_network : (permalink)
Poincare Project: Homeomorphisms
Previously we talked about bijections as a way of pairing up all the elements of two sets. Often this is done to express that one set is equivalent to another.
Once you have structure on the set, it isn't enough to just have a bijection. The elements of the two sets must be paired up in a way that maintains the structure before the two structured sets can be said to be equivalent.
Two topological spaces are equivalent if the bijection maintains the open sets. In other words, if the bijection maps open sets to open sets then our two spaces are topologically equivalent.
Another word for topologically equivalent is homeomorphic (note the 'e') and the topology-preserving mapping is called a homeomorphism.
A topological space is the most general space that has a notion of continuity, so two spaces that differ in terms of other structures (like distance between their points) might still be homeomorphic if continuity is preserved. One way to think about this is moulding a ball of clay...
Imagine taking a ball of clay and squashing it flat. If you think of the clay as a metric space, you've clearly changed the space quite a bit because distances between pairs of points are no longer the same. However, you haven't changed the topology. The open sets are still open sets in the squashed version. Squashing the clay is a homeomorphism. If you'd drawn a continuous line on your ball it would still be continuous after the squashing. Squashing hasn't ripped two points apart from one another.
But, now consider pushing your thumb through the clay to mould it into a doughnut-shape. To make the hole, you had to rip points apart from one another. This has altered the open sets. Two points that might have been very close (and hence in some very small open sets together) might now only share very large open sets in common. Because the topology is not preserved, the mapping from ball to doughnut is not a homeomorphism.
A topologist would say that the ball of clay is not homeomorphic to the doughnut-shaped clay.
We've reached an important milestone because the Poincare Conjecture has to do with whether one particular type of topological space is always homeomorphic to another particular type.
UPDATE: next post
by jtauber : Created on Nov. 29, 2004 : Last modified July 1, 2005 : Categories poincare_project : 0 comments (permalink)