Has it really been over 3 MONTHS since my last post here?!? Jesus f***ing Christ!!! So much for my commitment to the blog. I'm hoping to be more committed to it from now on. Not that it matters-I seem to be the only one reading it. Maybe I shouldn't have posted that pic of my adorable self so fast............?
My third and hopefully final semester of Master's degree work began at the CUNY Graduate Center and Queens College nearly a month ago.It was off to a horrific start due to the usual shenanagians with the Registrar. I'll talk about that in a post later this week.Frankly, I feel like I had no summer. I spent half of it at home watching what's left of my family wither as my brothers "sort of" live there and my mother cries remembering my late father. The other half I spent trying to learn differential manifold theory from Loring W.Tu's wonderful Introduction to Manifolds. Yeah, THAT Loring Tu, the student of Raoul Bott's who coauthored his classic Differential Forms In Algebraic Topology. Requiring only a semester of real analysis and abstract algebra and rolling in at a total of 350 pages, the book is everything you want in a differential topology text and more. I needed a couple of good books for self study on the subject to teach myself differential geometry to complete the problem sets for Josef Dodziuk's course taken-and never finished-in the spring. A lot of my friends prefer John M.Lee's Introduction To Smooth Manifolds . Lee's wonderfully written, but its' HUGE and I think works better as a reference. And of course, the old timers point you towards Spivak's magum opus on differential geometry. Spivak's wonderful if you have a year and half to work through it at your leisure-which I don't. I looked at a couple of other books, but I am so glad I took a chance and plunked down 30 bucks for this softcover. Everything is clearly presented with lots of examples and pictures and with a deep mastery of the subject by the author-and good exercises,too. The thing I probably came away with more then anything besides a deep understanding of manifolds and differential forms is that the fact that manifolds are locally homeomorphic to Euclidean space is just the beginning of the story of modern differential geometry.A differentiable manifold needs to be able to support calculus on it and so much of the structure we take for granted in Euclidean space that makes that possible-the vector space structure over R or C, the norms and thier resulting metrics, the linear maps that result from the topological vector space structure-simply doesn't exist on an arbitrary manifold.Differential manifolds and thier respective structures are unique in that regard-and an enormous amount of that structure is not topological/analytic, but ALGEBRAIC-specifically all kinds of R-modules and thier related maps. Tangent spaces, derivations,smooth algebras of functions and Lie groups and algebras-without them,manifolds cannot support calculus and modern differential geometry vanishes in a flash of ectoplasm. The more modern mathematics I learn,the more I realize modern algebra is the glue that holds the whole mammoth structure together and it is taken for granted by most physicists and nonalgebraicists.................
It was eerie in the common room at the Graduate Center mathematics department for most of the summer. The room was for the most part dead with the occasional faculty or graduate student dropping in. Then a week before the semester started-boom,all the familiar faces began trickling in. Lou Thrall and Satyanand Singh (Sam to his friends), arguing low dimensional topology and sheaf theory from Sullivan's notes as usual. Pretty Jeanne Funk, splitting all her time between getting ready for her PHD thesis on algebraic geometry and working as a teacher in the shithole we sadly call the New York City school system for what little pay she gets. Gangly Schlomo Ben-Har, whom I've known for the better part of a decade on and off at Queens College and now in his second year of PHD work at the Graduate Center-what little I see of him in either school.
Faculty we don't see too much of outside of classes and the seminars here-at least,not in the early going. Dennis Sullivan-the invisible god here at the CUNY Grad Center-makes his presence felt by his impact on the serious topology students like Lou and Sam who've associated with him closely enough to freely call him "Dennis" in open conversation-and speak of him in hushed tones. John Terilla-one of my mentors among the younger staff at Queens and a student of James Stasheff-speaks of him in the same hushed tones. I'll be sitting in on Sullivan's class this semester-hopefully becoming a regular. I dunno if I'll be attending the string theory seminar Dr.Sullivan runs this semester. Depends on if a certain well-known geometer is attending.The person in question currently teaches at Columbia and that's as specific as it gets here. I had one run-in with her at the string theory seminar last year. We broke for lunch. I introduced myself to her,she smiled-we had what I thought was a nice conversation going for a few minutes,telling her about myself and what she thought of the seminar,etc. Smiling,she suddenly tells me,"Excuse me,you're just a very annoying person and you're boring me-could you please stop talking to me?" All with a smile. I smiled back,politely excused myself and went to mingle with others. Honestly, I was too stunned to be offended.I don't know if I caught her on a bad day or what-the whole episode still mystifies me. Anyway,after that I put her on my "avoid if possible" list.
I just wished at that moment my father had been alive and bore witness to it. At that moment, she crystalized the very significant distinction between being bluntly honest and being wantonly cruel-blunt honesty is done for a higher purpose and never without consideration for its effects. What she did is done randomly,capriciously and strictly for the affector's own pleasure. That distinction was one I could never make my father understand while he was alive.....................
I'm taking the deformation theory seminar at the Graduate Center this semester headed by John Terilla and Thomas Tradler. It's so exciting to be learning living mathematics-not something that was done and buried by World War II. I really need to brush up on my tensor algebra, though. I'll also be reading a ton of papers on the subject-particularly Gerstenhaber's original papers,which the lectures will be leaning on heavily. Taking graph theory with Christopher Hanusa and probability theory with Stefan Ralescu, both at Queens this semester-and it looks like a full slate.
I hope to finally begin research this semester on a topic I made a couple of false starts on-the p-adic topologies, a family of topologies named on the integers by Kevin Broughan.The first such instance of a p-adic topology was used by Harry Furstenberg over 40 years ago to give a topological proof of the infinity of primes. I was talking with Lou the other day and he told me to stop talking about it,just pick up a paper and do it. Research is the currency of mathematical careers-anyone serious has to begin it as soon as possible. I doubt anything earth shaking will come of it-but if I could just produce one or 2 publishable papers-it will go a long way towards cementing my place in the order of things.And maybe I don't have to settle for the CUNY graduate center for a PHD.
My bed is calling me.I have 72 hours beginning tomorrow to learn 4 weeks of deformation theory. I hope the Gods of Logic And Inspiration are with me.
Until next time.........................
