Timeline for Riemann Surfaces of Infinite Genus and Transcendental Curves
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Mar 28, 2019 at 22:45 | comment | added | MCS | created by the zeroes of $T_{N}$ in that region will be included in the structure of $T_{M}$ for all $M\geq N$. That's what I was hoping for, at any rate. Part of the reason why I'm asking this question online is because my advisor doesn't know enough about these particular questions to be able to answer them. | |
| Mar 28, 2019 at 22:43 | comment | added | MCS | One can fudge a bit with the choice of they approximating polynomials (they don't need to be taylor polynomials, that was just an example), but I'd like to think that Hurwitz's Theorem (complex analysis) has some bearing here. For simplicity, assume that all the zeroes of $T$ are simple. Then, for any $r>0$, there will be an $N_{r}>0$ such that the number of zeroes of $T_{N}$ in the open disk of radius $r$ centered at $0$ will remain constant for all $N>N_{r}$. That is to say, for any bounded region in the plane, the topological structure | |
| Mar 28, 2019 at 22:32 | history | edited | MCS | CC BY-SA 4.0 |
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| Mar 28, 2019 at 22:07 | comment | added | user35370 | I also don't see how $C_{N}$ sits in $C_{N+1}$ (a torus does not sit inside a genus 2 surface) I don't think there is a natural way to go from conformal structure of two things to a conformal structure on the connect sum, which seems important if you are trying to do some riemann surface theory. Also note that not all infinite genus surfaces are homeomorphic, Loch Ness monster and the Cantor tree for example, so you have to be careful. | |
| Mar 28, 2019 at 22:05 | comment | added | user35370 | I asked because whether or not something is a good basis for a dissertation is more of a question for an advisor. I don't really do stuff in complex geometry/Riemann surfaces but thinking about infinite type surfaces as exhaustion/ limits of finite types is not exactly new(recently a lot of stuff on inffinite type surfaces has been done I think even some stuff around flat surfaces and ergodic theory). | |
| Mar 28, 2019 at 21:04 | comment | added | MCS | The one I was brave enough to contact in person flat-out said "no". The others... just silence. I'd be lying if I said I was't depressed, or that I haven't considered dropping out altogether (I'm in year 4 of my 5-year program. I've passed all my exams (written, oral).) But I feel like I know nothing, and every time I've thought I've spotted a light at the end of the tunnel, I've been dashed to pieces by false hopes and/or the terrifying depths of just how much I don't know. Ugh. | |
| Mar 28, 2019 at 20:59 | comment | added | MCS | And it looks like it, too, is already known. This marks the fourth problem in a row that I either rediscovered something already known, or where I thought I had a solution, but it turned out to be wrong (Collatz Conjecture and the p-Adic Lindemann-Weierstrass Conjecture). All the defeats and false hope have really worn me down. I've sent multiple letters of inquiry to professors at nearby universities, hoping to see if any of them might be willing to take me on, but none of them replied (even with my advisors sending out messages alongside mine). | |
| Mar 28, 2019 at 20:52 | comment | added | MCS | Yep. S. Kamienny (USC). He's very nice and very encouraging. My interests run more toward analysis and analytic number theory, but there's no one at USC who does analysis that doesn't deal with PDEs (Deterministic & Stochastic) or Ergodic Theory, so he's the best they could give me. He's got a fallback problem for me about torsion subgroups of elliptic curves, but we both agree it would be best if I found something of my own to do. First idea I tried, ended up already being proved (and I rediscovered the Tate Module in the process); my second idea was this. | |
| Mar 27, 2019 at 21:19 | comment | added | user35370 | Do you have an advisor? | |
| Mar 27, 2019 at 21:14 | history | edited | Gerry Myerson | CC BY-SA 4.0 |
typo in title
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| Mar 27, 2019 at 20:58 | history | asked | MCS | CC BY-SA 4.0 |