Below are the 20 most recent journal entries recorded in Robert's LiveJournal:

[ << Previous 20 ]

Thursday, July 9th, 2009
7:32 pm	Paper: fixed. The asymptotics we get for q^*_d(n) are not as good, but they're still sufficient. And, uh, the proof is ridiculously easy. Time to revise! Update: There. (1 Comment |Comment on this)
Wednesday, July 8th, 2009
8:56 am	Well, fuck. There's a mistake in the paper I submitted yesterday, and it's pretty critical, too. While I frantically work on fixing it, I may as well offer a five dollar reward to the first person who finds it. You see, I want people to read my math, even if it's incorrect :). Money will be paid the next time I see you. (10 Comments |Comment on this)
Tuesday, July 7th, 2009
4:19 pm	Yesterday, I finished another paper (available here) and got football tickets. Life is good. (1 Comment |Comment on this)
Thursday, June 11th, 2009
7:45 pm	Earlier today, we sent off our revised paper to a bunch of big names (most recent version here). It's kind of nerve wracking. We haven't received any useful comments yet, but that's to be expected. If we do receive something useful, it'll prolly be a few days from now. Sometime next week, we're going to submit it, probably to TAMS (Transactions of the American Mathematical Society). Ken thinks we should have a good shot of getting in. Here's hoping. I'm now entering the period between projects, although we still have to prepare our presentation for the Ottawa conference (did I mention I'm going to Ottawa at the end of June?). I'm not sure what I'm going to be doing next. Ken just submitted a paper about the parity of the partition function which establishes a good framework for tackling the problem. I may do something related to that, or I may do something else entirely. He's asked me to read Iwaniec and Kowalski during the next two weeks, which is a daunting task. It's a 600 page book and is, I believe, the most comprehensive book on modern analytic number theory. Despite being as long as it is, it's still pretty ideas dense. M'oh, well. I've been planning on reading it for a while, so I guess this is a good push. Gonna go nap for a bit, as my sleep schedule is absolutely shot right now. (6 Comments |Comment on this)
Sunday, June 7th, 2009
9:02 pm	Alder's Conjecture is proved! (Modulo computations which'll take about a month) We've finished the first draft of our paper. It's pretty dense - there a lot of tersely explained analytic estimates. That said, it resolves a conjecture that's been open for fifty years. The introduction and statement of results section should be accessible to people who're interested in math. I've put it up on my website here. Comments would be greatly appreciated. ***EDIT*** Err.. need to add an acknowledgements section and fix that damn table. That'll be done... later. (2 Comments |Comment on this)
Thursday, May 14th, 2009
1:01 pm	Long, long long, longest? Does anyone have a recommendation for a high, possibly arbitrary, precision integer data structure in C++? I could put together my own pretty easily, but I would bet that there's something faster out there. I need to be able to go up to values of about 10^1500, maybe 10^2000. I don't need to do anything fancy with them, just addition. It may be easier to code something myself, but I doubt it'd be the best option. (2 Comments |Comment on this)
Tuesday, April 14th, 2009
11:53 am	I'm confused by why I'm better, in some ways, at math when I'm falling asleep. Some people have dreams that reveal solutions to problems or new approaches that may be fruitful. I never do. Instead, I get a lot of ideas while I'm unable to sleep and have nothing better to do than think about math. Most of them are crap, as my ability to effectively execute ideas goes way down when I'm tired, but some of them are genuinely good. Of course, this may just be that I'm more relaxed while I'm trying to sleep, but it happens often enough and in such a way that I suspect there's something more. I'm not sure, but I'd suspect that my thought processes work slightly differently while I'm trying to sleep. Does anyone else experience this? Does anyone do the dreaming thing? ..and yes, this did come up last night with my topology homework. Twice. (4 Comments |Comment on this)
Thursday, March 26th, 2009
5:39 pm	I got a shiny new keyboard to replace the one I broke last night. It's got a slightly different click to it, so it'll take some time to adjust, but it should be OK. There's a conference in Illinois this weekend that I'll be going to. I found out about it this morning, so it's sort of a last minute deal, but it should still be fun. I'm going with Marie, one of the other people working on Alder's conjecture. There'll be a lot of number theory there, but not a lot that'll be similar to what's done at Wisconsin. We had our prospective student day on Tuesday, which was fun. It'll be interesting to see who chooses to come to Wisconsin. There were definitely some cool people that I hope come, but we'll have to see. Time to make some dinner. (Comment on this)
Wednesday, March 25th, 2009
10:59 pm	VBroke /m/n/y/ /keyvboard. (2 Comments |Comment on this)
Friday, March 13th, 2009
7:30 pm	Left anonymous so as to preserve the dignity of the person who said it, "When you do an image search for tiger scrotum you get some really strange shit." (Comment on this)
Thursday, March 12th, 2009
2:38 pm	Mmmm... spring break! I'm feeling really weird right now. I was very tired this morning, so in order to not fall asleep in class field theory (which is the class I enjoy the most), I had some coffee. Whatever, normally this would be fine. Today, though, it's made me really jittery. Sort of. Jittery with my eyes, where turning my head feels weird. Oh, well. My plans for spring break: Video game movie marathon: Steffen and I want to have a movie-based-on-video-games marathon, with a focus on bad movies. So basically, things like Super Mario Bros, Mortal Kombat, Street Fighter, whatever else. It should be a lot of fun. Others are welcome, although I'm not sure exactly when it's happening. Work on Alder's Conjecture: I really want to work on the combinatorial argument for this. I'm not sure I'm going to be able to come up with something, but at the very least I want to better understand the obstructions to doing this. At the very least, it should be possible to show it for sufficiently large n, where easy argument prove it for small n. I have some ideas which are promising but have some serious problems to be worked out. Study class field theory: There's no work for this class, and since I want to make sure I actually know the material well, it's important to study it. We're using Milne's notes, which are pretty good. The stuff we're doing (mostly looking at the local case so far, with a large emphasis on homological algebra) is pretty cool so I want to make sure I understand it. Relax, sleep, have fun: While I do want to do some math each day, the main thing I want to accomplish over break is, well, to get a break. Steffen'll be in town, which is great, and just yeah. I have not even come close to sleeping enough recently, so getting sleep is definitely a priority. So yeah, I think break'll be pretty awesome. I really want to put an emphasis on math, if you couldn't tell. I'm getting better about being responsible about doing work, but I really want to try to push myself. I still have no idea of how good I am at research level math, and, since I want to be a research mathematician, I want to push myself and find out. Also, I'm getting a lot of satisfaction out of studying this stuff, and I'm no where close to breaking. *Shrug* Off to catch a bus. Later, all. (4 Comments |Comment on this)
Saturday, March 7th, 2009
4:02 pm	Polynomials - Partial Solutions This is behind a cut so that you don't have to see it if you want to think about the puzzle aspect. There's discussion of the generalizations here, so if you're interested in the math, I'd recommend it. ( Spoiler Alert ) (Comment on this)
1:58 am	Polynomials This will be a very mathy post, but I think it should be pretty understandable for the most part. The first part is posed as a puzzle, the second part is me talking a bit about generalizations. Suppose I have some polynomial and you want to figure out what it is. What you know is that the coefficients are non-negative integers, and that's it. You don't know the degree of the polynomial. What you can do is give me some number, and I'll tell you what you get when you plug that number into the polynomial. So, if my polynomial was x^2 + 2 and you gave me 3, I'd tell you 11. What you want to do is figure out my polynomial in the minimum number of evaluations. What's the best you can do? You can base your later guesses on your earlier ones. There are two specific cases here, depending on what numbers you're allowed to plug in. If we allow all real numbers, or really any transcendental numbers, the answer is 1. If we restrict to algebraic numbers, or even down to integers, the answer is 2. (The fun, of course, is in figuring out how to do this) Solutions are definitely encouraged! So now, what happens if we change the conditions on the coefficients? I'm pretty sure that allowing the coefficients to be negative is bad (meaning an infinite number of evaluations would be necessary), although I'm not certain. If we allow the coefficients to be non-negative rational or real coefficients, there is a finite solution. What I'm currently unsure of is whether there is one independent of the degree of the polynomial. I have a way to do it in n+1 steps (remembering that we don't know n, this is not entirely trivial). My instinct would be that this is optimal, but I'm not sure my instinct can be trusted on this problem. (1 Comment |Comment on this)
Monday, March 2nd, 2009
4:56 pm	I've been feeling really disoriented the past week or so. It started last Wednesday when I woke up, turned off my alarm, went back to bed, and felt stupid for setting an alarm on a Saturday. This weekend in particular was really bad. I normally have a pretty good sense of what day it is and what I'm doing even right after I wake up. On the actual Saturday, I had to spend a couple minutes figuring out exactly what was going on. On the other hand, math is going pretty well. We're getting very close to the analytic and unsatisfying proof of Alder's conjecture, which is pretty cool. We hope to finish it up in a week. There was one messy part that took us a coupla weeks to figure out, but we eventually got it. We needed to bound a function away from 0, and it was very hard to get something non-zero. The bound we ended up getting is pretty small (approx. 10^(-4*(d+3)), where d is a positive integer). We haven't been spending much time looking for a more elegant (combinatorial?) solution, but I'm convinced something is out there. I'm much less convinced that it's easy or that we'll be able to find it, but I am hopeful. I'm certainly going to give it a pretty serious shot. In other news, I'm starting to feel better health-wise, which is definitely good. I'm not sure if being vegetarian is helping, but I'm sure it doesn't hurt. (Comment on this)
Tuesday, February 24th, 2009
10:38 am	Fucking Maple Note to self: If I'm ever involved in the development of a computer algebra system (Maple, Mathematica, etc), make sure it's consistent. God I hate Maple! If I want to evaluate a function, say at n=10, I shouldn't need to evaluate it at 11 to get a non-zero answer. Essentially, it's doing something like this: h(10); 0 h(11); 0 h(10); *the right answer* Seriously, what the fuck? Since I know about this right now, it's not so bad. But it's done this for things where I don't know what the right answer should look like! In other words, Maple is not remotely trustworthy, and yet it's useful to look at what I should be getting. Looks like I need to learn Magma, or Sage, as the math department supports those and Maple. (Comment on this)
Sunday, February 15th, 2009
8:11 pm	As I posted on facebook last night, I've more or less made up my mind to become a vegetarian. At least, I'm going to try it for a few weeks to make sure I'm not going to kill myself. If all goes well, it'll become permanent. There are two main reasons, both of which are pretty easily guessable. First, I've wanted to do it for a while. Part of me feels the need to put my money where my mouth is, I guess. Also, with where my Crohn's is at right now, I'm pretty sure meat is a bad idea for a little while. So basically, I think that since I have to be basically vegetarian for the next while anyway... I'd appreciate advice on food and such (Sam and Paresh, I'm thinking of you, although others are obviously encouraged). Most things are fair game, although bell peppers are terrible and mushrooms are only sort of food -- there's a huge psychological block for some reason. I'm probably going to do a lot of eggs and beans for protein. I've heard that oatmeal and cream of wheat are good sources of iron. Are there other things? I'd just as soon use this as a reason to eat more healthily in general, so getting a good mix of things is important. In other news, I still have this icon saved. May as well use it for something... (4 Comments |Comment on this)
Wednesday, February 4th, 2009
1:48 pm	Zee came to eat lunch with me and some of the other math grads today. She gave opinions of people based on how they present themselves and act. I thought it was pretty awesome. It occurs to me that you (Zee) might find it useful to look at facebook pages of people. If you want to do this, lemme know and I can give you my password or something. I continue to like math. I'm on fellowship this semester, so I don't have to teach, but Ken's given me a lot of stuff to work on so I still don't have very much free time. I've got three different problems to look at. The first is Alder's conjecture, which I'm working on with two other people. A classic result on partitions says that a number (positive integer) can be written as a sum of odd positive integers is the same as the number of ways that it can be written as a sum of distinct positive integers. So, for example, we can write 6=5+1, 3+3, 3+1+1+1, or 1+1+1+1+1+1, so there are four ways to write 6 as a sum of odd numbers. We can also find that 6=6, 5+1, 4+2, or 3+2+1. Thus, there are four ways to write 6 as a sum of distinct numbers. OK, sweet. The Rogers-Ramanujan identity gives another result related to this. If we look at partitions (writing a number as the sum of other numbers), the number of ways using numbers that are only +-1 (mod 5) is the same as the number of ways where the numbers must be at least 2 apart. Alder's conjecture generalizes this, and says that the number of partitions using parts that are +-1 (mod d+3) is always less than the number of partitions whose parts are at least d apart (d=1 and 2 are the two cases above). It's known that this is true eventually for a fixed d. We're taking the asymptotic estimates used and finding explicit error terms so that we can end up with a finite number of cases to check. Looking at the difference, however, there are some clear patterns, even though it's not monotonic. I haven't messed around with them yet, but it seems that standard combinatorial arguments don't work in the expected way. *shrug* I think the moral of the story here is that it's a nice problem with some nice patterns but will probably be solved using ugly methods. That's kind of unsatisfactory, but I'll live. Progress is fun, even if the methods are uninteresting. The other two problems are harder to explain, but I'll give it a shot. One I'm working on by myself. It's a conjecture concerning the zeros of a family of modular forms associated (somehow) to Virasoro algebras. The conjecture is that the zeros on the fundamental domain are all on the bottom arc, e^(i*theta) for pi/2 < theta < 2*pi/3. They arise as the quotients of Wronskians. The other conjecture concerns functions related to those used in the proof that zeta(3) is irrational. In particular, we're supposed to show that some particular sums reduce nicely modulo a power of a prime p. Essentially, it's some elementary sum that looks like it may not be approachable via elementary techniques. Ken has a paper that he thinks should helps us get these sums in a non-elementary manner, but I haven't put a lot of time into this problem yet. How many of you actually read these things? If you're confused by things, I'd love to explain them as best I can! (9 Comments |Comment on this)
Saturday, January 17th, 2009
3:15 am	Finally, not a math icon! A coupla months ago, Zee posted this thing on her journal and I've been thinking about it a lot. Read the following paragraph, and, if you can TRUTHFULLY do so, repost it on your journal with these instructions. (Actually, putting a similar notice on FB, especially if you're single or have an appropriate 'looking for' would be nice, too.) "If you and I are becoming involved (romantically or sexually or whatever) and you are transgendered, YOU DO NOT HAVE TO TELL ME. If you're trans, you've got plenty to worry about without trying to figure out when to tell a potential partner. If it's me, don't worry. You can be yourself with me. I won't think it's a betrayal of trust." It wasn't something I could immediately post because I hadn't seriously thought about it before. I didn't exactly ignored it, but I wasn't totally comfortable posting it. I've been thinking a lot about relationships recently, and it occurred to me that it just. doesn't. matter. *shrug* Sigh. I think gender and sexual orientations are weird. I understand from a basic biological standpoint why they exist, but it goes against my intuition, which says that most people should be essentially agendered and pansexual. Seriously. Failing that, I feel like sexual orientation should be normally distributed. In other words, the fact that I'm straight really goes against the grain and seems odd to me, even though I'm very comfortable with myself. Yeah, OK, I don't really have a connection to the other thing I was thinking about posting, so I'm just going to use this half-ass sentence to do it. During breaks, I tend to read a fair amount. That said, I don't read new things terribly often. The books I read as a kid (usually fantasy or sci fi of some sort) hold particular emotional value for me. I usually reread books not to relive the story, but to relive the feelings or, maybe more accurately, moods I've associated with them. Unfortunately, I'm not sure how to express exactly what these moods are, but the books are important to me. I'll tie this into another thought later, but right now, I feel like eating ice cream. (4 Comments |Comment on this)
2:36 am	I don't really have anything to say here, but apparently I'm supposed to post to Livejournal, at least according to lex_of_green. I hope she's satisfied. I've been thinking about teaching math recently. In particular, for certain subjects, I've started to develop particular ideas about the best way to teach them. I think the one I'm most particular about is linear algebra. I strongly believe that it should be the first serious introduction to working with axiomatic systems. In particular, most people are going to have some experience working with vectors and matrices from high school so there should be an intuition to motivate the axioms. I think this motivation is important. Without it, the stuff just doesn't matter. It makes proving things a natural thing to do. Um. I didn't intend to babble so much about this in particular, so I'm not going to say much more on this. The main point is that I'm starting to develop the opinion that there may be a best way to teach something rather than a number of good approaches. I think it's pretty similar to finding the book proof* for something. I'm not going to think too much on whether or not this approach is useful for engineers. I don't think I could be persuaded that this isn't the best way to actually understand the stuff. Anyway, I want to mess around with actually teaching Linear Algebra. It's not because I want to see if I'm right that it's the best way - that'll be evident either way - but because I want friends to understand the joy and the elegance of math. Sort of. It's more that math matters to me and I want people to be able to understand why. Does that make sense? I think I also like being able to bubble about math in a focused way. I mean, all these people have had me bubble at them about random things, but I'd like to have it be organized and purposeful and awesome. Apparently, I only post math things. I'll try to remedy this with a barrage of other posts tonight. I have two other things in mind, but I may try to work them together. *shrug* Monkeys. (2 Comments |Comment on this)
Friday, December 19th, 2008
12:14 pm	Coefficients I agreed to compute the coefficients of some modular forms for Ken. He has infinite series expressions for them and wanted an efficient way to compute them. He'd written everything up in Maple and then sent me an email asking if I knew C and could write something for him. That part wasn't too bad, as the only 'hard' thing was to find some code to approximate a Bessel function. It's only accurate to about fifteen decimal places (which is OK for most of the forms, but not all). I may end up rewriting it with arbitrary precision at some point, but I'd have to get a much better approximation of the Bessel function. I think there are infinite series expansions I should look into? Anyway, it's pretty cool. He's proven that the coefficients of certain modular forms are rational, with a bound on the denominator. Hence, there's some integer such that when you multiply the form by that integer, all the modified coefficients are integers. I'm now looking for that integer. Unfortunately, the code runs in O(n^2) time, and a pretty slow O(n^2) at that. Using 500 terms in the summation takes somewhere in the five to ten minute range. 5000 terms, which is what I care about, takes 20 hours and counting. Whee? I'm not sure I trust a connection with the math department's computers to last long enough for me to guarantee being able to run it elsewhere, so I'll just continue to run them here. I can do three at a time here pretty easily, so I should only need three more runs to get everything handled. In the mean time, I'm going to go hide under the covers some more. (7 Comments |Comment on this)