The Research Program for the Fall 2009

This fall is me finishing up the icing on the cake which is my project analyzing the complexity / tractability of  solving  various classes of models regarding social collaboration, as well as hopefully some results which identify which bits form the essential core of their respective tractability/hardness characters. This should be publicly available come early December in the form of a Technical Report/ my Senior Thesis!

At the same time, I’m also really excited because I’m beginning a project related in spirit (ie global properties arising from local interactions) over the fall over a class of problems which have a wee bit more mathematical structure, and I do hope I’ll have partial results by late fall too.

Hopefully with those powers  of self directed research combined, I’ll be able to do something fun next year!

My Research Present and Future, Summer and Fall 09

So I am literally within a week of completing a draft of the past year’s research that is suitable for external consumption! Before discussing that/anything, those of you who only care about math skip past the bump.

Ok, now that there aren’t any misanthropes reading, I’d like to make a few observations of varying utility and/or obviousness.

1. Avoidable or unjustifiable stress is bad, don’t let it crop up
2. Stress is bad, don’t have it
3. Sleep, at regular intervals after no more than 18 hours of being awake, is always good
4. Exercise is pleasant, if you’re not stressed.
5. For the dopamine addicts and their diametric opposites the novelty avoiding curmudgeons, if when you exercise regularly (which you should), it should be enough that you feel slightly sore, but assuredly less than that which causes any pain. Pain is bad, don’t do it, injuries tend to happen otherwise.
6. Eat healthy, or failing that exercise more. At least for me, I’ve found that exercising regularly and with reasonable intensity sensitives me to food tastes in a way that seemingly correlates with health. Ie more exercise means I enjoy (and am thence incentivized to follow through on) the consumption of healthy food
7. Did I mention that sleep is good? If you’re not well rested, the only things you can do are things whose thoughts have already been thought in your mind. Every single element of content in this article was not thought of as I was writing it, but was thunk [sic] a while ago, and it is only now amidst a bout of insomnia that I find myself with the time and lack of things to occupy my self that are safe to do when tired that I’m actually writing this blog entry. I was also going to include pictures of my cute aged 2 niece in this entry along with some mini cute thing about how family can be a fun break from whatever, but look, I got distracted!
8. Insomnia sucks, avoid having stress that may trigger it from all the worry.
9. having fun things to look forward to at regular intervals is great for snaity
10. Only if people really want shall I make cute niece/fam pictures available

And at their heart lay equation solving…

Wewf, taking an intensive language class while also doing mathy research is tiring.

I’ve starting to read up on how to apply linear programming style techniques to combinatorial optimization (this book) by a fellow named schrijver does a lovely job of talking about when they can be used to exactly solve problems, and I’m also perusing this book by vijay vazirani about what are similar techniques but applied to approximating intractable problems.

It’s already given me some nifty ideas, but first FINISHING my other work!

Cool stuff forthcoming

Do you know or care about any of the following?

• graphical nash games
• combinatorial auctions
• price of blah (where blah = anarchy, stability, etc)
• robust computational modelling of multiagent economic or sociological phenomena
• equation solving with a nonlinear twist and combinatorial pinch
• interesting variations on maximum weight matching

then you’ll want to read the draft of my research that I’ll have available in a few weeks. If you’re more into abstract algebra or pure math, you’ll have to wait another month or so for more suitable fair.

-Carter

Almost there…

I’ve been spending the past half month finishing up the last of my coursework, by hell or high water, I’ll be done by the 19th, and thence to resume my research and fun!
-carter

Combinatorics is at the root of everything

So earlier this semester I was sitting in my painfully computational galois theory class that was poorly exposited in lecture (and in the text ) and I asked my instructor whether for every lattice there’s a group who’s subgroup structure is isomorphic to that lattice.

Her response wasn’t terribly helpful, but I was recently able to find a paper that talks about that and related matters. this paper by Michael Aschbacher (of classification of finite groups fame) talks about that and related questions, and provides some references that seem to indicate that this is a pretty important open problem! I think its pretty cool, so once i’ve finished up the semester and have appropriately rested and written up my social optimization work, I think i’ll play with this family of problem!

mwhahahaha

Collaboration is hard, even if you care

Last spring, Tardos and Kleinberg published a really cute paper on how to compute equilibria for a collaboration game where each party simply wants to maximize the amount of money it receives, so that for vertex i (which is paired with some j), when given money , its utility is . I’ve been working since late august on the general case when is a function of and to answer some questions about the computational complexity of this problem.

I’ve proven some interesting stuff about the complexity of computing both any equilibrium and the socially optimal one, for a pretty general class of . A mini walk through on the results (or at least the bird’s eye view) is forth coming!

Why I'm glad I live in the modern era

I had the unfortunate foolishness to sit through half of a philosophy seminar several weeks past on a Tuesday afternoon. This seminar was on mathematics as it is connected to ancient greece and the writings of the philosophers of that time. Being the silly optimist I am, my working assumption entering the class was that the professor would be mathematically literate by modern standards and hence the discussion could focus on the conceptual elements of the topic.

I was wrong, painfully so. Is it too much to ask that when we’re looking at a exposition of a relatively elementary topic thats been thought about for another 2000+ years, that we can recast it in a more modern exposition that makes it possible direct the conversation to focus on the conceptually substantive points, instead of getting bogged down in boring details of an archaic proof notation.

Oh well, at least I live in a time when there are people who actually substantively understand these topics

First Post

Hello, this is my new blog. Hopefully I’ll get around to posting short little musings and nifty expositional bits here.

-Carter

some will be of general interest and others will be of the sort.