
Academic Guidance Academic Guidance  Academic guidance for those pursuing a college degree... what college? Grad school? PhD help? 
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April 14th, 2010, 06:53 PM  #1 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Research topics
Additive number theory: What numbers (Sloane's A156695) cannot be expressed as p + 2^a + 2^b for prime p and nonnegative integers a and b? Computability theory: Many results, starting with Richardson 1968, have explored classes of problems where equality is/is not decidable. If you include +, , *, /, pi, log 2, x, sin, exp, and abs, it is undecidable. It seems, oddly enough, that the problem is related to the use of nested sin functions. Explore the boundary between classes; experiment with other oscillatory functions beside sin. Prime number theory: Explore the consequences of GreenTao (see Granville's "Prime number patterns"). Differential algebra/Computer algebra: Implement the Risch algorithm (possibly starting with the partial implementation in Axiom). Computer science: Parallelize the linear algebra step in NFS factorization. Sieve theory: Explore small prime gaps using GoldstonPintzY?ld?r?m, Pintz, FouvryIwaniec, BombieriFriedlanderIwaniec, etc., unconditionally or otherwise. Mathematical economics: Tanaka 2003 proves a version of Arrow's theorem (via Wilson's improvement) that does not include IIA, the main assumption; explore replacing IIA in other versions like Pattanaik & Peleg 1986, Sen 1970, or Gibbard 1969/Campbell 1992. 
May 1st, 2010, 01:57 PM  #2 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Research topics
I don't know why you posted this, but if the reason was to be helpful... SUCCESS! Not that I would know how to research this stuff... (edit) And I see that I'm not the only "Chaz" out there 
May 1st, 2010, 03:01 PM  #3  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Research topics Quote:
These are topics that I have some connection to: ones I've wanted to see solved or ones where I've worked on some (much) simpler version. Quote:
 
May 1st, 2010, 03:10 PM  #4  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Research topics Quote:
 
May 1st, 2010, 03:20 PM  #5 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Research topics By the way: If you're interested in the first problem, I have what is probably the only digital copy of Crocker's thesis. I requested an interlibrary loan and they actually sent me the thesis itself... 

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