Dartmouth 2008 Kemeny Lecture Series MATH

 Daniel Goldston

San Jose State University


Primes and Twin Primes


Wednesday, April 23, 2008

7:00 -- 8:00 PM

008 Kemeny Hall


Abstract: I will talk about my recent work with Pintz and Yildirim where we proved that there are infinitely many pairs of primes much closer together than the average spacing between consecutive primes. This is still a long way from proving the ultimate conjecture that there are infinitely many pairs of primes differing by 2 -- The Twin Prime Conjecture. However, conditionally we are able to nearly come to grips with twin primes, and perhaps a proof is no longer out of reach.

 Usually mathematical research is a private activity generating no publicity, but this work was an exception. The initial work of Yildirim and me generated wide attention for a month before a mistake was discovered by Granville and Soundararajan.  Our proof crashed and burned, which did however generate further publicity. A new proof unexpectedly emerged with Pintz's help, and this generated a third and smaller round of publicity.  After the experience with Wiles and Fermat's Last Theorem in the '90s, the press takes this perhaps as the norm in mathematics, but personally I would recommend avoiding this cycle if at all possible.



Note: This talk is for a general audience and will be accessible to undergraduates.

NB: A PDF version of this announcement (suitable for posting) is also available.

Small Gaps Between Primes


Thursday, April 24, 2008

4:00 -- 5:00

007 Kemeny Hall 

(Tea 3:30 PM 300 Kemeny Hall)



Abstract: I will describe the ideas used by Pintz, Yildirim and me to prove the existence of pairs of primes very close together. The main idea is to use an approximation for prime tuples that comes out of sieve theory, and then a simple positivity argument to detect primes.

Note: This talk will be accessible to graduate students.

NB: A PDF version of this announcement (suitable for posting) is also available.


Problems Related to Primes


Thursday, April 24, 2008

2:30 - 3:30 pm

343 Kemeny Hall


Abstract:  I will discuss some questions on primes including large and small gaps, and related questions on almost primes. Recent joint work with Sid Graham, Pintz, and Yildirim provides an explicit solution to some conjectures of Erdos on consecutive values of arithmetic functions.


Note: This talk will be accessible to graduate students.