# 2008 Kemeny Lecture Series

## 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.