First Story. "Luminous."
I found this story to be OK, but not great. The idea involves the basics of mathematics being invalidated due to certain remote regions of formulation having been corrupted in the quantum craziness of the early universe...or something. Mainly the characters sit in front of a computer and talk about this, which is interesting enough, but it wasn't a truly engaging premise for me.
Second, "Riding the Crocodile"
This is an excellent story. Nominated for Hugo award. Did it win? I'm not sure. It probably should have. This story takes place in the terrifyingly distant future (more than a million years from now). Humans live as software and live in an interstellar community called the Amalgam. This coalition of worlds/species inhabits the outer ring of our galaxy, while the mysterious Aloof live in the galactic core. The Aloof are inaccessible by any known means. They turn back all visitors and reject all contact.
This is about a married couple attempting to make contact before they choose to die. The time scales involved are mind boggling, and Egan insists on a relativistic universe where it takes a thousand years to travel between stars. Of course, software sentience doesn't care about time the same way we do. This was an awesome story, and is set in the same world as two of his novels and another story in this collection.
Third, "Dark Integers"
This is the direct sequel to "Luminous." I liked it more. The mathematics (and the threat they represented to our reality) were more clearly expressed in this story. There was actually something at stake, rather than just a vague sense of being intellectually threatened.
Fourth, "Glory"
Set in the same universe as "Riding the Crocodile." This follows Joan, a xenomathematician, on a quest to uncover ancient mathematics by a 3,000,000 year old culture. The problem is that the planet where this culture lived is currently home to a sentient offshoot of their evolutionary tree who are at a somewhat similar level of technological/social development as we are.
The fact that their planet was once home to a 3,000,000 year old culture has numbed them to the idea of extraterrestrial contact, so that isn't an issue in and of itself; the warring nations on the planet, and each nations desire to use their interstellar visitor to gain a terrestrial advantage, however, is.
The opening to "Glory" is an astonishing feat of hard, hard science fiction. I've never read anything quite like it. The first two pages of the story take place on a subatomic scale...it's really something.
Fifth, "Oceanic"
I didn't think I'd like this story, but I was horribly wrong. It's the least sci-fi of the bunch. A civilization distantly descended from ancient colonists from earth. The people here have been genetically modified in various ways by their ancestors who they revere as "angels." A whole religion surrounds the idea that angels live on earth.
The narrator is indoctrinated into a fundamentalist sect known as the "drowners." They weight themselves underwater and proceed to almost drown and discover the divine love of their deity Beatrice. As the narrator comes of age he discovers sex (which is a strange and disturbing hermaphroditic affair that I still don't quite understand), and slowly sheds more and more of his religious convictions until the only thing important to his faith is the love of Beatrice inside him.
This can't last, and in college it is discovered that the religious ecstasy he experienced as a child is the result of a mild hallucinogen secreted by a particular variety of microorganism that is present in all the world's oceans.
Really it's a simple story of growing out of religious faith, but it was well told and the narrator was one of the more well drawn characters that Greg Egan has created so far. I still don't understand the sexual anatomy of these people...really confusing.