Finity is a science fiction novel set in an alternate universe, although by the end of the book it is actually set in quite a lot of alternate universes. The one we start in, however, is one where America lost World War II, and many Americans are living in expat communities around the world. The war was a long time ago, however, and most Americans, while they still identify with their traditional culture, are living quiet and uneventful lives in the year 2063.
One of these Americans, Lyle Peripart, is hired unexpectedly by an eccentric billionaire, Geoffrey Iphwin, who is interested in his rather obscure branch of probability. Lyle is a largely ignored academic who has made a hobby of a neglected branch of statistics, the study of what sampling frames are logically possible.
It emerges that something odd is happening across the globe -- people and things are quietly appearing, disappearing, and subtly changing, and there have been numerous cases where cause and effect appear to be reversed. One possibility is that there may be alternate universes interfering with theirs... which could be a very profitable line of investigation. On the other hand, allowing these events to continue to occur randomly would be very bad for business.
And so Geoffrey is carefully piecing together a team of scientists, mercenaries, and apparently random people to look for the source of the problem... which appears to be the former United States of America, a region that appears to have somehow dropped off the face of the Earth without anyone noticing.
This really isn't that great a book. It has some good aspects, including some clever ideas and a Heinlein-like feel, with earnest and sometimes over-involved exploration of the plot, hard-headed people with strong ideologies, and people fighting long odds in unpredictable and sometimes incomprehensible situations. On the other hand, the explanation of the science behind the story is poorly done, and sometimes quite confusing, even in cases where it's just a rehash of physics that we have all heard explained a dozen times before. The pacing of the plot is also poor -- the early chapters drag, and the ending moves too fast. There is an appearance of the constant plague of modern science fiction, the gratuitous rape scene. And finally, the ending is fairly unsatisfying.
As you might have gathered, I wouldn't particularly recommend this book. There are a lot better books out there, including some by John Barnes himself. A quick look at other reviews online finds that most people agree with me, although there are some reviewers that loved it, generally praising the setting and SF aspects despite a slow start and some difficulty figuring out what exactly is going on at times.