How would one use Lucene.NET to help implement search on a site like Stack Overflow?

The answers you are looking for really can not be found using lucene alone. You need ranking and grouping algorithms to filter and understand the data and how it relates. Lucene can help you get normalized data, but you need the right algorithm after that.

I would recommend you check out one or all of the following books, they will help you with the math and get you pointed in the right direction:

Algorithms of the Intelligent Web

Collective Intelligence in Action

Programming Collective Intelligence

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