The iPhone client is written in Objective C. The Android client is written in Java. The webserver appears to be written against the JVM, so it could either be

The iPhone client is written in Objective C. The Android client is written in Java. The webserver appears to be written against the JVM, so it could either be plain Java, JRuby, Jython, Scala, or any other language that runs in the JVM. DJ Patil however, was LinkedIn's Chief Scientist, and LinkedIn is a heavy Java/J2EE shop, so my guess is that it's vanilla J2EE.