One thing that I could have used when I started graduate school was a good list of survey articles and introductions to different modeling paradigms and mathematical ideas used in communications and signal processing. It would have done wonders to help me get up to speed on these widely-used ideas. As a grad student, you can’t take classes in everything, and a lot of these ideas are important in research but haven’t really made it into course curricula either. Hopefully people reading this will comment and suggest more titles — I’ll expand the list as more material is suggested.
Topics that it would nice to have (preferably at a level for early graduate school) : convex analysis, Fourier analysis, percolation (there’s a book but it’s a bit advanced for many I think), generating functions… anything else, really.
High dimensional convex geometry
- Keith M. Ball, An Elementary Introduction to Convex Geometry, Mathematical Sciences Research Institute Publications Vol. 31 : Flavors of Geometry, 1998.
- This is a survey article that requires a bit of mathematical maturity, but covers a lot interesting material on high dimensional balls, convex polytopes, volume ratios, and ends up in Dvoretsky’s Theorem. This is less of a “techniques” paper but is good for getting some intuition and facts straight about high dimensional convex things.
- A. Dembo, T. Cover, and J.A. Thomas, Information theoretic inequalities, IEEE Transactions on Information Theory, 37(6), 1991.
- This is the information-theoretic take on some isoperimetric inequalities.
Markov Chains and Mixing
- V. Guruswami. Rapidly Mixing Markov Chains: A Comparison of Techniques, May 2000.
- A nice readable survey that gets the main results across.
- Robert Gibbons, Game Theory for Applied Economists, Princeton University Press, 1992.
- This is a quick read that will get the basic terminology and ideas of game theory. It’s not as useful for learning the deeper stuff, but it’s definitely accessible and well-written.
Auctions and auction theory — used for network congestion and resource allocation
- Paul Klemperer, A Survey of Auction Theory, in Auctions: Theory and Practice, Princeton University Press, 2004.
- A non-technical introduction to the study of auctions and how they are modeled. Good for getting an idea of what all the fuss is about.
- Vijay Krishna, Auction Theory, Academic Press, 2002.
- This book covers the basics and reading the first few chapters should let you get a handle on the terminology so that you can read some of the network congestion and pricing papers.