The computational complexity of polynomial time problems

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Complexity theory aims at rigorously classifying problems solvable by computers on the basis of the amounts of resources needed to solve them. Computing time is considered as a resource, as well as memory, processors, random bits, communication bits, etc. The methodology of complexity theory rests on the definition of abstract computer models, on the development of algorithms solving a problem on such models, and (ideally) on the mathematical proof that the algorithms found are the best possible. In some cases, such a proof would have great practical value; for example, the security of cryptographic protocols in actual use today rests on the perceived --but as yet unproven-- difficulty of computational problems such as factoring a large number into two smaller numbers that multiply out to this number. The long-term goal of my research is to make progress towards elucidating the structure of the complexity class P of problems solvable in polynomial time. In particular, I'm interested in the 40 year-old open question of whether every problem in P can be solved using only a logarithmic amount of memory. That question and others like it hint at the main shortcoming of complexity theory: despite intensive efforts by a generation of researchers, very few satisfactory lower bounds on the complexity of specific problems within the complexity class NP (a class presumably larger than P) are known. My short-term goal is to determine the complexity of specific problems by (A) comparing this complexity with that of other well-studied problems, (B) proving lower bounds on restricted models of computation and (C) further investigating the connections between algebra, logic and complexity. The hope is that understanding the power of restricted models of computation and characterizing the latter in different ways will help developing proof technology that can eventually tackle problems of immediate practical concern (such as factoring).