Computability Theory and Its Applications

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). In this project, Antonio Montalban studies problems in both pure and applied Computability Theory. The problems in pure computability theory are part of the ongoing program to understand the structure of the Turing Degrees. As for applied computability theory, Montalban applies methods from computability theory to study the effective content and proof-theoretical strength of various areas of both classical mathematics and foundations of mathematics. He concentrates in problems related to linear orderings, Boolean algebras, and ordinals, but not exclusively. Computability Theory is the area of Logic that studies the notion of algorithm. Its applications are based on the idea that problems that can be solved using algorithms are simpler than the ones that cannot. This is used in various ways to measure the complexity of mathematical objects, constructions and proofs. It is usually the case that this analysis gives a better understanding of the subject under study.