Proof Theory is the branch of mathematical logic that investigates formal proof systems modelling mathematical reasoning. There are different types of proof systems, each having its own advantages and disadvantages. A certain class of proof systems (Gentzen-type systems) has the interesting property of exhibiting a geometric structure in proofs. Logical and proof-theoretic problems that are formulated in the context of these systems are thus susceptible to analysis by geometric and combinatorial methods. The recent years have seen a development of a great variety of such methods by different communities. The aim of the proposed project is the application of these geometric and combinatorial techniques to logical problems in proof theory. The proof-theoretic problems have been carefully selected in order to lend themselves well to the analysis by these methods. The scientific impact of this project will be twofold: On the one hand, the solutions of the problems will be of importance for proof theory and the foundations of mathematics in their own right. On the other hand, the combined application of combinatorial methods from different traditions will have a unifying effect on them, leading towards a general theory of proofs as combinatorial structures.
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