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## New Challenges in Reverse Mathematics (3 - 16 January 2016)

The central theme of Reverse Mathematics is calibrating the strength of classical mathematical theorems in terms of the axioms needed to prove them; this calibration also takes into account recursion-theoretic complexity measures and consistency strength.

The area is a very active one that investigates many topics in classical mathematics and involves all major branches of modern mathematics.  For its first two or three decades, that is from 1970s to 1990s, almost all of the mathematical theorems that were studied in Reverse Mathematics were found to be equivalent to one of five subsystems of second-order arithmetic (the so called "big five" systems).  Since about 2000, Reverse Mathematics has become much more diversified. The list of theorems that led to this new flowering began primarily with Ramsey’s Theorem for Pairs and related combinatorial principles. Since that beginning, many new axiom systems and techniques from other areas in mathematical logic as well as combinatorics have been introduced and fruitfully explored to study problems in Reverse Mathematics.  These investigations have presented us with important new challenges and opened several new frontiers for research in the field.

The following is a list of topics that are of great current interest:

1. Reverse Mathematics of standard uncountable mathematics including algebra, combinatorics and analysis. Topics within logic include Determinacy Axioms, (effective versions of) cardinal invariants and higher order Reverse Mathematics.

2. Applications of nonstandard models of arithmetic in Reverse Mathematics. For example, conservation results, first-order consequences of combinatorial principles, the study of base theories like RCA0 in Reverse Mathematics and proof-theoretic ordinals corresponding to various combinatorial principles.

3. Alternative approaches to classify the strength of axioms and theorems like Weihrauch degrees, typed reverse mathematics and connections between reverse mathematics and related areas like recursion theory and computational complexity.

• Collaborative Research and Workshop: 3 - 16 January 2016
There will be talks in the mornings and free discussions in the afternoons.

• Public Lecture
 Date: Wednesday, 6 January 2016 Venue: 6:30pm - 7:30pm Foundations of Mathematics: An Optimistic Message Stephen G. Simpson, Pennsylvania State University, USA

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