Reverse Mathematics of Combinatorial Principles (19w5111)


Damir Dzhafarov (University of Connecticut)

Peter Cholak (Notre Dame)

(University of Chicago)


The Casa Matemática Oaxaca (CMO) will host the "Reverse Mathematics of Combinatorial Principles" workshop in Oaxaca, from September 15, 2019 to September 20, 2019.

Mathematicians prove theorems from basic assumptions called axioms. Today, the subject benefits from having "firm foundations'', by which we usually mean axioms sufficient to prove virtually all of the theorems we care about. But given a particular theorem, can we specify precisely which axioms are needed to derive it? This is a natural question, and also an ancient one: over 2000 years ago, the Greek mathematicians were already asking it about the axioms of geometry. Reverse mathematics provides a modern approach to this kind of question. A striking empirical fact repeatedly demonstrated in this area is that the vast majority of mathematical propositions can be classified into just five main types, roughly corresponding to five general mathematical principles that crop up all across mathematics, regardless of whether we are looking at algebra, calculus, geometry, or many other areas.

But there are exceptions, and they include some very important mathematical theorems. One of these is a famous theorem due to F. P. Ramsey, which can be colorfully stated as follows: at any dinner party with infinitely many guests, it is possible to find either infinitely many of the guests that all know each other, or infinitely many of the guests none of whom knows any of the others. This is a profound result in the area of combinatorics, with numerous applications in mathematics and computer science. And as it happens, it falls outside the five main types mentioned above. Understanding why this theorem, and others like it, behave differently from the vast majority of others, sheds light on the capacities and limitations of different ways of reasoning in mathematics, particularly in combinatorics, and in so doing, gives us a better picture of the underpinnings of mathematics as a whole.

The Casa Matemática Oaxaca (CMO) in Mexico, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, are collaborative Canada-US-Mexico ventures that provide an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station in Banff is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT). The research station in Oaxaca is funded by CONACYT