This volume originated from the International Congress "ULTRAMATH: Applications of Ultrafilters and Ultraproducts in Mathematics", which was held in Pisa, Italy, from June 1-7, 2008. The volume aims to present the state-of-the-art of applications in the whole spectrum of mathematics which are grounded on the use of ultrafilters and ultraproducts. It contains two general surveys on ultrafilters in set theory and on the ultraproduct construction, as well as papers that cover additive and combinatorial number theory, nonstandard methods and stochastic differential equations, measure theory, dynamics, Ramsey theory, algebra in the space of ultrafilters, and large cardinals. The papers are intended to be accessible and interesting for mathematicians who are not experts on ultrafilters and ultraproducts. Greater prominence has been given to results that can be formulated and presented in non-special terms and be, in principle, understandable by any mathematician, and to those results that connect different areas of mathematics, revealing new facets of known important topics.
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