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Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working on problems like the metrization problem , topologists (including Steen and Seebach) have defined a wide variety of topological properties .
In topology, an irrational winding of a torus is a continuous injection of a line into a two-dimensional torus that is used to set up several counterexamples. [1] A related notion is the Kronecker foliation of a torus, a foliation formed by the set of all translates of a given irrational winding.
The Lichnerowicz conjecture usually refers to the generalization that locally harmonic manifolds are flat or rank-1 locally symmetric. It has been proven true for compact manifolds with fundamental groups that are finite groups (Szabó 1990) but counterexamples exist in seven or more dimensions in the non-compact case (Damek & Ricci 1992)
Counterexamples of the global Torelli theorem.", Invent. Math. , 63 (2): 287–304, doi : 10.1007/BF01393879 , MR 0610540 This algebraic geometry –related article is a stub .
In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. [1] For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are ...
Shortly after, Thomas Jenrich derived a 64-dimensional counterexample from Bondarenko's construction, giving the best bound up to now. [ 13 ] [ 14 ] Apart from finding the minimum number n of dimensions such that the number of pieces α ( n ) > n + 1 , mathematicians are interested in finding the general behavior of the function α ( n ) .
Kollár is known for his contributions to the minimal model program for threefolds and hence the compactification of moduli of algebraic surfaces, for pioneering the notion of rational connectedness (i.e. extending the theory of rationally connected varieties for varieties over the complex field to varieties over local fields), and finding counterexamples to a conjecture of John Nash.
Books about counterexamples in various areas of mathematics. Pages in category "Books about mathematical counterexamples" The following 3 pages are in this category, out of 3 total.
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