Polynomial rings

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  • The curriculum vitae of Alice Turner Schafer lists two specializations: abstract algebra (group theory) and women in mathematics. As early as her high school years Alice exhibited a love for mathematics and an interest in teaching as a career. As a mathematics educator she championed the full participation of women in mathematics.

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  • The Eisenstein irreducibility critierion is part of the training of every mathematician. I first learned the criterion as an undergraduate and, like many before me, was struck by its power and simplicity. This article will describe the unexpectedly rich history of the discovery of the Eisenstein criterion and in particular the role played by Theodor Sch¨onemann.

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  • We consider a specialization of an untwisted quantum affine algebra of type ADE at a nonzero complex number, which may or may not be a root of unity. The Grothendieck ring of its finite dimensional representations has two bases, simple modules and standard modules. We identify entries of the transition matrix with special values of “computable” polynomials, similar to Kazhdan-Lusztig polynomials. At the same time we “compute” q-characters for all simple modules. The result is based on “computations” of Betti numbers of graded/cyclic quiver varieties.

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  • Topological Hochschild homology and localization 2. The homotopy groups of T (A|K) 3. The de Rham-Witt complex and TR· (A|K; p) ∗ 4. Tate cohomology and the Tate spectrum 5. The Tate spectral sequence for T (A|K) 6. The pro-system TR· (A|K; p, Z/pv ) ∗ Appendix A. Truncated polynomial algebras References Introduction In this paper we establish a connection between the Quillen K-theory of certain local fields and the de Rham-Witt complex of their rings of integers with logarithmic poles at the maximal ideal.

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