The u-invariant of a field is the maximal dimension of a nonsingular anisotropic quadratic form over that field, whose order in the Witt group of the field is finite. By a classical theorem of Elman and Lam, the u-invariant of a linked field of characteristic different from 2 can be either 0, 1, 2, 4 or 8. The analogous question in the case of characteristic 2 remained open for a long time. We will discuss the proof of the equivalent statement in characteristic 2, recently obtained in joint work by Andrew Dolphin and the speaker.

Spectral geometry, among other things, asks the question `can one hear the shape of a drum?' To a mathematical object, say a Riemannian manifold, one can attach its spectrum and one is interested to know to what extent the object can be recovered from its spectrum. The spectral information can be encoded in terms of zeta functions, heat trace, or wave trace. Isometry invariants like volume and total scalar curvature can be obtained as special values of the spectral zeta function (Weyl's law). I shall give a quick introduction to these ideas and will end by giving the first example of two isospectral manifolds which are not isometric. The example, due to Milnor (using some deep work of Ernst Witt based on the theory of modular forms), exhibits two 16 dimensional flat tori which are isospectral but not isometric. This talk will be accessible to all grad students.