Daniel Grubb's Home Page

I am an associate professor at NIU and have been here since 1986. I got my PhD from Kansas State University from Karl Stromberg. My mathematical interests include Harmonic Analysis, Measure Theory, Topology, and Galois Theory (that last one is a bit out of place, isn't it?).

  • I started out studying sets of uniqueness in compact, 0-dimensional metric groups (also known as Vilenkin groups). It turns out that much of the pointwise theory concerning uniqueness of trigonometric expansions on the circle can be done in this context, often with some significant simplifications (due to the fact that the characters are finite-valued). Some of my results include the fact that a countable union of closed sets of uniqueness is again a set of uniqueness (a result due to Nina Bary for trig series) and the fact that a closed subgroup of infinite index must be a set of uniqueness.

  • My next major area of mathematical research had to do with what are known as topological measures (previously, quasi-measures). These give a type of integration where linearity is only assumed on singly-generated subalgebras of the ring of continuous functions. It turns out that many results from classical measure theory still hold, but there are some very interesting differences also. In particular, the connectedness of the underlying topological space is much more important for topological measures than it is for classical measures. Because of this, I have applied results from algebraic topology to study this form of integration.

  • Lately, I have become interested in the theory of polynomials and Galois theory. In particular, I am interested in questions surrounding the notion of real radicals. A classical result along the lines of my interests is the Causus Irreducibilis: If an irreducible cubic polynomial over the rationals splits over the reals, then none of its roots can be real radicals! This theory requires detailed study of the properties of Tchebyshev polynomials and their factorization.

Here are notes from several classes I have taught. They are definitely NOT in final form! New chapters will be added as they are written.

Please let me know if you find any mistakes!