We study the relationship between the geometry and the Laplace spectrum of a Riemannian orbifold O via its heat kernel; as in the manifold case, the time-zero asymptotic expansion of the heat kernel furnishes geometric information about O. In the case of a good Riemannian orbifold (i.e., an orbifold arising as the orbit space of a manifold under the action of a discrete group of isometries), H. Donnelly [10] proved the existence of the heat kernel and constructed the asymptotic expansion for the heat trace. We extend Donnelly's work to the case of general compact orbifolds. Moreover, in both the good case and the general case, we express the heat invariants in a form that clarifies the asymptotic contribution of each part of the singular set of the orbifold. We calculate several terms in the asymptotic expansion explicitly in the case of two-dimensional orbifolds; we use these terms to prove that the spectrum distinguishes elements within various classes of two-dimensional orbifolds.

Abstract. Let G be a closed, non-transitive subgroup of O(n + 1), where n ≥ 2, and let Q n = S n /G. We will show that for each n there is a lower bound for the diameter of Q n . If G is finite then Q n is an orbifold of constant curvature one and an explicit lower bound can be given. For Coxeter groups, the resulting lower bound is independent of dimension. Otherwise, Q n is a spherical Alexandrov space and we will show existence of a lower bound. In the process, we will compute some examples of quotient spaces and their diameters.

Let G ⊂ O(4) act isometrically on S 3 . In this article we calculate a lower bound for the diameter of the quotient spaces S 3 /G. We find it to be 1 2 arccos( tan( 3π 10 ) √ 3 ), which is exactly the value of the lower bound for diameters of the spherical space forms. In the process, we are also able to find a lower bound for diameters for the spherical Aleksandrov spaces, S n /G, of cohomogeneities 1 and 2, as well as for cohomogeneity 3 (with some restrictions on the group type). This leads us to conjecture that the diameter of S n /G is increasing as the cohomogeneity of the group G increases.

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The Simpsons is an ideal source of fun ways to introdu ce important mathemat ical concepts, moti vat e st udents, and red uce math anxiety. We discuss examples from The Simpsons related to calculus, geometry, and number theory t hat we have incorporated into t he class room . We exp lore st udent reactions and educational benefits and difficult ies encountered.KEYWORDS: The Sim psons, pop cult ure, redu cing math anxiety, Flatland , writ ing assign ments.