Fix a number field K with ring of integers O and a finite group G of odd order. If L/K is a Galois extension with group G, then a theorem of Erez says that the square root of the inverse different A_L/K of L/K is locally free over OG if and only if L/K is at most weakly ramified. In this case, A_L/K defines a class cl(A_L/K) in the locally free class group Cl(OG) of OG. In this talk, I will give an outline of how one can prove that the subcollection of classes realizable by tame extensions form a group when G is abelian. The main idea is to consider all Galois G-extensions instead of only field extensions and then use resolvends to characterize the realizable classes. The techniques I shall use are based on a paper of McCulloh, in which he proved the same result for the ring of integers. I will define all the terminologies that appeared in this abstract to make the talk more accessible.

       I shall discuss an approach to describing certain aspects of the relative Galois structure of tame rings of integers in terms of invariants arising via relative algebraic K-theory.

       We use the language of the hyperreal numbers to illustrate how the well-known Kummer congruences involving Bernoulli numbers is the shadow of congruences of infinite hyperintegers. We will also discuss Euler's intuitions on divergent series and show how the values of the Riemann zeta function at negative integers agree with the various values associated to divergent series. We do this by demonstrating that Ramanujan summation corresponds, in a certain sense, to taking the "standard part" of an infinite hyperreal number.

       Hirzebruch noticed "an amusing connection between continued fractions and class numbers" while computing the signature sign(X) of a Hilbert modular surface X attached to a real quadratic number field K. We will discuss how continued fractions show up in the resolution of cusp singularities in a compactification of X. The number of cusps is the class number of K, and, when K has no units of negative norm, sign(X) can be given in terms of class numbers of imaginary quadratic number fields.

       Two integral lattices of the same dimension and signature are said to be in the same genus if they are equivalent over the p-adic integers for every prime p. Each genus is a union of integral equivalence classes of lattices and reflects a general failure of local-to-global methods to characterize integral equivalence. By somewhat of a folk theorem, two lattices are in the same genus if and only if they become integrally equivalent after a direct sum with the rank-2 unimodular even lattice with bilinear form f(x, y) = x₁y₂ + x₂y₁. I will discuss the proof of this theorem, which relies on the spinor genus and the strong approximation theorem on rational orthogonal groups. Stable equivalence with respect to sums of the odd unimodular lattice with bilinear form f(x, y) = x₁² - x₂² arises naturally in physics through the classification of edge phases of two-dimensional fermionic systems having the same bulk topological order. While each genus consists entirely of even or odd lattices, these equivalence classes are unions of even and odd genera. I will discuss in what sense the theorem can be extended to cover this more general type of equivalence.