Christian G J Roettger Office 463 Carver Phone 294-8164 |

Amongst other things, I am interested in

- Asymptotic counting problems
- Discrete dynamical systems
- Kloosterman sums and uniform distribution
- Diophantine approximation
- Representations of finite groups
- Finite fields, Galois rings

**Galois Theory and the Ring of Sauron**

Combinatorics/Algebra Seminar, ISU, April 24, 2017
Consider a polynomial f(x) over a field k with degree n and discriminant D. The Galois group G of f(x) can be viewed a a subgroup of the symmetric group S_{n} via its action on the roots of f(x).
It is a well-known application of Galois Theory that G is contained in the alternating group A_{n} exactly if D is a square in the ground field k. Moreover, if this is not the case, then sqrt(D) generates the fixed field of the intersection of G with A_{n}.
We are looking for formulas that generalize this - i.e., for any transitive subgroup H of S_{n}, find expressions in the coefficients of a generic polynomial of degree n which are in the ground field exactly if G is contained in H, and which generate the fixed field of the intersection of G and H otherwise. At first, it is not even clear that such expressions have to exist. But they do, and they in principle provide a way to determine G. Practical computations are lengthy and best done by computer, but the theory is very appealing and natural. In particular, we use multivariate polynomial rings and classical representation theory of finite groups (in a very basic way).

Joint work with John Gillespie.

**Rashomon - Reconstructing Reality**
from inexact measurements

Joint work with H Hofmann, D Cook.

MAA section meeting, Graceland University, Lamoni, IA, Oct 3, 2015

Here are
slides.

The art installation 'Rashomon' was displayed on the Iowa State University campus during summer 2015. It consists of 15 identical, abstract sculptures. Artist Chuck Ginnever posed the challenge whether it is possible to display the sculptures so that no two of them are in the same position (modulo translation/rotation). We investigated the related question of reconstructing such a sculpture from (ordinary tape-measure) inexact measurements. Mathematics involved are the Cayley-Menger determinant, and the gradient method / Steepest Descent. We'll explain the mathematics with some simple examples and then show the results of our reconstruction. We will only assume elementary linear algebra (matrix - vector multiplication, determinants).

Math 105: Introduction to Mathematical Ideas.

We examine the mathematics of voting systems.Math 141/142 Trigonometry (and analytic geometry)

Math 151: Calculus for Business and Social Sciences

Math 166: Calculus II (recitations only)

Math 181: Calculus For the Life Sciences I

Math 195: Math for Elementary Education

Math 267: Differential Equations, Laplace Transform

Math 297: Intermediate Topics in Elementary Mathematics

Math 307: Matrices and Linear Algebra

Math 317: Theory of Linear Algebra

Math 350: Elementary Number Theory

Course notes for Analytic Number Theory

General advice, frequently asked questions

**My Favorite Quote:**"Should array indices start at 0 or 1? My compromise of 0.5 was rejected without, I thought, proper consideration." -
Stan Kelly-Bootle

Pros and cons of metric vs imperial system.

This webpage is http://www.public.iastate.edu/~roettger/homepage.html

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