Levin Ceglie

Hi there :)

My name is Levin Ceglie and I am currently studying Mathematics at ETH Zurich. This site serves as a public repository for my mathematical notes, teaching materials, and writing.

Teaching Assistant Resources

Here you can find the materials from the exercise classes I have taught at ETH. These hubs contain weekly lecture notes, exercise sheets, multiple-choice questions, cheat sheets, and other possibly helpful resources for students:

Expository Math Writing

I also occasionally write expository pieces exploring specific topics in greater detail. My recent writings include:

  • An Introduction to the Dynamics of Compact Group Automorphisms after Klaus Schmidt (Bachelor Thesis): This thesis provides an introduction to the theory of algebraic dynamical systems based on the work of Klaus Schmidt. We begin by establishing fundamental dynamical properties through the lens of actions of countable groups on compact metrizable groups by continuous automorphisms, which we call quasi-algebraic actions. Specifically, we introduce the notions of topological transitivity, ergodicity, mixing, and expansiveness, and illustrate them through four running examples. With these foundations in place, we explore the rigidity of quasi-algebraic actions, structural constraints imposed by dynamical conditions, and provide a spectral characterization of ergodicity and mixing. The culmination of this work is an “algebraic dictionary” that establishes a correspondence between algebraic -actions and countable modules over a Laurent polynomial ring, translating dynamical properties into algebraic ones and providing a unified framework for their study.
  • On the Irreducibility of Cyclotomic Polynomials: A detailed account of Landau’s elegant proof of the irreducibility of the cyclotomic polynomial over , building on basic Group, Ring, and Field Theory.
  • Modules over PIDs: An exposition presenting the fundamental structure theorems for modules over principal ideal domains and providing a selection of their applications. In particular, showing how the Smith Normal Form and the Jordan Normal Form can be derived from these structure theorems.