# Dylan Madden

**About me: **I am a fourth year PhD student, supervised by Dr Marco Schlichting. Previously, I attended the University of Glasgow, graduating in 2016 with an MSci in Pure Mathematics. I am originally from Glasgow. My workroom is B0.15, and my email address is d (dot) madden (at) warwick (dot) ac (dot) uk.

**Research interests: **My current research interests are in higher Grothendieck-Witt theory, also known as Hermitian K-theory, which is a generalization of algebraic K-theory. In algebraic K-theory, given a ring , one studies a series of abelian groups , which encode information about . In Hermitian K-theory, we consider rings with involution, and study a series of abelian groups which encode information about , including the structure of its involution. This has applications, for example, to A1 homotopy theory, which is a method for applying topological techniques to algebraic geometry; for instance, in A1 homotopy theory, the affine line plays a role analogous to that of the interval in topology.

In the past, during my Masters year at Glasgow, I worked on cyclic homology with Dr Ulrich Kraehmer.

**Talks:**

- Cyclic vs mixed homology, Postgraduate Seminar, University of Warwick, January 2018.
- Getting What You Pay For: An Introduction to Algebraic K-theory, Young Mathematicians Colloquium, University of Birmingham, April 2018.

**Papers:**

- Cyclic vs mixed homology (Joint work with Ulrich Kraehmer)
*Homology, Homotopy, and Applications,*vol.20 (1), 2018, pp.237-250

**Other writings:**

- First year report (A report on the first year of my PhD; it takes the form of an essay on localization theorems in algebraic K-theory, Witt theory, and Hermitian K-theory.)
- The Dold-Kan and Dwyer-Kan Correspondences (An essay on the titular results, written as a Masters dissertation at the University of Glasgow.)
- Introductory Categorification (An essay on categorification, written as a fourth year dissertation at the University of Glasgow.) (A winner of the Weiglhofer Prize.)