Finnur LárussonProfessor of Pure Mathematics
office: Ingkarni Wardli, room 649
My book Lectures
on Real Analysis was published by Cambridge University Press in mid-2012.
Click on the title or the picture of the front cover, and you can have a look inside the book on the CUP website.
The book is a rigorous introduction to real analysis for undergraduate students, starting from the axioms for a complete ordered field and a little set theory. The book avoids any preconceptions about the real numbers and takes them to be nothing but the elements of a complete ordered field. All the standard topics are included, as well as a proper treatment of the trigonometric functions, which many authors take for granted. The final chapters provide a gentle, example-based introduction to metric spaces with an application to differential equations on the real line. The exposition is brisk and concise, helping students focus on the essentials. Over 200 exercises of varying difficulty are included, many of them adding to the theory in the text. The book is intended for second-year undergraduates and for more advanced students who need a foundation in real analysis.
I will be grateful for any comments from readers. Please e-mail me if you find misprints or mistakes in the book.
I welcome enquiries from prospective honours or postgraduate students. Feel free to e-mail me or, if you are in Adelaide, stop by my office.
If you are interested in PhD or MPhil studies in Adelaide, I would be glad to discuss possible thesis topics with you. I mainly work in complex analysis and complex geometry, including pluripotential theory, and applications of abstract homotopy theory to these areas. Start by looking at my papers or read this to get an idea of what sorts of things I do. There are several other people in Adelaide working in related areas. We run an active seminar with talks by locals and visitors.
I am happy to supervise honours projects in complex analysis and the many areas of mathematics that interact with it. An honours topic that I particularly like is the theory of compact Riemann surfaces. It provides an opportunity to meet and apply important ideas from functional analysis, homological algebra, manifold theory, partial differential equations, and sheaf theory in an accessible geometric context. The first goal of a project in this area would be to get to the central theorem on compact Riemann surfaces, the Riemann-Roch Theorem. After that, the project could continue in various different directions.
There are many other options, including complex analysis in higher dimensions, algebraic geometry, differential geometry, topology, and category theory. In recent years I have supervised an honours project on wavelets, a newly developed area within real and functional analysis with applications to image compression; a project about the correspondence between planar trees and so-called Shabat polynomials, involving combinatorics, complex analysis, and topology; a project on one-dimensional complex tori, also known as elliptic curves; a project about category theory and toposes of graphs; a project on the so-called Oka principle, which mixes homotopy theory and complex geometry; a project on Belyi's famous theorem, characterising the compact Riemann surfaces that can be defined over a number field; and a project on 20th century generalisations of the Riemann mapping theorem.