Register for Professor Lasse Rempe's Inaugural Lecture: Friday, 21 November 2025, 12.00 - 1.00pm, Alan Turing Building, Department of Mathematics, G.107
This is an opportunity to come together and join your colleagues in celebrating as?Lasse Rempe gives his lecture on “Drawing maps on equilateral triangles”.
The event will conclude with light refreshments and informal discussions after the lecture has taken place from 1:00pm - 2:00pm.
Abstract:
The study of conformal (angle-preserving) maps has a long history. Since the earth is curved, any geographical map has to distort either angles or sizes. For navigation, conformality - preserving angles - is usually more important. In 1569, Flemish geographer Gerardus Mercator introduced a famous method for drawing a conformally correct map of the earth (without its poles) on a cylinder. This Mercator projection became the standard method used in navigation and is still used in internet mapping services. An even older conformal projection, stereographic projection, was already used in antiquity to produce maps of the night sky.
In 1965, Lee observed that one may draw an image of the earth conformally onto a tetrahedron - that is, we may draw a complete map on the earth on four equilateral triangles, fitting together along their edges, in such a way that angles are faithfully represented. We ask what happens when the sphere is replaced by another surface - when is it possible to make such an atlas consisting of equilateral triangles of equal sizes?
When the surface is closed (such as a sphere or a torus - the surface of a donut), the question was posed in 1986 in the context of string theory. In this case, there is a beautiful answer, going back to work of Belyi from 1979: most surfaces cannot be "equilaterally triangulated", and those that can have a natural description. Such surfaces have found wide applications across diverse areas of mathematics.
For open surfaces, the problem remained unsolved until we were able to solve it in recent joint work with Bishop (to appear in Inventiones Mathematicae). The answer is surprising: every such surface has a conformal representation on a (now infinite) collection of equilateral triangles.
In this lecture, I will begin with a historical overview and then discuss the above question and our result, as well as its consequences, in further detail. The lecture will be accessible to a general scientific audience, and in particular to undergraduate and postgraduate students in mathematics.
Biography:
Lasse Rempe studied mathematics and computer science in Kiel (Germany), Stony Brook (USA) and Orsay (France), and was awarded his doctoral degree in Mathematics by the University of Kiel in 2003. Prior to joining The University of Manchester, he held the Chair of Pure Mathematics (endowed 1882) at the University of Liverpool. Professor Rempe is a Fellow of the American Mathematical Society; his research concerns the geometry, analysis and dynamics of functions of one complex variable, and has been recognised by an LMS Whitehead Prize and a Philip Leverhulme Prize.
Passionate about communicating mathematics to broader audiences, Professor Rempe has organised several exhibitions and public lectures connected to his research area; in particular he narrated a BBC audio slideshow in 2009. He is an amateur musician and has collaborated with composers at the RNCM's Centre for Practice & Research in Science & Music (PRiSM) who use mathematical ideas in their creative processes.
