Manchester Geometry Seminar - Lasse Rempe
| Dates: | 2 March 2026 |
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| Times: | 15:00 - 16:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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| Who is it for: | University staff, External researchers, Current University students |
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Speaker: Lasse Rempe (Manchester)
Abstract. In 1965, cartographer L. P. Lee proposed using a conformal map of a tetrahedron to the sphere as the basis for a map projection with less dramatic area distortion than the classical projections. While this has not caught on, it suggests the following mathematical question: Which orientable surfaces can be conformally represented on a collection of equilateral triangles? Equivalently, which such surfaces can be built (up to a conformal change of coordinate) by glueing together a finite or infinite collection of copies of a closed equilateral triangle? Such surfaces are called *equilaterally triangulable*.
The answer in the compact case is given by a famous classical theorem of Belyi, which states that a compact Riemann surface is equilaterally triangulable if and only if it is defined over a number field. These *Belyi surfaces* - and their associated “dessins d’enfants” - have found applications across many fields of mathematics, including mathematical physics.
In joint work with Chris Bishop, we give a complete answer of the same question for the case of infinitely many triangles (i.e., for non-compact Riemann surfaces). In some sense, the talk follows on from my inaugural lecture last year, but attending that lecture is not a pre-requisite for attending the seminar talk, which should be accessible to a wide audience including postgraduate students. In the first half of the talk, I will summarise and review the main results, while in the second half I will discuss some of the ideas and techniques that go into the proof, including elementary constructions of "almost" equilateral triangulations, as well as results from the theory of quasiconformal mappings and Teichmüller theory.
Travel and Contact Information
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Frank Adams Room 1
Alan Turing Building
Manchester
