Fellow and Tutor in Pure Mathematics

Melanie Rupflin

  • I am Associate Professor in the Mathematical Institute.

  • My research is at the interface of Geometry and Analysis, and I am in particular interested in the study of minimal surfaces and of so-called geometric flows, which deform geometric objects towards an optimal state.

  • I hold a degree and PhD in Mathematics from ETH Zurich and have held postdoctoral positions at the University of Warwick, the MPI in Graviational Physics and the University of Leipzig.

Melanie Rupflin

Teaching

In Trnity, I teach the tutorials for most of the courses in pure mathematics for first years, namely for Introduction to University Mathematics, the three Analysis courses and the two courses in Linear Algebra. For second years, I teach the tutorials for Metric Spaces and Differential Equations 1 and for options related to Analysis such as Integration and Introduction to Manifolds.

In the Mathematical Institute, I will be lecturing the core second-year course Differential Equations 1 in the upcoming year, as well as a third-year course in Functional Analysis.

I currently supervise four DPhil students who are working on problems related to geometric flows, calculus of variation and spectral problems on manifolds.

Research

Many interesting problems, not only in mathematics but also in other sciences, can be formulated as partial differential equations.

In my research I work on problems in the theory of partial differential equations that describe geometric problems. I am in particular interested in questions that relate to surfaces which have minimal possible area (given some constraint such as a given boundary curve or prescribed enclosed volume), which can also be seen in nature as soap films or soap bubbles. I am also studying the properties of geometric flows, which are equations that are designed to evolve a geometric object, such as a surface, towards an optimal state. Over the past years and in joint work with P. Topping we have proven that a geometric flow which we defined in 2012 succeeds in changing any given surface in an arbitrary manifold into a collection of minimal surfaces, that is minimisers (or critical points) of the Area.

Selected Publications

Hyperbolic metrics on surfaces with boundary’, Journal of Geometric Analysis (2020)

M. Rupflin and P. Topping, ‘Global weak solutions of the Teichmüller harmonic map flow into general targets’, Anal. PDE 12 (2019), 815-842

N. Große and M. Rupflin, ‘Sharp eigenvalue estimates on degenerating surfaces’, Partial Differential Equations 44 (2019), no.~7, 573–612

M. Rupflin and M. Schrecker, ‘Analysis of boundary bubbles for almost minimal cylinders’, Calc. Var. Partial Differential Equations 57 (2018), Art 121, 34 pp

Teichmüller harmonic map flow from cylinders’, Mathematische Annalen, 368 (2017), 1227-1276

M. Rupflin and P. Topping, ‘Teichmüller harmonic map flow into non-positively curved targets’, J. Differential Geom.108 (2018), 135-184

R. Buzano and M. Rupflin,’Smooth long-time existence of Harmonic Ricci Flow on surfaces’, J. Lond. Math. Soc. 95 (2017), 277-304

M. Rupflin and P. Topping, ‘Flowing maps to minimal surfaces’, Amer. J. Math. 138 (2016), no. 4, 1095-1115

Subjects
Professor Rupflin
melanie.rupflin@trinity.ox.ac.uk

Many interesting problems, not only in mathematics but also in other sciences, can be formulated as partial differential equations.