Melanie Rupflin

In Trinity, 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 DPhil students who are working on problems related to geometric flows and geometric variational problems.

Many interesting objects and processes, not only in mathematics but also in other sciences, can be characterised as solutions of partial differential equations or as minimisers or gradient flows of a suitably defined quantity.

My research focuses on the theory of such partial differential equations and variational problems that are inspired by questions from geometry. I am in particular interested in the properties of 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, such as a minimiser of the area.

A particular focus of my current research is to try to give a precise answer to the natural question of whether knowing that an object has nearly minimal energy is enough to conclude that this object is very close to the absolute minimiser. Recently I have proven a new result for the classical Dirichlet energy of maps between spheres (for which the minimising states are the meromorphic functions discussed in the second-year undergraduate lecture on complex analysis), that shows that even for this simple model problem this is not always the case and have designed a partial differential equation that can capture the behaviour of such maps at all relevant scales.

'Sharp quantitative rigidity results for maps from S2 to S2 of general degree', arXiv:2305.17045 (2023)

'Low energy levels of harmonic maps into analytic manifolds', arxiv:2303.00389 (2023)

A. Malchiodi, M. Rupflin and B. Sharp, 'Łojasiewicz inequalities near simple bubble trees', to appear in Amer. J. Math.

‘Hyperbolic metrics on surfaces with boundary’, Journal of Geometric Analysis 31 (2021), 3117-3136

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

‘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