Summer 2026
Together with Kostas Zemas, we offer the course
- V5B5 - Advanced Topics in Analysis and Calculus of Variations:
Quantitative Isoperimetric-Type Inequalities
We will have two lectures per week, on Mondays (in SR 0.011) and Tuesdays (in N 0.008), in both cases from 16:15 to 18:00.
Info: the eCampus webpage is now open!
Material: you can download a syllabus of the course here.
Below a quick report of the lectures:
Lecture 1 (on 13.04.2026): the Dido's problem, Hurwitz's proof, De Giorgi's formulation of the isoperimetric problem, Bennesen-type inequalities, the isoperimetric deficit, Fuglede's result and ideas, Fraenkel's asymmetry, Hall's conjecture. [notes]
Lecture 2 (on 14.04.2026): outer measures, Lebesgue measure, measurable sets, Borel and Borel regular measures, Radon measures, inner and outer approximations, restrictions and support, Hausdorff measure, properties and example, $\mathcal{H}^1=\textit{length}$. [notes], [addendum]
Lecture 3 (on 20.04.2026): Riesz's Theorem, vector-valued outer measures, weak* convergence and compactness, Besicovitch covering, Lebesgue-Besicovitch differentiation, density of sets, the Area formula for linear maps. [notes]
Lecture 4 (on 21.04.2026): set of Jacobian zero, Lipschitz linearization lemma, proof of the Area formula, $\mathcal{H}^k$-rectifiable sets, decomposition into regular Lipschitz images, existence of an approximate tangent space for regular Lipschitz images. [notes]
Lecture 5 (on 27.04.2026): approximate tangent space for $\mathcal{H}^k$-rectifiable sets, sets of finite perimeter, Gauss-Green measure, perimeter and relative perimeter, the role of the divergence theorem, examples, relations with BV functions. [notes], [addendum]
Lecture 6 (on 28.04.2026): local convergence of sets, lower semicontinuity of the perimeter, examples, topological boundary and support of Guass-Green measure, regularization via mollification, coercivity of the perimeter. [notes]
Lecture 7 (on 04.05.2026): the coarea formula, examples and discussion, approximation of sets of finite perimeter with smooth (and bounded) sets, the constraint isoperimetric problem, reduction to minimization among smooth sets. [notes]
Lecture 8 (on 05.05.2026): optimality of the ball (not proved yet), proof of the isoperimetric inequality, comments and discussion (the planar case), the reduced boundary and (measure-theoretic) outer normal vector, blow-ups of sets of finite perimeter and their tangential properties, De Giorgi's structure theorem. [notes]
Winter 2025/2026
Together with Prof. Angkana Rüland, I was organizing the graduate seminar
- S5B3 - Graduate Seminar of New Developments in PDE:
Variational Methods and Inverse Problems
Have a look at the Group Seminar webpage for more info.
Winter 2024/2025
Together with Lennart Machill, I was organizing the graduate seminar
- S5B3 - Graduate Seminar of New Developments in PDE:
Variational Methods and Inverse Problems
Have a look at the Group Seminar webpage for more info.