Teaching Responsibilities 2019/20: None
Most of my research sits at the intersection between Partial Differential Equations, Geometric Measure Theory, Harmonic Analysis, and the Calculus of Variations. In particular, I am interested in oscillation and concentration phenomena and what can be rigorously proved about their "shape".
Most of my research is funded by the European Research Council and the Lloyds Register Foundation (previously also EPSRC). Our group maintains the website www.ercsingularity.org, which contains details on our research, publications, and recent preprints:
- Concentration versus oscillation effects in brittle damage (with J.-F. Babadjian, F. Iurlano), to appear in Comm. Pure Appl. Math., arXiv:1906.02019.
- Dimensional estimates and rectifiability for measures satisfying linear PDE constraints (with A. Arroyo-Rabasa, G. De Philippis, J. Hirsch), Geom. Funct. Anal. 29 (2019), pp 639-658, Online version.
- Liftings, Young measures, and lower semicontinuity (with G. Shaw), Arch. Ration. Mech. Anal. 232 (2019), 1227-1328, Online version.
- On the structure of A-free measures and applications (with G. De Philippis), Ann. of Math. 184 (2016), 1017-1039, Online version.
- Directional oscillations, concentrations, and compensated compactness via microlocal compactness forms. Arch. Ration. Mech. Anal. 215 (2015), 1-63. Online version.
- Lower semicontinuity for integral functionals in the space of functions of bounded deformation via rigidity and Young measures, Arch. Ration. Mech. Anal. 202 (2011), 63-113. Online version.
- Characterization of generalized gradient Young measures in W1,1 and BV (with J. Kristensen), Arch. Ration. Mech. Anal. 197 (2010), 539-598. Online version.
A complete list of publications can be found in my CV.
I usually teach Analysis courses at the University of Warwick (e.g. PDEs, Calculus of Variations, Complex Analysis).
The book Calculus of Variations is based on my lectures at the University of Warwick on that topic and appeared with Springer in 2018:
See www.calculusofvariations.com for details.