Broadly speaking my interests focus around Partial Differential Equations and the Calculus of Variations. Here is a list of topics (of course somewhat overlapping):
Turbulence and the Euler equations

1. High-dimensionality and h-principle in PDE
(Bulletin AMS 2017)

2. Weak solutions of the Euler equations: non-uniqueness and dissipation
(JÉDP 2015)

3. From Isometric Embeddings to Turbulence
(HCDTE Lecture Notes 2014)
Version including corrections thanks to :

4. The h-principle and the equations of fluid dynamics
(Bulletin AMS 2012)

5. Counterexamples to elliptic regulariy and convex integration
Analysis and geometry in their interaction
(AMS Contemp. Math. 2006)

A fundamental problem of the theory of turbulence is to find a satisfactory mathematical framework linking the Navier-Stokes equations to the statistical theory of Kolmogorov. One of the cornerstones of the statistical theory is the famous 5/3 law, predicting the power law of the energy spectrum in turbulent flows. Although this law concerns the Navier-Stokes equations, a very closely related conjecture was made by Onsager regarding the critical Hölder regularity of weak solutions of the Euler equations which preserve the energy. Recently, in joint work with Camillo De Lellis, I have devoted a lot of effort in constructing weak solutions of the Euler equations at the critical 1/3 Hölder exponent. Based on "Mikado flows" and Phil Isett's gluing technique, we can now reach any uniform exponent less than 1/3.

1. Onsager's conjecture for admissible weak solutions
with Tristan Buckmaster, Camillo De Lellis and Vlad Vicol

2. Dissipative Euler flows with Onsager-critical spatial regularity
with Tristan Buckmaster and Camillo De Lellis
(CPAM 2015)

3. Transporting microstructure and dissipative Euler flows
with Tristan Buckmaster, Phil Isett and Camillo De Lellis
(Annals 2015)

4. Dissipative Euler flows and Onsager's Conjecture
with Camillo De Lellis
(JEMS 2014)

5. Dissipative continuous Euler flows
with Camillo De Lellis
(Inventiones 2013)

A central difficulty in the Euler equations is the inherent non-uniqueness and pathological behaviour of weak solutions. This non-uniqueness, rather than being an isolated phenomenon, turns out to be directly linked to the celebrated construction of Nash and Kuiper of rough isometric embeddings and, more generally, to Gromov's h-principle in geometry. The same phenomenon appears in several other equations from fluid dynamics as well.

1. Non-uniqueness and h-principle for Hölder-continuous weak solutions of the Euler equations
with Sara Daneri
(ARMA 2017)

2. Nash-Kuiper theorem for C1,1/5 immersions of surfaces in 3D
with Camillo De Lellis and Dominik Inauen
(Rev. Mat. Iberoamericana)

3. Equidimensional isometric maps
with Bernd Kirchheim and Emanuele Spadaro
(Comm. Mat. Helv. 2015)

4. Weak solutions to the stationary incompressible Euler equations
with Antoine Choffrut
(SIAM J. Analysis 2014)

5. Non-uniqueness for the Euler equations: the effect of the boundary
with Claude Bardos and Emil Wiedemann
(Uspheki Mat. Nauk. 2014)

6. Relaxation of the incompressible porous media equation
(Ann. l'ENS 2012)

7. Young Measures Generated by Ideal Incompressible Fluid Flows
with Emil Wiedemann
(ARMA 2012)

8. Weak Solutions to the Incompressible Euler Equations with Vortex Sheet Initial Data
(C.R. Math. Acad. 2011)

9. Weak-strong uniqueness for measure-valued solutions
with Yann Brenier and Camillo De Lellis
(CMP 2011)

10. The h-principle and rigidity for C1,alpha isomeric embeddings
with Sergio Conti and Camillo De Lellis
(Nonlinear Partial Differential Equations, Proc. Abel Symp. 2010)

11. On admissibility criteria for weak solutions of the Euler equations
with Camillo De Lellis
(ARMA 2010)

12. The Euler equations as a differential inclusion
with Camillo De Lellis
(Annals 2007)

1. Laminates meet Burkholder functions
with Nicholas Boros and Alexander Volberg
(J. Math Pures Appl. 2013)

2. Uniqueness of normalized homeomorphic solutions to nonlinear Beltrami equations
with Kari Astala, Albert Clop, Daniel Faraco and Jarmo Jääskeläinen
(IMRN 2011)

3. Convex Integration and the L^p theory of elliptic equations
with Kari Astala and Daniel Faraco
(Ann. Sc. Norm. Super. Pisa 2008)

1. T5 configurations and non-rigid sets of gradients
with Clemens Förster
(submitted 2017)

2. Laminates supported on cubes
with Gabriella Sebestyén
(J. Conv. A. 2016)

3. On the gradient set of Lipschitz maps
with Bernd Kirchheim
(J. Reine Angew. Math. 2008)

4. Tartar's conjecture and localization of the quasiconvex hull
with Daniel Faraco
(Acta Math. 2008)

5. On quasiconvex hulls in symmetric 2x2 matrices
Ann. I.H.P. Anal. Non Lineare 23 (2006)

6. On the local structure of rank-one convex hulls
(Proc. AMS 2006)

7. Simple proof of two-well rigidity
with Camillo De Lellis
(C. R. Math. Acad. Paris 2006)

8. Rank-one convex hulls in R2x2
(Calc. Var. PDE 2005)

1. The regularity of critical points of polyconvex functionals
(ARMA 2004)

Non-uniqueness and h-principle
Quasiregular maps and Beltrami equations
Quasiconvexity, Rank-one convexity and Rigidity
Polyconvexity and regularity in the Calculus of Variations
Surveys and lecture notes
Lisa Beck
Camillo De Lellis
Camillo De Lellis
László Székelyhidi Jr.
Surveys and lecture notes
Turbulence and the Euler equations
Non-uniqueness and h-principle
Quasiregular maps and Beltrami equations
Quasiconvexity, rank-one convexity and rigidity