Profile
Profile

Prof. Dr. László Székelyhidi Jr.

Former Bonn Junior Fellow
Current position: Professor (W3), University of Leipzig

E-mail: szekelyhidi(at)math.uni-leipzig.de
Homepage: http://www.math.uni-leipzig.de/~szekelyhidi
Institute: Institute for Applied Mathematics
Research Areas: Research Area A
Research Area B
Date of birth: 17.Apr 1977
Mathscinet-Number: 733106

Academic Career

2003

Dr. rer. nat., University of Leipzig

2003 - 2004

Postdoc, Institute for Advanced Study, Princeton, NJ, USA

2004 - 2005

Postdoc, ETH Zürich, Switzerland

2005 - 2007

Heinz Hopf Lecturer, ETH Zürich, Switzerland

2007 - 2011

Professor (W2, Bonn Junior Fellow), University of Bonn

Since 2011

Professor (W3), University of Leipzig

Contribution to Research Areas

Research Area B
The main focus of my work in this research area is weak solutions of the Euler equations of ideal fluid flow using geometric methods. In [1], we developed a version of convex integration, originally due to Nash and his work on isometric embeddings, which can be used to construct a large class of weak solutions of the Euler equations exhibiting turbulent behaviour. As neither of us had a strong background in fluid dynamics and turbulence to start with, we profited substantially from the interaction with Ch. Doering (Ann Arbor) and Y. Brenier (Nice), who were in Bonn in 2007 as guests of RA B.
This initial interaction with experts from the fluids community was essential for our research programme, which subsequently resulted in [2] and a joint paper with Brenier.

The connection of turbulence to geometry, more specifically to the theory of bending surfaces, resulted in further collaboration within RA B. Motivated by the famous conjecture of Onsager concerning energy conservation for the Euler equations, we looked at the analogous problem of C^{1,<br>alpha} embeddings in the Nash-Kuiper setting. There is a well-developed theory of bending surfaces from elasticity theory and the calculus of variations, and thus it was natural to cooperate with Sergio Conti in RA B for this project. Thus, we developed a rather general framework for extending convex integration to Hölder-continuous derivatives. Our hope now is to transfer these ideas back to Euler.

The collaboration with C. De Lellis allowed us to attract him to Bonn for an extended stay in May 2009 for a lecture series.
Research Area A
My previous work on compensated compactness and Tartar's conjecture [3] is related to the celebrated conjecture of Morrey regarding quasiconvexity and rank-one convexity. The essence of the conjecture asks for a characterization of possible microstructures arising in planar deformations.

Although this conjecture is within the setting of the calculus of variations, an interesting connection arises with harmonic analysis and singular integral operators via the Beurling-Ahlfors transform. This connection leads to several exciting new developments in the field, in particular a possibly fruitful way to approach Morrey's conjecture via stochastic optimal control. In this direction, which forms a joint interest with Herbert Koch in RA A, we organized a Winter School in February 2010 with T. Iwaniec (Syracuse), M. Struwe (Zürich) and A. Volberg (Michigan), leading experts in the area, in order to jump-start the activity on this research topic in Bonn. The school was highly successful, with over 30 participants.

A. Volberg, one of the pioneers in the approach via stochastic optimal control, stayed in Bonn for 3 weeks for further discussions. Although we haven't achieved definite results so far, further cooperation with Volberg is planned, with the hope of combining techniques from the calculus of variations (laminates) with techniques from stochastic optimal control (martingale transforms).

Furthermore, we were able to win back M. Struwe, who is well-known for delivering lectures with a very high degree of clarity, for another extended stay in June 2010. He delivered a series of lecture on geometric variational problems and quantization.

Selected Publications

[1] Camillo De Lellis, Jr., László Székelyhidi
The Euler equations as a differential inclusion
Ann. of Math. (2) , 170: (3): 1417--1436
2009
DOI: 10.4007/annals.2009.170.1417
[2] Camillo De Lellis, Jr., László Székelyhidi
On admissibility criteria for weak solutions of the Euler equations
Arch. Ration. Mech. Anal. , 195: (1): 225--260
2010
DOI: 10.1007/s00205-008-0201-x
[3] Daniel Faraco, László Székelyhidi
Tartar's conjecture and localization of the quasiconvex hull in \Bbb R2× 2
Acta Math. , 200: (2): 279--305
2008
DOI: 10.1007/s11511-008-0028-1
[4] Kari Astala, Daniel Faraco, Jr., László Székelyhidi
Convex integration and the Lp theory of elliptic equations
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) , 7: (1): 1--50
2008
[5] Bernd Kirchheim, Jr., László Székelyhidi
On the gradient set of Lipschitz maps
J. Reine Angew. Math. , 625: : 215--229
2008
DOI: 10.1515/CRELLE.2008.095
[6] Jr., László Székelyhidi
Counterexamples to elliptic regularity and convex integration
The interaction of analysis and geometry
of Contemp. Math. : 227--245
Publisher: Amer. Math. Soc., Providence, RI
2007
DOI: 10.1090/conm/424/08104

Publication List

Awards

2000

Gibbs Prize, Oxford University, England, UK

2008

Elected member of the Junge Akademie (BBAW / Leopoldina)

2010

Oberwolfach Prize

2011

ERC Starting Grant

Selected Invited Lectures

2004

Analysis and Geometry (in honour of Yu. Reshetnyak), Novosibirsk, Russia

2008

Geometric Analysis, Elasticity and PDE (in honour of John M. Ball), Edinburgh, Scotland, UK

2010

Andrejewski Day, Bonn

2011

Applied Mathematics from Waves to Fluids (in honour of Claude Bardos), University of Nice, France

Offers

2007

Lecturer, Oxford, England, UK

2009

Associate Professor, EPF Lausanne, Switzerland

2010

Professor (W2), University of Bonn

Supervised Theses

  • PhD theses: 2, currently 2
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