Address:

Department of Mathematics,
Polytechnic Institute of NYU
6 Metrotech Center
Brooklyn, New York, 11201 USA.

Phone:

+1 (718) 260 3666

E-mail:

see the blackboard ...

Web-page:

Department of Mathematics

Published papers

• G. Berck, A. Bernig & C. Vernicos, Volume entropy of Hilbert geometries. Pacific J. Math. 245 (2010), n. 2, 201-225
  
  It is shown that among all plane Hilbert geometries, the hyperbolic plane has maximal volume entropy. More precisely, it is shown that the volume entropy is bounded above by 2/ (3−d) < = 1 , where d is the Minkowski dimension of the extremal set of the convex K. An explicit example of a plane Hilbert geometry with non-integer volume entropy is constructed. In arbitrary dimension, the hyperbolic space has maximal entropy among all Hilbert geometries satisfying some additional technical hypothesis. To achieve this result, a new projective invariant of convex bodies similar to the centro-affine area is constructed.
  
  
• G. Berck, Convexity of L_p-intersection bodies. Adv. Math. 222 (2009), n. 3, 920-936
  
  The family of Lp-intersection bodies of a centered convex body is an important concept of convex geometry, in particular in the theory of bodies-valued valuations. Up to normalization these bodies are also the polar p-centroids bodies used by Lutwak and Zhang to obtain new affine isoperimetric inequalities.
  The convexity of these bodies was known for p bigger than 1, as well as for p= -1. This last case was proved by Busemann and has deep consequences in his theory of areas in Finsler spaces. We give in this paper a unified proof of the convexity of these bodies, and this for the whole range of p's. The geometric core of the proof is an extension of Brunn's theorem to moments of convex bodies, and the distributions are used to unify the a priori different cases.
  
  
• G. Berck, Minimality of totally geodesic submanifolds in Finsler geometry. Math. Annalen 343 (2009), n. 4, 955--973
  
  Using the symplectic definition of the Holmes-Thompson volume we prove that totally geodesic submanifolds of a Finsler manifold are minimal for this volume. Thanks to well suited technics the minimality of totally geodesic hypersurfaces and 2-dimensional totally geodesic surfaces had already been proved. However the corresponding statement for the Hausdorff measure is known to be wrong even in the simplest case of totally geodesic 2-dimensional surfaces in a 3-dimensional Finsler manifold.
  
  
• J.C. Álvarez & G. Berck, What's wrong with the Hausdorff measure in Finsler spaces. Adv. Math. 204 (2006), n. 2, 647--663
  
  We construct a class of Finsler metrics in three-dimensional space such that all their geodesics are lines, but not all planes are extremal for their Hausdorff area functionals. This shows that if the Hausdorff measure is used as notion of volume on Finsler spaces, then totally geodesic submanifolds are not necessarily minimal, filling results such as those of Ivanov do not hold, and integral-geometric formulas do not exist. On the other hand, using the Holmes-Thompson definition of volume, we prove a general Crofton formula for Finsler spaces and give an easy proof that their totally geodesic hypersurfaces are minimal.
  
  

Preprints

• J.C. Álvarez, J.C. & G. Berck, Finsler surfaces with prescribed geodesics.

Notes

• G. Berck, Isoperimetrix and floating bodies.
  
  In this short note, we prove that up to normalization the isoperimetrix of Busemann's area is the limit of the family of floating bodies of the unit ball. This is quite straightforward and is not intended to be published. For a matter of taste, we have presented everything intrinsically, avoiding the use of any Euclidean structure.

Positions held

• October 2007- : Postdoc at the University of Fribourg
• October 2006-September 2007: Postdoc at the Centro de Giorgi, Scuola Normale Superiore, Pisa
• July 2006-August 2006: Postdoc at the University of Fribourg
• October 2005-May 2006: Postdoc at the University of Neuchâtel
• September 1998-July 2005: Assistant at the Université Catholique de Louvain

Education

• December 2004: Ph.D. thesis at the UCL (Belgium), Minimalité des sous-variétés totalement géodésiques en géométrie de Finsler. Advisors: Juan Carlos Álvarez Paiva and Pascal Lambrechts.
• June 1999: Master thesis (DEA) at the UCL, Connexions principales et problèmes d'équivalence. Advisor: Juan Carlos Álvarez Paiva.
• June 1998: Senior thesis at the UCL, Invariants de Vassiliev et représentation intégrale. Advisor: Juan Carlos Álvarez Paiva.

Teaching experience

• Fribourg
  • Supervision of the Free seminar: Combinatoric of Polytopes
  • Supervision of the Thematic seminar: Characteristic Classes
  • Exercise sessions for futur school teachers.
  • Exercise sessions : Analysis3, Vector and Complex analysis. (2nd year math)
  • Exercise sessions : Analysis4, Complex analysis, Distributions theory. (2nd year math)
• UCL: Exercise sessions for the following lessons (about 10h a week):
  • MATH1126: Affine and Euclidean Geometry, Differential Geometry of curves and surfaces. (1st year math students)
  • MATH2480: Introduction to Differential Geometry: manifolds, vector fields, differential forms, embeddings, ... (3rd year math students)
  • MATH1160A-B: One and two variables Real Analysis, Linear Algebra. (1st year natural sciences students)
  • FSA1304: Complex Analysis. (1st year civil engineer)
  • Math1-2: One variable Real Analysis, Linear Algebra. (1st year financial engineer)
  • Math3: Several variables Real Analysis. (2nd year financial engineer)
  • SESP1171: One variable Real Analysis. (1st year economical sciences students)

Divers

Organizer with E. Fernandes of the Concours Galileo at the UCL, a mathematical contest on the internet for high school students. There were about a hundred competitors at the two sessions.

Conference

Organizer with Andreas Bernig of the conference Integral & Finsler geometry which took place in Fribourg from the 21st to the 23rd of January 2009.

Workgroup

Organizer of the workgroup on the Busemann-Petty problem in Fribourg. A list of references may be downloaded here.

Povray

The background image of this webpage as well as the poster of the conference in Fribourg have been drawn using the free software Povray. A huge quantity of examples and explanations on this software may be found on Lohmüller's website.

You may download the scripts of these images clicking on the following links: webpage and poster, or simply the images themselves: background image and poster (2,5 Mo).

Dictionary

A very useful French-English dictionary on the web: here.