Table Of ContentProgress in Mathematics
Volume 169
Series Editors
H. Bass
J.Oesterle
A. Weinstein
European Congress
of Mathell1atics
Budapest, July 22-26, 1996
Volume II
A. Balog
G .O.H. Katona
A. Recski
D. Sza'sz
Editors
Springer Basel AO
Editors:
A. Balog O.O.H. Katona
Mathematical Institute Mathematical Institute
Hungarian Academy of Sciences Hungarian Academy of Sciences
Realtanoda str. 13-15 Realtanoda str. 13-15
H-I053 Budapest Hungary H-1053 Budapest Hungary
A. Recski D. Sza'sz
Mathematical Institute Mathematical Institute
Technical University of Budapest Hungarian Academy of Sciences
H-1521 Budapest Hungary Realtanoda str. 13-15
H-1053 Budapest Hungary
1991 Mathematics Subject Classification 00B25
A CIP catalogue record for this book is available from the Library of Congress,
Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
European Congress of Mathematics <2, 1996, Budapest>:
European Congress of Mathematics: Budapest, July 22 - 26, 1996 IA. Balog ...
ed. - Base! ; Boston; Berlin: Birkhăuser.
ISBN 978-3-0348-9819-5 ISBN 978-3-0348-8898-1 (eBook)
DOI 10.1007/978-3-0348-8898-1
This work is subject to copyright. All rights are reserved, whether the whole or part ofthe
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© 1998 Springer Basel AG
Originally published by Birkhiiuser Verlag,Base1, Switzerland in 1998
Softcover reprint ofthe hardcover Ist edition 1998
Printed on acid-free paper produced of chlorine-free pulp. TCF 00
ISBN 978-3-0348-9819-5
987654321
Table of Contents ofVolume I
Speeches
Address by J.-P. Bourgignon IX
Address by G. Demszky XI
Address by A. Goncz xu
Address by G. Katona . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xiii
List of talks xv
Prizes of the European Mathematical Society xvii
Contributions
N. Alon
Randomness and pseudo-randomness in discrete mathematics 1
L. Ambrosio
Free discontinuity problems and special functions with
bounded variation 15
K. Astala
Recent connections and applications of planar quasiconformal
mappings 36
R. Benedetti
A combinatorial approach to combings and framings
of 3-manifolds 52
Ch. Bessenrodt
Algebra and combinatorics 64
F. Bethuel
Some recent results for the Ginzburg-Landau equation 92
P. Bjorstad
Mathematics, parallel computing and reservoir simulation 100
E. Bolthausen
Large deviations and perturbations of random walks
and random surfaces 108
J. Bricmont, A. Kupiainen
Renormalization group for fronts and patterns 121
vi Table of Contents of Volume I
D. Burago
Geometry of tori: Riemannian versus Finsler? 131
L. Caporaso
Counting curves on surfaces: A guide to new techniques
and results 136
U. Dierkes
Minimal surfaces in singular spaces 150
1. Dynnikov
Surfaces in 3-torus: Geometry of plane sections 162
L.H. Eliasson
One-dimensional quasi-periodic Schrodinger operators -
dynamical systems and spectral theory 178
W.T. Cowers
Banach spaces with few operators 191
H. Hedenmalm
Recent developments in the function of the Bergman space 202
A. Huber
Extensions of motives 218
J. Kaczorowski
Boundary values of Dirichlet series and the distribution of primes 237
J. Kollar
Low degree polynomial equations: Arithmetic, geometry
and topology 255
D.O. Kramkov, A.N. Shiryaev
Sufficient conditions of the uniform integrability of
exponential martingales 289
C. Lescop
On the Casson invariant 296
R. Marz
EXTRA-ordinary differential equations: Attempts to an
analysis of differential-algebraic systems 313
Table ofContents ofVolume II
Contributions (continued)
J. Matousek
Geometric set systems 1
D. McDuff
Recent developments in symplectic topology 28
A.S. Merkurjev
K-theory and algebraic groups 43
V. Milman
Surprising geometric phenomena in high-dimensional
convexity theory 73
St. Muller
Microstructures, phase transition and geometry 92
T. Nowicki
Different types of non-uniform hyperbolicity for interval maps
are equivalent 116
E. Olivieri, E. Scoppola
Metastability and typical exit paths in stochastic dynamics 124
v. P. Platonov
Rationality problems for group varieties 151
L. Polterovich
Precise measurements in symplectic topology 159
J. Poschel
Nonlinear partial differential equations, Birkhoff normal forms,
and KAM theory.................................................... 167
L. Pyber
Group enumeration and where it leads us 187
N. Simanyi
Studying dynamical systems with algebraic tools 200
J.P. Solovej
Mathematical results on the structure of large atoms 211
A. Stipsicz
Geography of irreducible 4-manifolds 221
viii Table of Contents of Volume II
G. Tardos
TI-ansversals of d-intervals - comparing three approaches.. . . . . . . ... .. 234
J.-P. Tignol
Algebras with involution and classical groups 244
A.P. Veselov
Huygens' principle and integrability 259
E. Zuazua
Some problems and results on the controllability of partial
differential equations 276
Round Tables
(A) Electronic literature in mathematics
B. Wegner (chair); A. DeKemp, A. Bardelloni, J.-P. Allouche 315
(B) Mathematical Games
D. Singmaster (chair); A. Fraenkel, M.E. Larsen, T. Szentiv6nyi .... 338
(D) Women and mathematics
K. Hag (chair); S. Paycha, R. Piene, D. McDuff, R. Miirz 347
(E) Public image of mathematics
R. Bulirsch (chair); M. Chaleyat-Maurel, Gy. Staar,
St. Deligeorges 376
(G) Education
V.L. Hansen (chair); Ch. Mauduit, J.-P. Boudine,
M. Laczkovich, L. P6sa 380
Progress in Mathematics, Vol. 169, © 1998 Birkhiiuser Verlag Basel/Switzerland
Geometric Set Systems
JIM MATOUSEK*
Department of Applied Mathematics, Charles University
Malostranske nam. 25, 118 00 Praha 1, Czech Republic
e-mail:
2 Jifi Matousek
For example, it is not difficult to check that the VC-dimension of the set
system H is 3 (no 4-point set can be shattered). Determining the VC-dimension of
T exactly requires some work, but using simple tools presented below it is easily
seen that this dimension is bounded by a constant. Similarly set systems defined
by other simple geometric figures in a Euclidean space have typically a bounded
VC-dimension (a precise formulation will be given later).
On the other hand, the system of all convex sets in the plane, say, has an
infinite VC-dimension.
The notion now commonly called VC-dimension was introduced by Vapnik
and Chervonenkis [VC71] 1. Numerous applications and extensions of the VC-
dimension concept have been developed in statistics (in the theory of so-called em-
pirical processes; some relevant references are [Vap82], [Dud84]' [GZ84]' [Dud85],
[AT89], [PoI90]), in learning theory (where VC-dimension is one of the main con-
cepts; e.g., [BEHW89], [AB92], [Hau92], [KW93], [ABCBH93], [DHS94]), but also
for example in program testing [RV96].
In the combinatorics of hypergraphs, set systems of VC-dimension dean
be viewed as a class of hypergraphs with a certain forbidden subhypergraph (the
complete hypergraph on d+ 1 points), which puts this topic into a broader context
of extremal hypergraph theory (see for instance [Fra83], [WF94]' [DSW94]). Here
we do not consider these areas.
This survey is mainly focused on the directions of the author's own work;
we review some general results of a combinatorial nature about set systems of
bounded VC-dimension, and present applications in geometric discrepancy the-
ory, combinatorial geometry, and computational geometry. We also mention more
geometric notions and results (which have no good analogue for arbitrary set sys-
tems of bounded VC-dimension), namely cuttings (sec. 4.) and simplicial partitions
(sec. 5.).
2. Set systems of bounded VC-dimension
Shatter functions. These are parameters of a set system related to VC-dimension
but often giving more information and easier to work with.
The primal shatter function of a set system (X, S) is a function, denoted by
Jrs, whose value at m (m = 0,1,2, ... , IXI) is defined by
Jrs(m) = max ISIAI.
A~X, IAI=m
In words, Jrs(m) is the maximum possible number of distinct intersections of the
sets of S with an m-point subset of X. Yet another way to understand the primal
shatter function is via the incidence matrix, M, of the set system S (with rows
indexed by sets of S and columns indexed by points of X): Jrs(m) is the maxi-
mum number of distinct rows appearing in any m-column submatrix of M. An
d
alternative definition of VC-dimension of S is then sup{d; Jrs(d) = 2 }.
1Under different names, this also appears in other papers ([Sau72),[She72]) but the work
[Ve7l] was probably the most influential for the subsequent developments.
