30 research outputs found
Approximate algebraic structure
We discuss a selection of recent developments in arithmetic combinatorics
having to do with ``approximate algebraic structure'' together with some of
their applications.Comment: 25 pages. Submitted to Proceedings of the ICM 2014. This version may
be longer than the published one, as my submission was 4 pages too long with
the official style fil
The apparent structure of dense Sidon sets
The correspondence between perfect difference sets and transitive projective
planes is well-known. We observe that all known dense (i.e., close to
square-root size) Sidon subsets of abelian groups come from projective planes
through a similar construction. We classify the Sidon sets arising in this
manner from desarguesian planes and find essentially no new examples, but there
are many further examples arising from nondesarguesian planes. We conjecture
that all dense Sidon sets arise in this manner. We also give a brief bestiary
of somewhat smaller Sidon sets with a variety of algebraic origins, and for
some of them provide an overarching pattern.Comment: 16 page
A bound on the multiplicative energy of a sum set and extremal sum-product problems
In recent years some near-optimal estimates have been established for certain
sum-product type estimates. This paper gives some first extremal results which
provide information about when these bounds may or may not be tight. The main
tool is a new result which provides a nontrivial upper bound on the
multiplicative energy of a sum set or difference set.Comment: 13 page
The structure theory of set addition revisited
In this article we survey some of the recent developments in the structure
theory of set addition.Comment: 38p
Discrete Geometry
The workshop on Discrete Geometry was attended by 53 participants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Abstract regular polytopes: recent developments. (Peter McMullen) Counting crossing-free configurations in the plane. (Micha Sharir) Geometry in additive combinatorics. (József Solymosi) Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) • Forbidden patterns. (János Pach) • Projected polytopes, Gale diagrams, and polyhedral surfaces. (Günter M. Ziegler) • What is known about unit cubes? (Chuanming Zong) There were 16 shorter talks in the afternoon, an open problem session chaired by Jesús De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (Jürgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants
Combinatorics and Probability
For the past few decades, Combinatorics and Probability Theory have had a fruitful symbiosis, each benefitting from and influencing developments in the other. Thus to prove the existence of designs, probabilistic methods are used, algorithms to factorize integers need combinatorics and probability theory (in addition to number theory), and the study of random matrices needs combinatorics. In the workshop a great variety of topics exemplifying this interaction were considered, including problems concerning designs, Cayley graphs, additive number theory, multiplicative number theory, noise sensitivity, random graphs, extremal graphs and random matrices