Introduction to additive combinatorics

Od wtorku do czwartku odbędzie się (w ramach wspólnego seminarium z Analizy Funkcjonalnej i Rachunku Prawdopodobieństwa) miniseria wykładów o addytywnej kombinatoryce i jej zastosowaniach., którą wygłosi prof. Tom Sanders (Cambridge University) Wyklady odbeda sie
- wtorek 11.05  godz. 15.15 w sali 106 IMPAN;
- sroda 12.05  godz. 12.15 w sali 5850 MIMUW;
- czwartek 13.05 godz. 12.15  w sali 5850 MIMUW.

Streszczenie:

Additive combinatorics is a relatively young field which has developed from a quantitative perspective, with many of the tools solving problems in number theory as well as providing new information on reviously qualitative results in harmonic analysis.  Our objective over this series of lectures is to give brief introduction to some of its main tools.

In the first lecture we plan to discuss Roth's theorem on arithmetic progressions.  This is one of the central problems of the field and provides a natural way to introduce many of the Fourier analytic techniques used.  There are many different proofs of this result applicable in different levels of generality and we hope to get as far as the regularity lemma which has wide ranging implications in combinatorics.

In the second lecture we plan to develop Freiman's theorem which provides a structure theory of approximate groups.  Approximate groups share many of  the properties of their
exact cousins but are far more populous, and so very useful for developing inductive arguments which seem on the face of it inaccessible.

Finally, in the third lecture we shall discuss some very recent non-abelian analogues of the preceding techniques and how they can be used to prove a quantitative version of the non-abelian idempotent theorem.  Throughout our emphasis shall be on techniques
so as to make many of the ideas of the area *as accessible as possible*.