Approximate groups and projective geometries.

Add to your list(s) Download to your calendar using vCal

• Emmanuel Breuillard (Cambridge)
• Friday 17 May 2019, 13:00-13:50
• MR3 Centre for Mathematical Sciences, level -1.

If you have a question about this talk, please contact HoD Secretary, DPMMS.

The structure of finite subsets A of an ambient algebraic group G, which do not grow much under multiplication, say |AA|<|A|^{, is well understood after the works of Hrushovski, Pyber-Szabo and Breuillard-Green-Tao on approximate subgroups of algebraic groups. A more general question, tackled by Elekes and Szabo, asks for the structure of Cartesian products A_1 \times … \times A_n of finite subsets of size N of an arbitrary d-dimensional algebraic variety W, with large (i.e. >N}{\dim V/d}) intersection with a given subvariety V \leq W^n (the case n=3, W=G, A_i=A, V={(x,y,xy)} corresponds to the above mentioned approximate group problem). In joint work with Martin Bays, we completely characterize the algebraic varieties V that can admit a (general position) family of such finite Cartesian products with large intersection. We show that they are in algebraic correspondence with a subgroup of a commutative algebraic group endowed with an extra structure arising from a certain division ring of group endomorphisms. The proof makes use of the Veblen-Young theorem on abstract projective geometries, generalized Szemeredi-Trotter bounds and Hrushovski’s formalism of pseudo-finite dimensions.

This talk is part of the SEEMOD Workshop 9 series.

This talk is included in these lists:

• All CMS events
• CMS Events
• DPMMS info aggregator
• Hanchen DaDaDash
• Interested Talks
• MR3 Centre for Mathematical Sciences, level -1
• SEEMOD Workshop 9
• bld31

Note that ex-directory lists are not shown.