Asymptotic Geometric Analysis, Convex and Discrete Geometry, and Combinatorics

The proposed CRG is expected to serve as a major impetus for deepening connections among our individual research programmes, which oftentimes exist at intersections of the main areas. Therefore it is natural to look for a more organized opportunity to solidify and/or jump-start collaborations, which then have all the needed potential to become longterm.

This can also be seen from the research objectives associated with the proposed CRG, which we described in full detail above. In summary, these include:

• development, from essentially an `infancy' level, of new lines of results on Discrete Structures and settings (Lattice Sets, Discrete Tomography), which will be inspired from, and echoing, very classical (and very well understood) results in Convex Geometry and/or Asymptotic Geometric Analysis;
• a systematic study of some of the techniques whose applicability has already been shown to traverse the main areas they come from, so that we can expand on them and refine them further (with  specific, suitable problems that each of us has worked on as a guide and motivation): e.g. probabilistic arguments for covering and packing, Stochastic Analysis techniques, Fourier Analysis techniques that are relevant to classical Convex Geometry problems, as well as to discrete counterparts of those;
• a systematic analysis of the role of symmetry in problems from AGA, DG and CG, where we observe major gaps in known conclusions between symmetric and non-symmetric objects, contrary to conjectured outcomes and/or our intuition, with the purpose of developing some methodological frameworks here (as appropriate in each main area).

It is clear that any concrete progress on specific problems in any of these broad directions will also benefit other colleagues of ours, junior or senior, in our wider communities.