Special vector configurations in geometry and integrable systems

The main objects of study of the thesis are two classes of special vector configurations appeared in the geometry and the theory of integrable systems. In the first part we consider a special class of vector configurations known as the V-systems, which appeared in the theory of the generalised Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. Several families of V-systems are known, but their classification is an open problem. We derive the relations describing the infinitesimal deformations of V-systems and use them to study the classification problem for V-systems in dimension 3. In particular, we prove that the isolated cases in Feigin-Veselov list admit only trivial deformations. We present the catalogue of all known 3D V-systems including graphical representations of the corresponding matroids and values of ν-functions. In the second part we study the vector configurations, which form vertex sets for a new class of polyhedra called affine B-regular. They are defined by a 3-dimensional analogue of the Buffon procedure proposed by Veselov and Ward. The main result is the proof of existence of star-shaped affine B-regular polyhedron with prescribed combinatorial structure, under partial symmetry and simpliciality assumptions. The proof is based on deep results from spectral graph theory due to Colin de Verdière and Lovász.