Galois groups of Schubert problems via homotopy computation

Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois groups in the Schubert calculus. For example, we show by direct computation that the Galois group of the Schubert problem of 3-planes in C^8 meeting 15 fixed 5-planes non-trivially is the full symmetric group S_6006.Comment: 17 pages, 4 figures. 3 references adde

Factor-Group-Generated Polar Spaces and (Multi-)Qudits

Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce gradually necessary and sufficient conditions to be met in order to carry out the following programme: Given a group \vG, we first construct vector spaces over \GF(p), $p$ a prime, by factorising \vG over appropriate normal subgroups. Then, by expressing \GF(p) in terms of the commutator subgroup of \vG, we construct alternating bilinear forms, which reflect whether or not two elements of \vG commute. Restricting to $p=2$, we search for ``refinements'' in terms of quadratic forms, which capture the fact whether or not the order of an element of \vG is $\leq 2$. Such factor-group-generated vector spaces admit a natural reinterpretation in the language of symplectic and orthogonal polar spaces, where each point becomes a ``condensation'' of several distinct elements of \vG. Finally, several well-known physical examples (single- and two-qubit Pauli groups, both the real and complex case) are worked out in detail to illustrate the fine traits of the formalism.Comment: 20 pages, 6 figures, 1 table; Version 2 - slightly polished, updated references; Version 3 - published version in SIGM

On some differential-geometric aspects of the Torelli map

In this note we survey recent results on the extrinsic geometry of the Jacobian locus inside $\mathsf{A}_g$. We describe the second fundamental form of the Torelli map as a multiplication map, recall the relation between totally geodesic subvarieties and Hodge loci and survey various results related to totally geodesic subvarieties and the Jacobian locus.Comment: To appear on Boll. UMI, special volume in memory of Paolo de Bartolomei