In this talk I will explain how I construct algebraic varieties using the computer algebra system Magma. Most examples will be about plane curves and surfaces in 3-dimensional space.

# Seminars

[This is joint work with Conceição Nogueira and M. Lurdes Teixeira.] The semidirect product is a fundamental operation in the theory of pseudovarieties of semigroups. In turn, the pseudovarieties of the form V*D, where D is the pseudovariety of all finite semigroups whose idempotents are right zeros, are among the most studied semidirect products. The concept of tameness of a pseudovariety was introduced by Almeida and Steinberg as a tool for proving decidability of the membership problem for semidirect products of pseudovarieties.

Framed sheaves over surfaces first appeared as a generalisation of framed SU(r) instantons on S4. Later, Uhlenbeck provided a compactification for the moduli space of framed SU(r) instantons which through the Donaldson correspondence and the work of Li and Morgan produced the so called Donaldson-Uhlenbeck compactification of the moduli space of framed sheaves. We want to focus on a further generalisation to the moduli space of framed sheaves on Deligne-Mumford stacks and construct an analogous compactification. This is work in progress with U. Bruzzo.

Symplectic geometry and combinatorics are strongly intertwined due to the existence of Hamiltonian torus actions. These actions are associated with a special map (called the moment map) which "transforms" a compact symplectic manifold into a convex polytope. We will concentrate on the special class of reflexive polytopes which was introduced by Batyrev in the context of mirror symmetry and has attracted much attention recently. In particular, we will see how the famous "12 and 24" properties in dimension 2 and 3 can be generalized with the help of symplectic geometry.

Branes are special submanifolds of hyperkähler manifolds that play an important role in string theory, particularly in the Kapustin–Witten approach to the geometric Langlands program, but which also are of intrinsic geometric interest. More precisely, a brane is a submanifold of a hyperkähler manifold which is either complex or Lagrangian with respect to each of the three complex structures or Kähler forms composing the hyperkähler structure.

Taking as departure point an article by Cameron, Gadouleau, Mitchell and Peresse on maximal lengths of subsemigroup chains, we introduce the subsemigroup complex H(S) of a finite semigroup S as a (boolean representable) simplicial complex defined through chains in the lattice of subsemigroups of S. The rank of H(S) is the above maximal length minus one and H(S) provides other useful invariants concerning the lattice of subsemigroups of S. We present a research program for such complexes, illustrated through the particular case of combinatorial Brandt semigroups.

Given a totally nonholonomic distribution of rank two ∆ on a three-dimensional manifold M, it is natural to investigate the size of the set of points Xx that can be reached by singular horizontal paths starting from a same point x ∈ M. In this setting, the Sard conjecture states that Xx should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero.

The moduli space M_C of Higgs bundles over a complex curve X admits a hyperkaehler metric: a Riemannian metric which is Kaehler with respect to three different complex structures I, J, K, satisfying the quaternionic relations. If X admits an anti-holomorphic involution, then there is an induced involution on M_C which is anti-holomorphic with respect to I and J, and holomorphic with respect to K. The fixed point set of this involution, M_R, is therefore a real Lagrangian submanifold with respect to I and J, and complex symplectic with respect to K, making it a so called AAB-brane.

Game theory studies the interactions among agents that try to maximize their payoffs by choosing from a set of actions. When this interaction occurs over time, the agents can change their choice of action depending on what they believe is the best action at any given time. This process constitutes a learning mechanism and can be described by various dynamical systems. Different dynamical systems arise as a result of different forms of belief update.

Let X be a smooth projective curve of genus at least two. Let G be an almost simple affine algebraic group. The universal principal G-bundle is called the Poincaré bundle. It is a principal G-bundle on the product of the curve and the moduli stack of principal G-bundles. We prove that it is stable with respect to any polarization (joint work with I. Biswas and N. Hoffmann).