A new scheme for proving pseudoidentities from a given set Σ of

pseudoidentities, which is clearly sound, is also shown to be complete

in many instances, such as when Σ defines a locally finite variety, a

pseudovariety of groups or, more generally, of completely simple

semigroups. Many further examples when the scheme is complete are

given when Σ defines a pseudovariety V which is σ-reducible for the

equation x = y, provided Σ is enough to prove a basis of identities

for the variety of σ-algebras generated by V. This gives ample

# Seminars

The notion of structural stability is one of the oldest concepts in dynamical systems. Structurally stable diffeomorphisms are conjugate to all nearby diffeomorphisms and are known to be strongly related to dynamics with persistence of the shadowing property.

Cluster maps are birational maps arising from mutation-periodic quivers. These quivers are associated to (mutation-periodic) cluster algebras, introduced in 2002 by Fomin and Zelevinsky.

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.

[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.