Consider a vector field F in R^n that is definable in some o-minimal expansion S of the real field and has an isolated singularity at the origin. What can we say about the expansion of S by one or more trajectories of F? For example, if n=2 and there is a trajectory of F approaching the origin, then all (germs of) trajectories are definable in the pfaffian closure of S; if there is no spiralling trajectory, we can still say quite a few things in the analytic setting. But if n > 2, little is known in general: for one thing, what generalizes the notion of "spiralling" trajectory? I will give some elementary definitions and results in the case of 2-dimensional systems of linear ordinary differential equations.

Iterating the triple construction applied consecutively to n Boolean algebras, we introduce two finitely axiomatizable subclasses SAi(n) and SAs(n) of the class SA(n) of all Stone algebras of degree n with all the structure homomorphisms in their P-product representation injective or surjective, respectively. Then the class of all Post algebras of degree n is definitionally equivalent to the intersection of SAi(n) and SAs(n). We show that for each n greater than 2 the class SAi(n) is hereditarily undecidable while SAs(n) is decidable. As a consequence we obtain several (un)decidability results for various axiomatic classes of Stone algebras: among them the decidability of the class of all Stone algebras of degree n which are dually pseudocomplemented and form a dual Stone algebra under the operation of dual pseudocomplement, and undecidability of the class of all Stone algebras with Boolean dense set. On the other hand, the class of all finite members in SA(n) is decidable.

In a recent work, A.Conversano and A. Pillay have proved that for the topological universal cover of SL(2,R), where R is the real field, certain model-theoretic connected components are different. I will explain the meaning of this fact and present a generalization of their result to a central extension of SL(2,K) by integers, corresponding to some 2-cocycles (symplectic Steinberg symbols), where K is an arbitrary ordered field. This is a join work with K. Krupinski.

(Joint work with Roman Kontchakov, Yavor Nenov and Michael Zakharyaschev)
We consider the quantifier-free languages, Bc and Bc0, obtained by augmenting the signature of Boolean algebras with a unary predicate representing, respectively, the property of being connected, and the property of having a connected interior. These languages are interpreted over the regular closed sets of Euclidean space and, additionally, over the regular closed polyhedral sets of Euclidean space. The resulting logics are examples of formalisms that have recently been proposed in the Artificial Intelligence literature under the rubric "Qualitative Spatial Reasoning". I shall explain, in broad outline, why the satisfiability problem for Bc is undecidable over the regular closed polyhedra in all dimensions greater than 1, and why the satisfiability problem for both languages is undecidable over both the regular closed sets and the regular closed polyhedra (polygons) in the Euclidean plane. I shall present a contrasting result, however: the satisfiability problem for Bc0 is NP-complete over the regular closed sets in all dimensions greater than 2, while the corresponding problem for the regular closed polyhedra is ExpTime-complete. These facts indicate that Qualitative Spatial Reasoning is harder than its early proponents conceivably envisaged.

Voevodsky has introduced a new axiom, the Univalence Axiom, to be added to Per Martin-Lof's intensional dependent type theory (MTT), thereby initiating Homotopy Type Theory (HoTT). It is claimed that HoTT can provide a new approach to the foundations of mathematics, Univalent Foundations, based on an amalgam of intuitions from type theory, higher dimensional category theory and homotopy theory.
In my talk I will review MTT in order to give a precise statement of the Univalence Axiom. But first I will try to motivate the axiom mostly via a `Structure Identity Principle' which expresses that isomorphic structures are structurally identical; i.e. have the same structural properties.

Perfect exponential fields are a special class of structures that mimic complex exponentiation on one side, but on the other side feature nicer model-theoretical properties, such as having an uncountably categorical (infinitary) axiomatisation, or being quasi-minimal, and more easily understood algebraic behaviour, as they satisfy the statement of Schanuel's conjecture. Zilber conjectured that complex exponentiation is actually the perfect exponential field of power of the continuum.
A perfect exponential field can be explicitly constructed by manually defining the exponential function so that all compatible systems of equations have solutions, while Schanuel's conjecture is satisfied. After recalling the definition of perfect exponential field, and describing its relationship with other exponential fields, I will give some details about one of these construction methods, and I will sketch how it can be tweaked in order to prove that perfect exponential fields have involutions, i.e., automorphisms of order two akin to complex conjugation.

By replacement of real closed fields we can sometimes prove globally problems on real algebraic geometry when they are already proved locally by algorithm. This is the case for the second Lojasiewicz inequality. Let f be a polynomial function on R^n. Then there exist a semialgebraic neighborhood V of f^{-1}(0) in R^n and a number p such that 0< p< 1 and |f(x)|^p\le\sum_{i=1}^n|\partial f(x)/\partial x_i| for x in V.