Pure Maths Seminar Archive - Abstracts 2004
In 1988, Ehud Hrushovski invented a remarkable construction which produced counterexamples to several longstanding conjectures in model theory. Initially mysterious and seemingly produced `from nowhere,' these counterexamples are now seen as having connections with other parts of mathematics (random graphs and complex analytic functions, to name two). In this talk, I will give a survey of some aspects of these constructions and then describe a new, and very elementary, approach to them.
In model theory, stable structures are ones which carry a well-behaved notion of independence, generalising the notions of linear independence in a vector space, and algebraic independence in an algebraically closed field. There is a hierarchy of complexity of this independence, starting off with 1-basedness and CM-triviality. We will discuss under which conditions reducts of 1-based (respectively CM-trivial) stable structures are 1-based (respectively CM-trivial).
Following results of N.Ito and K.Ishikawa on finite groups with two sizes of conjugacy classes, we study how the size of the set of centralizer dimensions of a finite dimensional Lie algebra affects its structure. This is joint work with Marty Isaacs.
We survey a history and latest developments concerning extensions of Hilbert's Tenth Problem to subrings of number fields and the connection to some conjectures by Barry Mazur on topology of rational points.
Developing a model for an industrial strength software system (ideally during design time) is as essential as having a blueprint for large buildings. In addition, it is vital to be able to ensure the correctness of the design at the earliest stage possible. The most widely used currently practised methods for design validation is still largely simulation and testing. Although useful in the early stages of debugging, when there are still multiple bugs, it is less effective as the design becomes cleaner. Model checking is an automatic technique which allows us to formally check a model of a concurrent, communicating system by checking the state space of the model. Essentially this involves checking that a property - specified in a temporal logic - holds at every state that the system can be in. However, creating the model in the first place and describing the properties in temporal logic are non-trivial. Also there is the problem of state space explosion: the size of the state space grows exponentially with the number of components. Our research has involved modelling and verifying various protocols for fixed sized networks (telecommunications, (IEEE) FireWire tree identify, email) and developing mathematical techniques to generalize results to networks of any size and to reduce the state space for fixed size models. The first of these techniques uses a combination of abstraction and induction. The second involves symmetry reduction of the state space. If you think of a state space as a graph Γ(V,E), where V is the set of states and E the transitions between states, symmetry reduction involves model checking over the quotient graph &Gamma'(V',E'), where V' is the set of orbit representatives.
I shall present a set of axioms defining an 'analytic Zariski structure', as a generalisation of Hrushovski and Zilber's Zariski structures, and then I shall explore some consequences of the axioms.
I will then present a simple example constructed using Hrushovski's method of free amalgamation - in fact the first counterexample to Zilber's conjecture on strongly minimal sets.
The example will then be shown to be a non-trivial example of an analytic Zariski structure, proving that the notion on an analytic Zariski structure is a proper extension of that of a Zariski structure. This suggests that the example, previously thought to be non-classical, may have a prototype in some analytic space (i.e. be the zero set of some non-algebraic analytic function).
Let V be a countably infinite dimensional vector space over the finite field F. Then V is omega-categorical, and so are the projective space PG(V) and the projective symplectic, unitary and orthogonal spaces on V. Using a reconstruction method developed by M.Rubin we prove the following result: let M be one of the spaces above, and let N be omega-categorical and such that Aut(M) and Aut(N) are isomorphic as abstract groups. Then M and N are bi-interpretable.
My talk will have three parts:
1. Lie algebras (including Kac-Moody algebras)
2. Groups (including Coxeter groups)
3. Conformal algebras (including free associative and Lie conformal algebras).
"A Riemann surface X of positive genus with a given group G of symmetries can be deformed preserving the symmetries in 3g-3+n dimensions, where g is the genus of the quotient curve, Y=X/G and n is the number of branch points on Y." I will explain what this statement (essentially due to Riemann) means and what it should be replaced by for algebraic curves over fields of positive characteristic.
We prove a canonical partition theorem for certain classes of trees using a new universal method for this type of results. In particular we prove that for any function f on a given tree T there is a large subtree S of T such that the restriction of f on S is either constant, one-to-one or the level function. We consider this a generalization of Erdös-Rado canonical partition theorem for integers.
Given a number field k and a projective algebraic variety V, an age-old goal in number theory is to relate properties of the set V(k) of k-rational points on V to the intrinsic geometry of V. Whenever V(k) is Zariski dense in V, one intriguing aspect of this problem is revealed by trying to estimate the cardinality of points in V(k) that have height at most B, for given B>1. The purpose of this talk is to discuss a rather crude conjecture in the field, together with recent progress that has been made upon it.
This is a report on joint work with Manfred Einsiedler, where we show that many algebraic actions of higher-rank abelian groups on zero-dimensional groups are mutually disjoint. The proofs exploit differences in the entropy geometry arising from subdynamics and a form of Abramov-Rokhlin formula for half-space entropies.
This is a report on joint work with Paul Broussous, on a description of the Bruhat-Tits building of a p-adic classical group (with p not 2) in terms of lattices and on embeddings of such buildings. I will try to describe what these terms mean and why they are interesting.
A central problem in the area of mathematical physics known as "quantum chaos" is to understand the spacings between eigenvalues of the Laplacian on compact negatively curved surfaces. An analogous and related problem is to understand the spacings between lengths of closed geodesics on the surface. The purpose of the talk will be to discuss the two cases and present new results in the geodesic case. (This is joint work with Mark Pollicott.)
This will be an account of recent progress towards showing that every finitely generated recursively enumerably defined lattice-ordered group can be embedded in a finitely presented one. In the absence of any kind of amalgamation property, this is achieved by an exercise in infinite permutation groups and techniques.
There have been several different attempts to find the "right" context for studying model theory of analytical structures (e.g. classes of Banach or Hilbert spaces with additional structure), such as Henson's logic for Banach spaces, Ben-Yaacov's compact abstract theories, and others. We present a logic, which on one hand generalizes all these approaches (as well as the first order logic), and on the other hand keeps certain "good" properties of the first order logic, e.g. compactness, which allows us to develop (a topological version of) model theory, similar to the classical one. Our version of continuous logic captures many examples that were not accessible to the first order logic: Banach spaces, probability algebras, representations of a given locally compact group, etc.
Phylogenetic trees are used by biologists to represent evolutionary relationships between species. Although such trees have been used ever since Darwin introduced his theory for evolution, the possibility of constructing phylogenetic trees from DNA and related data has recently fuelled a cross-disciplinary interest in these structures. Indeed, various techniques in mathematics, computer science, and statistics are now regularly used in phylogenetics and, as a result, a new theory for phylogenetics is rapidly emerging. In this talk we will introduce some basic facts concerning phylogenetic trees, and discuss some recent results concerning their combinatorics.
Holomorphic correspondences are multivalued maps z→w from the Riemann sphere to itself, defined by polynomial relations p(z,w)=0. Examples are rational maps (and their inverses) and Kleinian groups. There are particularly striking families of correspondences known as matings between rational maps and Kleinian groups, which have behaviour under iteration conjugate to that of maps and groups on complementary regions of the sphere. I shall state structure theorems for such correspondences and show computer plots illustrating some of the dynamical phenomena exhibited - such as `circle-packing implosion'.
I will explain a new method for constructing Anosov diffeomorphisms on certain nilmanifolds. More specifically, I will prove by p-adic methods that the group SLn(Z) contains elements g such that the eigenvalues e1,...,en of g are positive real numbers and generate multiplicatively a free abelian group of rank n - 1. The result holds more generally for arithmetic subgroups of the Lie group SLn(R).