Pure Maths Seminar Archive - Abstracts 2003
The structure of the set of Salem numbers remains mysterious, despite prolonged study. The problem of constructing Salem numbers that have negative trace was first considered by Chris Smyth. He produced an infinite family of Salem numbers of trace −1, but none of trace −2 or below. Moreover, he showed that any Salem number having trace below −1 would necessarily have degree at least 20.
This talk describes some recent joint work with Chris Smyth in which we construct explicit families of Salem numbers (and Pisot numbers), including the first known examples that have trace below −1.
The talk briefly covers some of the early work in the development of Fourier Series and proceeds to describe a specific application of Fourier Analysis to the field of Molecular Structure.
An original example is presented to show how, given certain constraints, the inverse Fourier Transform may be utilised to refine the original Fourier Series Coefficients and so provide a more precise representation of the Series.
The question is posed as to the possible extension of the above to Pure Mathematics.
The latter part of the talk extends the above to a discussion of the Fourier Transform and Wiener-Kintchine equations in the context of Number Theory.
Lattices in Rn such as the E8 lattice and the Leech lattice admit various actions from rings of quadratic integers. We study some of these actions in detail, and obtain the amusing result that the Leech lattice is not a free module under a natural action of the integers of Q(√-23).
We describe a more conceptual approach to the construction of Euler characteristics of nearly perfect complexes which was recently introduced by Chinburg, Kolster, Pappas and Snaith. We then discuss certain applications of our approach in the context of Lichtenbaum's theory of Weil étale cohomology.
In order to study the stochastic behaviour of a deterministic dynamical system, one first needs to choose an invariant measure. Many dynamical systems have many different invariant measures, and it is not clear which one to use. A similar problem appears in statistical physics, where one needs to choose a measure (aka `ensemble') before one can calculate thermodynamic averages. The thermodynamic formalism adopts the postulates of statistical physics to the dynamical context. This theory was developed for compact topological Markov shifts by Sinai, Ruelle, Bowen and others. I will discuss the non-compact case, where new phenomena exist (infinite equilibrium measures, phase transitions, slow decay of correlations etc.).
Motivated by the importance of commuting mappings (for instance in applications in cryptography) one is led to look for maximal abelian subgroups of permutation groups. For the full symmetric group this question can easily be solved. But there are remarkable phenomena which get clearer if one considers groups of mappings respecting additional structure. A good example to illustrate such phenomena is the group of order preserving permutations on the real line. In this talk the classification (up to conjugation) of all maximal abelian subgroups is presented.
The subject of my talk is finite-dimensional Lie algebras over a field k of characteristic zero which admit a commutative polarization (CP). Among the many results and examples, it is shown that, if k is algebraically closed, the nilradical N of a parabolic subalgebra in An and Cn has such a CP. Using this fact a simple closed formula is derived for the index of N.
I shall define the modular homology of simplicial complexes. For shellable complexes we have an embedding theorem which gives a good description in the case of many important classes of complexes, including Coxeter complexes and buildings. Much is joint work with Valeriy Mnukhin and my former PhD students Steven Bell and Phil Jones. Valeriy's talk will follow on.
Among shellable complexes a certain class has maximal modular homology, and these are the so-called saturated complexes. A survey of their properties will be given and saturated complexes will be characterized via p-ranks of incidence matrices and via structure of their links. I will show that rank-selected subcomplexes of saturated complexes are also saturated.
Infinite designs are quite rare in the literature - most books on design theory have no mention of them at all. In this talk I will present a brief history of infinite designs, give a precise definition, and compare some properties of finite and infinite designs.
In this talk we present a new construction in simple theory and in particular show that two notions of strong types and Lascar strong types are different.
Two problems on Jordan block structure of the images of unipotent elements in modular irreducible representations of the classical algebraic groups and their connections with recognizing representations and linear groups will be considered.
1. Computing the second maximal Jordan block for elements of prime order p in arbitrary irreducible representations in characteristic p. The goal is to show that elements with a big difference (>6) between the sizes of the maximal Jordan block and the second maximal one are rare. For groups of rank >1 actually only the images of the natural representations contain such elements.
2. Finding all block sizes (or all sizes of a certain parity) for root elements and a wide class of p-restricted representations with highest weights locally small with respect to p. A similar problem for regular unipotent elements of a naturally embedded subgroup of type A2, odd block sizes, and representations of groups of type Ar. These are joint results with Anna Osinovskaya. Here the class of locally small weights with respect to p depends upon the type of a group and upon elements considered. For root elements in all cases locally small weights are determined in terms of certain linear functions of their values on two simple roots linked at the Dynkin diagram of a group.
All the results discussed can be easily transferred to representations of finite Chevalley groups in describing characteristic.
In set theory, combinatorial principles are statements about the structure of infinite ordinals. I will discuss some well-known principles as well as some generalisations that concern a partially ordered set known as Pκ(λ).
Somos-4 is the sequence with initial values 1,1,1,1 defined by the recursion sn+2 sn-2 = sn+1 sn-1 + sn2, for n∈Z. Surprisingly, this is an integer sequence, and there are some 1992 conjectures by Robinson about its properties when considered modulo a prime power. We show how these conjectures can be proved using the machinery of elliptic divisibility sequences.
The finite polylogarithms were introduced by Elbaz, Vincent and Gangl following Kontsevich (who defined the finite (1)-logarithm) as the function from Z/p to itself given by the polynomial lin(z)=∑k=1p−1zk/kn. Kontsevich noticed that the finite logarithm satisfied a 4-term functional equation, known as the fundamental equation of information theory, and that this equation is also satisfied by the "derivative" of the complex dilogarithm. It was later checked by Elbaz-Vincent and Gangl that a similar relation exists between the finite 2-logarithm and the derivative of the complex 3-logarithm.
Kontsevich made the conjecture that (an appropriate version of) the p-adic n+1-logarithm would have the property that its derivative would have a reduction modulo p that would be exactly the finite n-logarithm and hoped that this conjecture explains the relation between the two functional equations. Recently we have proved this conjecture. In the talk we will explain the complex p-adic and finite polylogarithms and the known functional equations. we will give the precise version of the theorem we proved and discuss the proof (which is very simple).
Let K be a complete and algebraically closed valued field of characteristic 0. Then the set of rational integers is positive existentially definable in the field M of meromorphic functions on K in the language Lz* of rings augmented by a constant symbol for the independent variable z and by a symbol for the unary relation ``the function x takes the value 0 at 0''. Consequently, the positive existential theory of M in the language Lz* is undecidable. In order to obtain these results we need to have a complete characterization of all analytic projective maps (over K) from an elliptic curve E minus a point to E, for any elliptic curve defined over the field of constants.
I will outline new results about the largest prime divisor P(n! + f(n)) of n! + f(n) for a nonzero polynomial f ∈ Z[x]. In particular I give a sketch of a completely elementary proof of the inequality limsupn→∞ P(n! + f(n))/n ≥ 2.5, which improves and generalises the lower bound limsupn→∞ P(n! + 1)/n > 2 of Paul Erdös and Cameron Stewart.
We review various structures in the theory of integrable systems, in particular Lax pairs, spectral curves and Hirota bilinear equations. The connection is then made between fourth order quadratic recurrence relations and integrable discrete dynamical systems corresponding to dynamics on an elliptic curve, and an associated functional equation. A conjecture for higher order recurrence sequences is presented.
Distance functions on C[x] are real-valued functions which measure the complexity of polynomials. There are many examples of distance functions on C[x], including the height and length, as well as the Mahler measure. We consider analytic and geometric properties of multiplicative distance functions including the Mahler measure and some variants.
I shall discuss so called rigid linear representations of a free group into GL(n,F), where F is the algebraic closure of a finite field. I shall outline a method to compute the character field of the representation which allows therefore to realize the representation over this subfield. This will next be applied to decide which groups PSL(5,q) and PSU(5,q) are Hurwitz. Recall that a finite group is called Hurwitz if it is generated by two elements x,y of order 3 and 3, respectively, such that xy is of order 7. Determining Hurwitz groups is one of the mysterious problems in group theory.
Mathematicians have hypnotised themselves into thinking that series should only be summed one way. Engineers want to sum the big terms first. Sometimes this works and sometimes it does not. This talk only requires a first course in mathematical methods.
A sequence of natural numbers is called exactly realizable if it occurs as the sequence of the number of periodic points for some map. Following Puri and Ward we give the conditions for a sequence to be exactly realizable and give some examples and number theoretical consequences. Finally we show that if a sequence is realizable by a map then it is realizable by a diffeomorphism of the 2-torus. Time permitting some partial analogues for flows will be given.
The guarded fragment of first order logic was introduced by Andreka, van Benthem and Nemeti in 1995. Each quantifier in the guarded fragment is restricted by an atomic formula containing all the variables which are free in the scope of the quantifier. Unlike first order logic, the guarded fragment of first order logic is decidable. I will sketch a proof of decidability for a slightly simpler fragment than the one introduced by Andreka, van Benthem and Nemeti. Then I will give a survey of results obtained for guarded fragments of other logics, some examples of what you can express in the guarded fragments, and some applications.
A Conjugate Quadrature Filter is a continuous function f :T → C that satisfies |f(z)|2 + |f(ωz)|2 + ... + |f(ωm-1z)|2 = 1 where m is an integer >1 and ω is a primitive mth root of unity. We discuss the problem of approximating a CQF by a Laurent polynomial CQF that preserves specified zeros of f in T. This problem arises in filter design and wavelet construction. We first develop relationships between CQFs and loop groups that consist of continuous functions from T into SU(m) under pointwise multiplication. Then we prove that such approximations are always possible by combining loop group properties with the differential geometry concept of jets and the topological concept of the degree of a mapping of a sphere into itself.