Pure Maths Seminar Archive - Abstracts 2006
The zeta function of a dynamical system is an important invariant which features prominently in studies of orbit growth. Some useful periodic point calculations will be highlighted which allow explicit calculation of zeta functions for some algebraic systems. Also, for certain algebraic actions of higher rank, periodic point formulae can be established to enable meaningful comparison of zeta functions arising from elements of the action.
Hurwitz groups are the finite groups attaining the upper bound of 84(g-1) for the number of automorphisms of a compact Riemann surface of genus g > 1; equivalently, they are the nontrivial finite images of the (2,3,7) triangle group, which is the even subgroup of the rank 3 Coxeter group [3,7], a group of isometries of the hyperbolic plane. It appears that the analogous groups in dimension 3 are the finite quotients of the rank 4 Coxeter group [3,5,3], which acts on hyperbolic 3-space. I shall describe joint work with Alexander Mednykh and with Cormac Long on the finite quotients of this and some other rank 4 Coxeter groups, and on the 3-manifolds (such as the Poincaré dodecahedral space and the Weber-Seifert space) associated with them.
Phan's theorems, alongside with the Curtis-Tits theorem, were an important identification tool in the classification of finite simple groups. We describe a geometric approach that allows easier and more conceptual proofs of Phan's original theorems and expanding this theory to new classes of groups.
The Brauer algebras were introduced by Brauer in the study of the representation theory of the symplectic and orthogonal groups. In this talk, I will consider the Brauer algebras as a tower of algebras and explain how we can easily obtain a labelling set for the simple modules in this way. I will then expose a combinatorial condition which describes precisely when two simple modules belong to the same block of the Brauer algebra. (Joint work with P.Martin and A.Cox).
Let G be a finite group, acting linearly on the polynomial ring A := F [ x1, ... ,xd ] over a modular field F of characteristic p dividing the group order |G|. The ring of invariants AG := { f ∈ A | g(f) = f } is the central object of study in invariant theory. Many questions to which the answer is known in the "classical case" where F has characteristic zero, are wide open in the modular case. In the talk I will deal with some of those questions in particular dealing with the constructive complexity of AG. Among other things I will report on a new construction procedure for AG using ideas from number theory.
The philosophy of the talk is finding symmetry in chaos. More precisely, finding large symmetric monochromatic subsets in colorings of algebraic or geometric objects. In this respect we discover that the classical oriental "yin-yan" in a sense is an extremal coloring of the disk.
Philosophers are generally somewhat wary of the hints of number mysticism in the reports about the beliefs and doctrines of the so-called Pythagoreans. It's not clear how much Pythagoras himself (as opposed to his later followers) indulged in speculation about numbers, or in more serious mathematics. But the Pythagoreans whom Aristotle discusses in the Metaphysics had some elaborate stories to tell about how the universe could be explained in terms of numbers - not just its physics but perhaps morality too. Was this just fanciful speculation? Is it muddled as a theory of causality, as Aristotle suggests? I shall try to rehabilitate the passion for numbers by linking it with the notions of harmony and proportion in other thinkers who have a higher credibility factor in the philosophical stakes, and by showing that the desire to reduce quality to quantity, and to discover an exact science that can explain human life and meaning, is a serious philosophical passion that doesn't easily go away. And, after all, why should it?
That a connection exists between random matrix theory and number theory has been known ever since the 70s when H.L.Montgomery and F.J.Dyson discovered over a cup of tea that the statistics of the zeros of the Riemann zeta function calculated by the former agree with the eigenvalue distribution of random matrices examined by the latter. This has led to important and recent advances in number theory, through the use of random matrix theory, and a selection of these will be discussed in this talk.
The correlations of the eigenvalues of unitary matrices, chosen uniformly with respect to Haar measure on U(N), are observed to closely match the correlations of the zeros of the Riemann zeta-function and other L-functions. In addition, the distributions of the eigenvalues of other compact classical matrix groups approximate with high accuracy those of the zeros of various families of L-functions. Furthermore, the characteristic polynomials of the matrices provide a good model of the L-functions themselves. These ideas have lead to the introductions of techniques that have allowed to make predictions and obtain important results which could not have been achieved otherwise. In this talk we are concerned with a particular class of L-functions that have a functional equation but do not obey the Riemann hypothesis. Thus, not all their zeros will lie on a line. We suggest that such L-functions can be modelled by random self-reciprocal polynomials and make predictions on the correlations of sets of zeros that lie on the critical line.
The mathematical study of dependence (like the dependence of a field of a database on other fields) leads to difficult questions that cannot be solved by algorithmic means. This is true even if the concept of an algorithm is vastly generalized. I formulate the problem in the framework of set theory and discuss maximality principles that lead to partial solutions.
Let F be a p-adic field, and let D be a finite dimensional central division algebra over F. The study of the unitary dual of GL(n,F) has been achieved by Marko Tadic in 1986. Then Tadic gives a conjectural classification of the unitary dual of the group GL(m,D) and a description of the unitary Jacquet-Langlands correspondence, which relates the unitary dual of GL(n,F) to the one of GL(m,D). This description is based on five statements denoted by U0,...,U4. Statements U1,...,U4 have been proved. In this talk, I will give a proof of the remaining Conjecture U0, which asserts that any irreducible unitary representation of a Levi subgroup of GL(m,D) induces irreducibly to GL(m,D). This proof is based on type theory of Bushnell and Kutzko.
Arguably, the two most celebrated achievements of the 20th Century in the field of the Diophantine equations have been Baker's theory of linear forms in logarithms, and Wiles' proof of Fermat's Last Theorem. We explain how ingredients from both of these can be combined to solve several infamous exponential Diophantine equations. For example we show that the only perfect powers in the Fibonacci sequence are 0, 1, 8, 144, and the only solutions to the equation x2+7=ym have x=1,3,5,7,11,181. This is joint work with Bugeaud and Mignotte.
The Mahler measure of an integral polynomial of several variables is the geometric mean of its absolute value over the multiplicative d- dimensional torus. For a monic 1-variable polynomial, the definition reduces to the product of the absolute values of those zeros that lie outside the unit circle. In 1933, D. H. Lehmer asked whether there are integral polynomials with Mahler measures arbitrarily close to, but greater than 1. The closest known value is approximately 1.17628, the Mahler measure of a degree 10 polynomial discovered by Lehmer. A classical invariant associated to a link of d components is its d-variable Alexander polynomial. Lehmer's polynomial arises as the Alexander polynomial of a knot, and many links with nice properties have Alexander polynomials of small Mahler measure. Using results of Lind, Schmidt and Ward about algebraic dynamical systems, we give a topological interpretation to the Mahler measure of the Alexander polynomial in terms of homology of finite covers. (Joint work with Daniel Silver)
The Mahler measure of an n-variable polynomial P is the integral of log|P| over the n-dimensional unit torus Tn with the Haar measure. For one-variable polynomials, this is a natural quantity that appears in different problems such as Lehmer's question. While the algebraic nature of the values of the Mahler measure for one-variable polynomials with integral coefficients is well understood, the knowledge of the several-variable case is reduced to a collection of examples. In this talk I will show how to produce a family of n-variable polynomials whose Mahler measure we can relate to combinations of values of the Riemann zeta function and Dirichlet L-series.
I will present the basic concepts of differential Galois theory, an analogue of the usual Galois theory where polynomial equations are replaced by linear differential equations. I will describe the analogue of Galois extensions, the construction of the Galois group, and show how one can state precisely (and prove) statements about non-solubility of particular differential equations in terms of "elementary functions".
We introduce the idea of an iterated function system (ifs) and define self-similar and self-affine sets as attractors of certain ifs. For self-similar sets satisfying a separation condition, the Hausdorff dimension is well known. We show that the situation for self-affine sets is much more complicated. There is a formula introduced by Falconer which gives the dimension of a 'large class' of self affine sets, assuming the norms of the functions are bounded by ½. Jointly with Mark Pollicott and Karoly Simon we show that, by adding random errors at each stage, we can relax this assumption.
Difference sets and finite projective planes are related combinatorial structures which have been studied for a very long time. A famous and long-standing conjecture is that a projective plane which admits an automorphism group that is transitive on points is "Desarguesian" and, hence, known. In a remarkable paper in 1998, Chat Yin Ho proved that if a projective plane admits more than one abelian point-transitive group then that plane is Desarguesian. We examine Ho's proof in a bid to generalise from the abelian to the nilpotent situation. In the process we encounter a number of fascinating concepts from group theory, including the Generalized Fitting Group and the Bender-Suzuki theorem.
The talk will not assume any deep group theoretic or combinatorial background!
The set desc(α) of descendants of a vertex α in a directed graph (digraph) is the set of vertices reachable by a directed path from α. We investigate the set desc(α) in an infinite and highly arc transitive digraph D whose automorphism group is vertex-primitive. From unpublished work of Peter Neumann and Rögnvaldur Möller it was known that if the out-valency of D is a prime number p then the rooted digraph desc(α) must be a rooted tree.
We formulate three conditions which the digraph desc(α) must satisfy and show that if a digraph Γ satisfies these conditions and has out-valency p2, where p is a prime number, then it must be a (rooted) tree. Moreover, we create examples of digraphs Γ of out-valency p3 which satisfy the three conditions but are not trees; and then, by modifying the methods of David Evans, we show that these can be realised as the set of descendants in a vertex-primitive highly arc transitive digraph D. The same can be done with digraphs of out-valency 6 that we create.
Also, we show that there are only countably many digraphs satisfying the three conditions; and we show that if Γ is a digraph satisfying the three conditions then it has infinitely many ends.
Suppose R is the ring of algebraic integers in an algebraic number field k. The Kubota symbol is a homomorphism from a subgroup of finite index in SLn(R) to the group of roots of unity in R. This map is related to both the reciprocity law for k and the congruence subgroup problem for SLn(R). I'll explain some of the background to this and describe a new way of defining the Kubota symbol, which makes its properties more transparent.
Let P(x,y) be a positive polynomial. Following Hilbert, P(x,y) is a sum of four squares in R(x,y). However, nobody knows how to effectively determine whether or not a given polynomial is a sum of three squares in R(x,y).
In this talk we give a family of positive polynomials of degree 8 in y that are not a sum of three squares in R(x,y). This problem can be reformulated as the search for an antineutral point of the jacobian variety associated to some hyperelliptic curve C.
Following a method invented by Cassels, Ellison and Pfister, we check the nonexistence of an antineutral torsion point of the jacobian associated to C, and we use a 2-descent to show the triviality of the R(x)-Mordell-Weil rank of this jacobian variety.
Say you wanted to find primes with your telephone number embedded in the digits in some order. Is it possible? If you are writing numbers to base 10 the final digit must be 1, 3, 7 or 9 to get a prime, so you must be careful if your phone number ends in any other digit. Forget the 0 at the beginning of your number and you are likely to get a 10 digit number. Is it true that given k, sufficiently large, you can preassign any 10 digits (with the aforementioned caveat on the final digit) and find primes? We show this is true: indeed given any integer t there is a k = k(t) such that we can preassign any t digits so long as the final digit (if preassigned) is 1, 3, 7 or 9.
We prove that if f is a function definable in an o-minimal expansion of a real closed field and the structure (C,+,f) is strongly minimal then, up to conjugation by a real matrix f is a constructible function (i.e. - after correcting finitely many accidental values, f is conjugate to an algebraic function). The talk will not assume background in model theory.
Let C0 and W be two first-order structures and π0:C0→W a free finite cover with kernel K0. One of the most significant problems concerning this area of model theory, which is called the Lifting Problem, is to distinguish which closed subgroups of K0 arise as kernels of finite covers. The topology on the automorphism groups of first-order structures we are considering is that of pointwise convergence.
In the case where K0 is an abelian group, K0 becomes naturally a profinite continuous permutation module. In order to face the problem we mentioned above, we change our point of view and we look at finite covers from the point of view of group extensions. Making use of the second continuous cohomology groups, which in our case still retain their familiar applications, we give a criterion for the Lifting Problem in the case where K0 is abelian.
Then we take a specific non-split free finite cover, which none of the previously developed methods was able to handle, and we show, as an application of the criterion, that it is minimal.