Jim Langley, School of Mathematical Sciences, University of Nottingham.

Talks may be held in either Loughborough or Nottingham.

AUTUMN SEMESTER 2003-4

All talks this semester will be at Loughborough, on Thursdays at 1.00 in room W.143, David Davies Building, unless otherwise stated.

October 9th Jim Langley (Nottingham) Critical points of some discrete potentials in space.

The talk describes recent joint work with Rossi (Blacksburg). The equilibrium points of the electrostatic field generated by an infinite set of positive point charges in space correspond, subject to convergence conditions, to critical points of a certain positive superharmonic potential. We discuss sufficient conditions for the existence of such critical points.

October 23rd Risto Korhonen (Loughborough) Growth estimates for solutions of complex linear differential equations.

The talk discusses relations between the growth of the coefficients and that of the solutions for linear differential equations with coefficients analytic in the unit disc.

November 5th Ilpo Laine (Joensuu) Growth of Painleve transcendents in the complex plane.
(Wednesday, Mathematical Physics Seminar, 4.00 in W.145)

The talk describes recent joint work of the speaker and Aimo Hinkkanen (Illinois) on the minimal growth of solutions of the Painleve equations.

December 4th Eleanor Clifford (Nottingham) Nonvanishing derivatives and normal families.

If k is an integer greater than 1 and F is a family of analytic functions f on a domain D such that f and its k'th derivative have no zeros in D then Schwick proved that the logarithmic derivatives f'/f form a normal family on D. The talk discusses some generalisations to meromorphic functions and related differential operators.

SPRING SEMESTER 2003-4

Talks will be Mondays at 1.00, in either Nottingham or Loughborough. Nottingham talks will be in C27 (Maths/Physics Building). Loughborough talks will be in W145.

February 16th (at Nott'm) Rod Halburd (Loughborough) Integrable equations and complex analysis

This talk will be a survey of some of the ways in which complex analysis is used in the solution and study of several integrable equations. In particular, we will describe the role played by Riemann-Hilbert problems in the solution of integrable equations such as the Korteweg-de Vries equations and the Painlev\'e equations. Recent applications of Nevanlinna theory as a detector of integrability of difference equations will be presented.

1st March, Loughborough University Holger Dullin, Symplectic invariants: from the pendulum to the hypergeometric equation

8th March, Nottingham Vladimir Eiderman (Moscow State Civil Eng. Univ) Uniqueness theorems for bounded analytic functions

15th March, Nottingham Two short talks by Nottingham PhD students:

22nd, March, Loughborough (W143) Dmitri Chelkak (St. Petersburg State University) The inverse problem for the harmonic oscillator perturbed by a potential. Characterization
Abstract: Consider the perturbed harmonic oscillator $Ty=-y''+x^2y+q(x)y$ in $L^2({\mathbb R})$, where the real potential $q$ belongs to the Hilbert space ${\bf H}=\{q', xq\in L^2({\mathbb R})\}$. The spectrum of $T$ is an increasing sequence of simple eigenvalues $\lambda_n(q)=1+2n+\mu_n$, $n\ge 0$ such that $\mu_n\to 0$ as $n\to\infty$. Let $\psi_n(x,q)$ be the corresponding eigenfunctions. Define the norming constants $\nu_n(q)=\lim_{x\to\infty}\log |{\psi_n (x,q)}/{\psi_n (-x,q)}|$ (if $q(x)\equiv q(-x)$, then $\nu_n(q)=0$ for all $n\ge 0$). It is known that the spectral data $(\{\lambda_n(q)\}_0^\infty\,,\{\nu_n(q)\}_0^\infty)$ uniquely determine the potential $q(x)$. We give the complete characterization of the set of spectral data which corresponds to the space of potentials ${\bf H}$. In the proof we use ideas from the inverse spectral theory for the operator $-y''+py$, $p\in L^2(0,1)$, with Dirichlet boundary conditions on the unit interval.

26th April, Loughborough Janya Ferapontov (Loughborough), POSTPONED

10th May, Nottingham Jim Langley (Nottingham), Entire functions taking integer values at the same points

Abstract: An old result of Polya says that 2^z is the slowest growing transcendental entire function which takes integer values at 0, 1, 2, ..... It was conjectured by Osgood and Yang in 1976 that if f and g are entire and T(r, f) = O( T(r, g) ), and if f(z) is an integer whenever g(z) is an integer, then f is a polynomial in g. It turns out that this is true, as is rather more. We also discuss some applications of these ideas to differential equations.

SPRING SEMESTER 2004-5

11th February at 1.00, Nottingham S.P.Tsarev (visiting Loughborough), Identities for the polygamma functions and the problem of rational definite summation

Abstract: Recently an interesting link between the problem of finding finite rational sums $\sum_{k=0}^{n-1}\frac{P(k,n)}{Q(k,n)}$ in closed form and the problem of description of all possible linear identities for the polygamma functions was established. Using some well-known facts from the complex analysis we prove for example that there are no new identities of the form $\sum_i \beta_i(n)\psi^{(t_i)}(\gamma_i(n)) = 0$ with algebraic $\beta_i(n)$, $\gamma_i(n)$. This allows us to construct an algorithm for computing finite rational sums in the case of "slowly growing" denominators $Q(k,n)$. In this talk we give the necessary background material form the theory of definite summation and the necessary facts about the polygamma functions, formulate the obtained results, and discuss the possible aplications of the more advanced results form Nevanlinna theory to description of wider classes of polygamma identities and related finite rational sums.

9th March at 1.00, Loughborough Sasha Pushnitski (Loughborough), Spectral problems for the Sturm-Liouville equation and entire functions.

SPRING SEMESTER 2005-6
All talks this semester will take place at Loughborough, apart from that by Walter Hayman on 13th March 2006.

21st February at 1.00 Edmund Chiang (Hong Kong University of Science and Technology), Linear difference equations from the perspective of Nevanlinna theory.

9th March at 12.00 Rob Trickey (Nottingham), Critical points and zeros of certain discrete potentials.

13th March at 4.00 (Nottingham, Pope Building C17) Walter Hayman (Imperial), A generalisation of Stirling's formula.

AUTUMN SEMESTER 2006-7

8th September at 11.30 (Nottingham, Maths/Phys C5) Walter Bergweiler (Kiel), Fixpoints and periodic points of entire and quasiregular mappings
Fixpoints and periodic points play a key role in the iteration of entire and meromorphic functions. The speaker proved conjectures of Gross and Baker concerning the existence of such points, and the talk will describe some more recent extensions, including to quasiregular mappings, which form a natural generalisation of analytic functions to n-dimensional real space.

This page maintained by: Prof. J.K. Langley, School of Mathematical Sciences, University of Nottingham, NG7 2RD.

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