Colloquia Spring 2019
19-01-28 in Pascal
Jonathan Lenells (KTH) Long-time asymptotics for nonlinear integrable PDEs
For a certain class of nonlinear PDEs, called integrable PDEs, it is possible to derive remarkably precise asymptotic formulas for the long-time behaviour of the solution by solving inverse problems coming from scattering theory together with a nonlinear version of the steepest descent method. I will give an introduction to this circle of ideas using the sine-Gordon equation as an illustrative example. If time permits, I will present some new results on the topological charge and on the interaction between the asymptotic solitons and the radiation background for the sine-Gordon equation.
Slides_Lenells_2019.pdf 19-02-25 in Pascal
Søren Fournais (Århus)
Mathematics of the Bose Gas
The experimental realization of Bose-Einstein condensation in the laboratories in the 1990’s has motivated much new mathematical work on the cold Bose gas. A Bose-Einstein condensate is a new state of matter, where quantum phenomena become macroscopically apparent. In this talk, I will give an introduction to some of the involved mathematics and indicate both the difficulty and the importance of understanding this fundamental model in Quantum Mechanics. Finally, I will overview the state-of-the-art concerning the ground state energy of the model.
Slides_Fournais_2019.pdf19-04-04 in Euler
Sylvie Paycha (Potsdam) Are locality and renormalisation reconcilable?
According to the principle of locality in physics, events taking place at different locations should behave independently, a feature expected to be reflected in the measurements. The latter are confronted with theoretic predictions which use renormalisation techniques in order to deal with divergences from which one wants to derive finite quantities. The purpose of this talk is to confront locality and renormalisation.
Sophisticated (co)algebraic methods developed by physicists makes it possible to keep track of locality while renormalising. They mostly use a univariate regularisation scheme such as dimensional regularisation. We shall present an alternative multivariate approach to renormalisation which encodes locality as an underlying algebraic principle. It can be applied to various situations involving renormalisation, such as divergent multizeta functions and their generalisations, namely discrete sums on cones and discrete sums associated with trees.
This talk is based on joint work with Pierre Clavier, Li Guo and Bin Zhang
19-04-15 in Pascal
Per Salberger (Chalmers/GU)
On Nevanlinna's theory of meromorphic functions
The Weierstrass factorization theorem asserts that an entire function can be expressed as a product of certain elementary functions involving its zeroes. This result was refined and given a more canonical form for entire functions of finite order by Hadamard and then extended by Nevanlinna to meromorphic functions. The aim of the talk is to describe a new binary operation for holomorphic and meromorphic functions, which is related to the Hadamard convolution of their logarithmic derivatives. This operation is very natural when dealing with Weierstrass products and gives new insight to the multiplicative theory of holomorphic and meromorphic functions. One obtains for example new ring structures on the meromorphic functions of any given order.
19-05-20 in Pascal
Johanna Pejlare (Chalmers/GU)
Calculating π in the 18th century
Anders Gabriel Duhre (c.1680-1739), an important mathematician and mathematics educator in Sweden during the 18th century, contributed with two textbooks in mathematics, one in algebra and one in geometry. Among others, he treats infinitesimals based on Nieuwentijts’ theories from Analysis infinitorum and infinite series based on Wallis’ method of induction from Arithmetica infinitorum. Based on these results, Duhre develops an ingenious method to determine the area enclosed by curves by constructing a corresponding curve. He applies his method to the circle in order to find an expression of π as an infinite series. The series he finds is a modified version of the Gregory-Leibniz’ series. We consider in detail Duhre’s presentation in order to investigate the influence upon him as well as his influence on the Swedish mathematical society of his time.