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.
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.
Colloquia Autumn 2019
19-08-28 in Pascal
Elin Götmark (Chalmers/GU)
Mathematics for navigation
What is the shortest travel distance between two cities as the crow flies? How can you determine your position on the Earth by means of the sun or stars? The answer relies on spherical trigonometry. This is old mathematics that first began to develop in Hellenistic times, but most of us do not encounter it in our university courses today. This talk will be accessible to undergraduate students.
19-09-23 in Euler
Richard Schoen (University of California, Irvine)
New perspectives on scalar curvature
This will be a talk for a general audience explaining the dual role that the scalar curvature plays in differential geometry and in general relativity. Both subjects motivate questions about the geometric structure of spaces with scalar curvature bounded from below. We will describe some of these questions by analogy with similar questions for surfaces where the answers are largely known. In the physical case of dimension three we only have partial knowledge and many questions remain.
19-10-21 in Euler
Jeff Steif (Chalmers/GU)
Noise Sensitivity of Boolean Functions and Critical Percolation
I will introduce and discuss the notion of noise sensitivity for Boolean functions, which captures the idea that certain events are very sensitive to small perturbations. While a few examples will be given, the main example which we will examine from this perspective is so-called 2-dimensional critical percolation from statistical mechanics. There will also be connections to combinatorics and theoretical computer science. The mathematics behind the story includes, among other things, Fourier analysis on the hypercube, hypercontractivity and a stochastic version of the solution of the Bieberbach conjecture/theorem from geometric function theory (the so-called Schramm-Löwner evolution). Obviously, many of these topics will only be very briefly touched upon.
No background concerning percolation, Fourier analysis, hypercontractivity or geometric function theory will be assumed. [Slides]
19-11-18 in Pascal
Pär Kurlberg (KTH)
Repulsion in number theory and physics
Zeros of the Riemann zeta function and eigenvalues of quantized chaotic Hamiltonians appears to have something in common. Namely, they both seem to be ruled by random matrix theory and consequently should exhibit "repulsion" in the sense that small gaps between elements are very rare. More mysteriously, while zeros of different L-functions (i.e., generalizations of the Riemann zeta function) are "mostly independent" they also exhibit subtle repulsion effects on zeros of other L-functions.
We will give a survey of the above phenomena. Time permitting we will also discuss repulsion between eigenvalues of "arithmetic Seba billiards", a certain singular perturbation of the Laplacian on the 3D torus ℝ3/ℤ3. The perturbation is weak enough to allow for arithmetic features from the unperturbed system to be brought into play, yet strong enough to provably induce repulsion.
19-12-16 in Pascal
Aila Särkkä (Chalmers/GU) Sequential point processes - models for eye movements and sweat gland activation
Sequential point processes can be used to model point patterns, where the points appear one after the other,and also to construct regular, so-called hardcore point patterns. I will give a general definition of sequential point processes and discuss two examples of point patterns that can naturally be modelled by using sequential point processes. In the first example, eye movements on a painting are followed and the points where the gaze rests are modelled by using sequential point processes. In the second example, sequential point processes are used as models for activation of sweat glands.