Events

Of great interest in plasma physics is to determine whether excited charged particles in a non-equilibrium state will relax to neutrality or transition to a nontrivial coherent state. Due to the long range interaction between particles, the self-consistent generating electric field oscillates in time and disperses in space like a Klein-Gordon wave, known in the physical literature as plasma oscillations or Langmuir’s oscillatory waves. Landau in his original work addresses the decay of such an electric field, namely the energy exchange between the oscillatory electric field and charged particles, in a linearized setting. This talk will provide an overview of the recent mathematical advances in the nonlinear setting. The talk should be accessible to graduate students and the general audience. ___________________________________ Invited by Prof. Phan Thành Nam

We study the exchange of stability in scalar reaction-diffusion equations which feature a slow passage through a pitchfork type singularity in the reaction term, using a novel adaptation of the geometric blow-up method. Our results are consistent with known results on bounded spatial domains which were obtained using comparison principles like upper and lower solutions. However, from a methodological point of view, our approach is motivated by the analysis of a closely related ODE problem which was analyzed using geometric blow-up techniques by Krupa and Szmolyan in 2001. After applying the blow-up transformation, we obtain a system of PDEs which can be studied in local coordinate charts. Importantly, the blow-up procedure resolves a spectral degeneracy in which the spectrum is ‘pushed back’ so as to create a spectral gap in the linearization about particular steady states. This makes it possible to extend slow-type invariant manifolds into and out of a neighbourhood of the singular point using center manifold theory, in a manner which is conceptually analogous to the approach adopted by Krupa and Szmolyan in their analysis of the aforementioned ODE problem. We expect that the approach can be adapted and applied to the study of dynamic bifurcations in PDEs in a wide variety of different contexts.

Random matrices are amazing mathematical objects as they combine various mathematical fields, from linear algebra to probability, differential geometry, group theory, etc. Due to this richness, it is not surprising that they are versatile tools employed in Engineering, Physics, Statistics, and many more. Modern applications can be found in machine learning, quantum information, and wireless telecommunications. In my presentation, I will show you the basic concepts of random matrices, what kinds of objects are usually studied, and which important theorems have been proven. To the end I will turn to the relation of Harmonic Analysis and Random Matrix Theory and how it has helped to solve some contemporary problems.