Asymptotics, Operators, and Functionals


Leading the discussion in 2020:

Away: [13 January Operator Theory, Analysis and Mathematical Physics, IIMAS-UNAM, México City]

Away: [20 January BUC-XVI: Recent developments in wave propagation and their application to new materials, CIMAT, Mérida]

Away: [27 January ITT11: Exploring limits in wave transmission, Bath]

3 February Feodor Borodich (Cardiff) (Topic: Extensions of the JKR contact theory to elastic punches of arbitrary shapes, 2D Membranes and thin layers) Abstract

10 February Alexander Kiselev (IIMAS-UNAM, México) (Topic: Non-selfadjoint operators, their spectra, and functional models)

Away: [17 February Quantum Mechanics of Artificial Material Structures, Sochi]

2 March Kirill Cherednichenko (Bath) (Topic: Dispersion relations for stratified media with resonant layers)

9 March Pranav Singh (Bath) (Topic: New ideas in the analysis of convergence in numerical methods based on the Magnus expansion)

16 March Kirill Cherednichenko (Bath) (Topic: Functional model for extensions of symmetric operators and applications to scattering theory) Abstract: I shall discuss the functional model for extensions of symmetric operators and its applications to the theory of wave scattering. In terms of Boris Pavlov's spectral form of this model, we find explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator with equal deficiency indices. On the basis of these formulae, we are able to derive a new representation for the scattering matrix for pairs of such extensions. We use this representation to explicitly recover the coupling constants in the inverse scattering problem for a finite non-compact quantum graph with delta-type vertex conditions. This is joint work with A. V. Kiselev and L. O. Silva.

23 March Pierluigi Cesana (Kyushu) (Topic:Mathematical models and ideas for disclinations) Abstract: In this talk, we describe some recent results on the modeling of rotational mismatches at the level of a crystal lattice and on the modeling of self-similar-type martensitic microstructure, two phenomena which appear to be strongly interconnected. First, we introduce an energy functional defined over a triangular lattice accounting for nearest-neighbor interactions. We design special rotational-type boundary value problems on the lattice so that the minimizers necessarily exhibit non-homogeneous rotations. We are interested in the asymptotics of the energy minima and minimizers as the lattice spacing vanishes which we compute exactly with Gamma-convergence. We perform some numerical calculations for the discrete model and show that both the shape of the solutions as well as the values of the energies are in agreement with classical results for positive and negative disclinations. This is a collaboration with P. van Meurs (Kanazawa). Then, depending on time we present a probabilistic model for the description of martensitic microstructure as an avalanche process. A martensitic phase-transformation is a first-order diffusionless transition occurring in elastic crystals and characterized by an abrupt change of shape of the underlying crystal lattice. It is the basic activation mechanism for the Shape-Memory effect. Our approach to the analysis of the model is based on an associated general branching random walk process. Comparisons are reported for numerical and analytical solutions and experimental observations. This is a joint project with John M. Ball and Ben Hambly (Oxford).

Away: [6 April Basque Center for Applied Mathematics, Bilbao]


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