Asymptotics, Operators, and Functionals

Speaking in 2021:

25 January Giacomo Canevari (Verona) Title: Effective free energies for nematic colloids. Abstract: Nematic colloids are composite materials, obtained as mixtures of microparticles in a liquid crystal host. The presence of the inclusions induces, by surface anchoring effects, a deformation in the arrangement of the nematic molecules. As a consequence, the dopant microparticles have a measurable effect on the macroscopic properties, even in the dilute regime. In this talk, we will discuss a homogenisation limit for a variational model of nematic colloids, based on the Landau-de Gennes theory. Using variational methods, we will prove convergence results for local minimisers in the dilute regime. The talk is based on a joint work with Arghir D. Zarnescu (BCAM, Bilbao, and Simion Stoilow Institute of the Romanian Academy, Bucharest).

1 February Alexander Pushnitski (KCL) Title: The spectrum of some Hardy kernel matrices. Abstract

8 February Charles Smart (Chicago) Title: Localization and unique continuation on the integer lattice. Abstract: I will discuss results on localization for the Anderson-Bernoulli model. This will include my work with Ding as well as work by Li-Zhang. Both develop new unique continuation results for the Laplacian on the integer lattice.

15 February Roberto Paroni (Pisa) Title: A non-linear model for inextensible elastic ribbons. Abstract: In this talk we consider plates whose energy density is a quadratic function of the difference between the second fundamental form of the deformed configuration and a "natural" curvature tensor. This tensor either denotes the second fundamental form of the stress-free configuration, if it exists, or a target curvature tensor. In the latter case, residual stress arises from the geometrical frustration involved in the attempt to achieve the target curvature: as a result, the plate is naturally twisted, even in the absence of external forces or prescribed boundary conditions. Here, starting from this kind of plate energies, we rigorously derive a new variational one-dimensional model for naturally twisted narrow ribbons. Our result generalises, and corrects, the classical Sadowsky energy to geometrically frustrated anisotropic ribbons with a narrow, possibly curved, reference configuration. The talk is based on a joint work with L. Freddi, P. Hornung, and M. G. Mora.

22 February Andrew Comech (Texas A&M and IITP Moscow) Title: Virtual levels and virtual states of linear operators in Banach spaces, and applications to Schrödinger operators. Abstract

8 March Nicolas Raymond (Angers) Title: On the Dirac bag model in strong magnetic fields. Abstract: This talk is devoted to two-dimensional Dirac operators on bounded domains coupled to a magnetic field perpendicular to the plane. It will be focused on the MIT bag boundary condition. I will describe recent results about accurate asymptotic estimates for the low-lying (positive and negative) eigenvalues in the limit of a strong magnetic field. This is a joint work with J.-M. Barbaroux, L. Le Treust and E. Stockmeyer.

15 March Maria Giovanna Mora (Pavia) Title: Equilibrium measures for nonlocal interaction energies: The role of anisotropy. Abstract: Particle systems subject to long-range interactions can be described, for large numbers of particles, in terms of continuum models involving nonlocal energies. For radially symmetric interaction kernels, several authors have established qualitative properties of minimizers for this kind of energies. But what can be said for anisotropic kernels? Starting from an example that describes dislocation interactions in metals, I will discuss how the anisotropy may affect the equilibrium measure and, in particular, its dimensionality.

22 March María Eugenia Pérez-Martínez (Cantabria) Ttile: Homogenization for alternating boundary conditions: spectral problems and stationary local problems. Abstract: We consider a spectral homogenization problem for the elasticity operator posed in a bounded domain of the upper half-space, a part of its boundary being in contact with the plane. We assume that this surface is free outside of small regions, where we impose Robin-Winkler boundary conditions. These regions are periodically placed along the plane while its size is smaller than the period. The Robin-Winkler condition links stresses and displacements by means of a symmetric and definite positive matrix and a Robin-parameter that can be large as the period tends to zero. We highlight the role of the stationary local problems obtained by means of asymptotic expansions. We show the spectral convergence depending on the size and the Robin-parameter while we compare results and techniques with those for the Laplace operator. Part of the talk is based on joint works with D. Gómez and S. A. Nazarov.

12 April Daniel Faraco (UAM-ICMAT) Ttile: Homogenization and inverse problems. Abstract: I will review some conditional stability results and recovery algorithms in various inverse problems with irregular coefficients. The arguments use quasiconformal mappings and a connection with the non-elliptic time-dependent Schrödinger equation. Then I will discuss in detail how G-convergence can be utilized to create instabilities in inverse problems. The latter is joint work with Y. Kurylev and A. Ruiz. If time allows, I will sketch part of a program with Guijarro, Kurylev and Ruiz on homogenization of elliptic equations on parallelizable manifolds.

19 April David Damanik (Rice) Title: Schrödinger operators with potentials generated by hyperbolic transformations. Abstract: We discuss discrete one-dimensional Schrödinger operators whose potentials are generated by sampling along the orbits of a general hyperbolic transformation and present results showing that the Lyapunov exponent is positive away from a small exceptional set of energies for suitable choices of the ergodic measure and the sampling function. These results apply in particular to Schrödinger operators defined over expanding maps on the unit circle, hyperbolic automorphisms of a finite-dimensional torus, and Markov chains. (Joint work with Artur Avila and Zhenghe Zhang)

10 May Inwon Kim (UCLA) Title: Dynamics of congested crowds and incompressible fluids. Abstract: We will study various transport equations where the density is transported under a height constraint. The motivation comes from crowd motion in congested environment, and also from various tumor growth models. The problem usually features a congested region, where the density is saturated at maximum value, which evolves over time with Darcy's law. This rather simple problem leads to many interesting phenomena and provides link to several classical fluid interface problems, such as Hele-Shaw and Muskat problem. We will discuss recent progress on these problems which have been very actively studied in the past decade in different communities.

28 June Alexander Figotin (UCI) -- Please note unusual starting time 17:15 BST. -- Title: Synthesis of lossless electric circuits based on prescribed Jordan forms. Abstract: We present an algorithm of the synthesis of lossless electric circuits such that their evolution matrices have the prescribed Jordan canonical forms subject to natural constraints. Every synthesized circuit consists of a chain-like sequence of LC-loops coupled by gyrators. All involved capacitances, inductances and gyrator resistances are either positive or negative with values determined by explicit formulas. A circuit must have at least one negative capacitance or inductance for having a nontrivial Jordan block for the relevant matrix. The mathematics involved in our analysis is intimately related to the canonical forms for quadratic Hamiltonians.

11 October Kirill Ryadovkin (St.Petersburg) Title: Branching random walks on periodic graphs. Abstract: We will discuss a continuous symmetric branching random walk on a periodic graph with a periodic set of particle generation centers, i.e., branching sources. Despite the probabilistic background of the problem, the talk essentially will be about spectral theory. The main object of our interest will be the operator of the evolution of the average number of particles. We will calculate the leading term of an asymptotic behaviour for the mean number of particles at an arbitrary point for large times. This is a joint work with Mariya Platonova.

18 October Habib Ammari (ETH Zürich) Title: Functional analytic methods for discrete approximations of subwavelength resonator systems. Abstract: In this lecture, the speaker will review mathematical and computational frameworks to elucidate physical mechanisms for manipulating waves in a robust way at scales beyond the diffraction limit using subwavelength resonator systems. In particular, he will demonstrate large-scale effective parameters with exotic values. He will also show that these systems can exhibit localized and guided waves on very small length scales. Using the concept of topologically protected edge modes, such localization can be made robust against structural imperfections.

25 October Horia Cornean (Aalborg) Title: How to discretize (pseudo)differential operators with operator norm convergence in the mesh size. Abstract: We prove norm estimates for the difference of resolvents of operators and their discrete counterparts, embedded into the continuum using orthogonal Riesz sequences. The operators are typically a sum of a Fourier multiplier and a multiplicative potential, but also more general pseudodifferential operators are considered. As a side-product, the Hausdorff distance between the spectra of the resolvents of the continuous and discrete operators decays with the same rate in the mesh size as for the norm resolvent estimates. The same result holds for the spectra of the original operators in a local Hausdorff distance. This is joint work with H. Garde and A. Jensen.

1 November Uzy Smilansky (Weizmann Institute; currently David Parkin Visiting Chair at Bath) Title: The distribution of delay times in the scattering of wave packets of short duration. Abstract: The distribution of delay times, observed when short pulses of radiation are scattered from complex targets will be defined both in the classical limit and the wave (quantum) description. The general expression for the delay-time distribution will be derived and used to discuss the well-known expression for the mean delay time derived by Wigner as well as the modification of the pulse shape due to the interaction. Finally, the scattering from a semi-infinite one dimensional array of random scattering centers will be used to show the effect of Anderson localization on the distribution of delay times.

8 November Valeria Chiadò Piat (Politecnico di Torino) Title: An extension theorem from connected sets and homogenization of non-local functionals. Abstract: Extension operators are a classical tool to provide uniform estimates and gain compactness in the homogenization of integral functionals over perforated domains. In this talk we discuss the case of non-local functionals of convolution type. The results are obtained in collaboration with Andrea Braides and Lorenza D'Elia.

15 November Yi Sheng Lim (Bath) Title: Topics in spectral theory. Abstract: In this session, I will cover several topics of interest from the book "Unbounded Self-adjoint Operators on Hilbert Space" by Konrad Schmüdgen. These include, in the first half of the session, 1. the spectral theorem (and its proof), 2. decomposition of the spectrum, and in the second half, 3a. rank 1 perturbation theory of Aronszajn–Donoghue, and 3b. Krein's trace formula for rank 1 perturbations, with key ideas needed to extend to trace class perturbations.

22 November Luis Silva (IIMAS-UNAM) Title: Oversampling on a class of symmetric regular de Branges spaces. Abstract: The oversampling property for symmetric regular de Branges spaces is studied by means of canonical systems with diagonal Hamiltonians. We give conditions on the Hamiltonian that guarantee the occurrence of the oversampling property in the corresponding de Branges space.

29 November Alexander Kiselev (St.Petersburg)

6 December Elisa Davoli (TU Wien) Title: Homogenization of high-contrast composites under differential constraints. Abstract: In this talk, we will derive, by means of variational techniques, a limiting description for a class of integral functionals under linear differential constraints. The functionals are designed to encode the energy of a high-contrast composite, that is, a heterogeneous material which, at a microscopic level, consists of a periodically perforated matrix whose cavities are occupied by a filling with different physical properties. Our main result provides a Γ-convergence analysis as the periodicity tends to zero, and shows that the variational limit of the functionals at stake is the sum of two contributions, one resulting from the energy stored in the matrix and the other from the energy stored in the inclusions. As a consequence of the underlying high-contrast structure, the study is faced with a lack of coercivity with respect to standard topologies which we tackle by means of two-scale convergence techniques. In order to handle the differential constraints, instead, we establish new results about the existence of potentials and of constraint-preserving extension operators for linear, k-th order, homogeneous differential operators with constant coefficients and constant rank. This is joint work with Martin Kruzik and Valerio Pagliari.

24 January 2022 Sven Gnutzmann (Nottingham)

14 February 2022 Petr Siegl (TU Graz)