This series of lectures is named after Gunter Lumer, a great mathematician with extensive achievements and a deep curiosity for applied functional analysis and operator theory, and is a joint scientific activity of all Action's Working Groups. The lectures will be delivered by carefully selected distinguished scientists whose research interests are related to our Action.

Gunter Lumer was born in Frankfurt, Germany in 1929. With Nazism on the rise, the Lumer family left Germany in 1933 and settled in France, and then in Uruguay. Gunter graduated in 1957 with a degree in electrical engineering from the University of Montevideo, he then received a Guggenheim fellowship to study at the University of Chicago. There he received his Ph.D. in Mathematics in 1959. Although Gunter Lumer's professional focus was on functional analysis, partial differential equations, and evolution equations, he nourished a broad interest for almost all areas of mathematics and for science in general. He published more than one hundred papers and edited many books. Probably his best known result is the celebrated Lumer-Phillips theorem, which gives necessary and sufficient conditions on an operator to generate a strongly continuous semigroup of contractions on a general Banach space. This result, published in the Pacific Journal of Mathematics in 1961, is a key contribution to the theory of operator semigroups. The research areas of our Action are closely related to some of Gunter's research activities in the 80's, and also brings together many of his collaborators and students. In a sense he is at the origin of the topics developed in this Action.

More details about Gunter Lumer's life and activities can be found on the following link.

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Marcel Filoche received his Ph.D. in 1991 from the Université Paris-Sud Orsay. He then worked for the France Telecom R&D center, before joining the CNRS in 1997. Since 2008 he is CNRS Research Director at the Condensed Matter Physics lab at École Polytechnique. Since 2018 he is also among the PIs of a research collaboration on Localization of Waves, sponsored by the Simons Foundations. He is the author of over 180 articles on biophysics, condensed matter, and wave phenomena in mathematics and theoretical physics.

**Abstract:** In disordered systems or complex geometry, standing waves can undergo a strange phenomenon that has puzzled physicists and mathematicians for over 60 years, called “wave localization”. This localization, which consists of a concentration (or a focusing) of the energy of the waves in a very restricted sub-region of the whole domain, has been demonstrated experimentally in mechanics, acoustics and quantum physics. During this talk, we will present a new theory which brings out an underlying and universal structure, the localization landscape, solution of a Dirichlet problem associated with the wave equation [1]. In quantum systems, this landscape also allows us to define an “effective localization potential” which predicts the localization regions, the energies of the localized modes, the density of states, as well as the long-range decay of the wave functions. This theory holds in any dimension, for continuous or discrete systems. We will present the major mathematical properties of this landscape. Finally, we will review applications of this theory in mechanics, semiconductor physics, as well as molecular and cold atom systems.

[1] M. Filoche & S. Mayboroda, Proc. Natl Acad. Sci. (2012) 109:14761-14766.

Wolfgang Arendt received his Ph.D. in 1979 and his habilitation in 1985 from the University of Tübingen, Germany. From 1981 until 1987 he was Assistant Professor at the same university, and from 1987 until 1995 Full Professor of Applied Mathematics at the University of Besançon, France. From 1995 until his retirement in 2018 Wolfgang Arendt was Full Professor and head of the Applied Mathematics Institute at the University of Ulm, Germany. He was visiting professor in USA (Berkley, Stanford), Australia (Canberra, Sydney), England (Oxford), France (Nancy), Italy (Lecce), New Zealand (Auckland), Poland (Torun). He supervised 34 Ph.D. thesis. He was awarded the Merckle research Prize in 1999. He is the author of over 140 articles and 4 books on evolution equations and spectral theory of one-parameter semigroups. He is editor and associate editor of several international journals.

**Abstract:** We start explaining how the holomorphic version of the Lumer Phillips Theorem leads to the theory of forms. Then we study perturbations by weakly continuous forms. We obtain essentially coercive forms which generate asymptotically compact (= quasi compact) semigroups. Elliptic differential operators and Schrödinger operators are examples where this perturbation theory is efficient.

Essential coercivity occurs also in a completely different situation: it yields a topological characterization of the universal Galerkin property. This property means that the Galerkin approximation associated with the form and an arbitrary sequence of approximating subspaces converges (and this as fast as possible).

After being a student at Ecole Normale Supérieure in Paris, Laurent Desvillettes received his Ph.D. in 1990 at Univ. Paris 7 under Prof. Claude Bardos' supervision, his manuscript was entitled "Mathematical and numerical study of kinetic equations from Physics". He became Full Professor in 1994, successively at Univ. Orléans, ENS Cachan and since 2015 at Univ. Paris. He provided major contributions about mathematical and numerical properties of PDEs, in particular kinetic and reaction-diffusion equations with singularities, as much as on their applications in fluid mechanics, statistical physics and population dynamics. He is author of more than 120 papers, and serves as an editor and associate editor of several international journals. Upon invitation, he did many research visits among which the ones in Italy (Rome, Torino, Parma, Pisa, Messina), Austria (Wien), Spain (Barcelona, Bilbao, Grenade), Germany (Munich, Bonn, Mannheim, Göteborg), Netherlands (Leiden), Romania (Iasi, Bucharest), Switzerland (Zürich), UK (Cambridge), USA (Princeton, Brown, Stanford, UT Austin, Courant, Maryland), Canada (Laval), Japan (Kyoto), China (Beijing), Hong Kong, Singapore.

**Abstract:** Landau proposed in the late 30s to model the collisions between charged particles in a plasma by a bilinear operator sharing some similarities with the Boltzmann operator of rarefied gases, but containing second order terms (and sharing similarities with the Fokker-Planck equation). We propose to explain some of the properties of this operator and of the corresponding equation (describing the relaxation towards thermodynamical equilibrium of an homogeneous plasma). We will also present the analogies of this equation with other systems appearing in mathematical physics, such as the Navier-Stokes system for 3D incompressible fluids.