Plenary Lectures
Jesús Clemente-Gallardo (Universidad de Zaragoza)
Title: A geometric formulation of Ehrenfest molecular dynamics
Abstract: It is a very well known fact that the exact quantum description of heavy atoms and molecules is an impossible task and that suitable approximations are necessary to capture their main properties in an efficient way.
One of the best known approximations corresponds to the hybrid quantum-classical model where the nuclei and inner electrons are treated as classical particles, the valence electrons are considered to be quantum, and the interaction of these quantum and classical particles is encoded in Ehrenfest equations. The geometric description of Quantum Mechanics can be used to build a tensorial version of that framework and prove that Ehrenfest equations are Hamiltonian with respect to a suitable hybrid symplectic form. This Hamiltonian nature allows us to define an invariant volume form for the phase space and, with it, it is possible to build a Statistical Mechanics to describe more general situations. In this talk, we will summarize the main properties of this approach and some interesting applications obtained by our group in the last years.
Giancarlo Garnero (Università degli Studi di Bari Aldo Moro)
Title: Quantum Boundary Conditions
Abstract: Boundary conditions are ubiquitous in every area of physics. The analysis of a physical system, indeed, usually discriminates the behaviour of the bulk from the surrounding environment. In this sense, boundary conditions are a crucial ingredient, interpreting the interaction between confined systems and the environment.
In this talk I will discuss how quantum boundary conditions emerge in the description of bounded non relativistic quantum systems. In particular, I am going to discuss the case of a quantum particle confined into a cavity. I will present different examples and dynamical applications, underlying the relation between the physical and geometrical aspects of the problem.
Magdalena Caballero (Universidad de Córdoba)
Title: On the spacelike hypersurfaces with the same Riemannian and Lorentzian mean curvature
Abstract: Spacelike hypersurfaces in the Lorentz-Minkowski space $\mathds{L}^n$ can be endowed with another Riemannian metric, the one induced by the Euclidean space $\mathds{R}^n$. Those hypersurfaces are locally the graph of a smooth function $u$ satisfying $|Du|< 1$. If in addition they have the same mean curvature with respect to both metrics, they are the solutions to a certain partial differential equation, the $H_R = H_L$ hypersurface equation. It is well known that the only surfaces that are simultaneously minimal in $\mathds{R}^3$ and maximal in $\mathds{L}^3$ are open pieces of helicoids and of spacelike planes, (O. Kobayashi 1983). Similar results have been obtained more recently for timelike surfaces (Kim-Lee-Yang 2009), and also for spacelike surfaces in the product spaces $\mathds{S}^2\times\mathds{R}$ and $\mathds{H}^2\times\mathds{R}$ (Kim-Koh-Shin-Yang 2009). In this talk we consider the general case of spacelike hypersurfaces with the same mean curvature with respect to both metrics, paying special attention to the minimal and maximal case. Firstly in the Lorentz-Minkowski space, and afterwards in other ambient settings.
Janusz Grabowski (Institute of Mathematics, Polish Academy of Sciences)
Title: Solvability implies integrability
Abstract: Integrability of a vector field means that you can find in an algorithmic way its flow in a given coordinate system. The existence of additional compatible geometric structures may play a relevant role and it allows us to introduce other concepts of integrability (e.g. Arnold-Liouville integrability).
Our aim is to develop the study of integrability in the absence of additional compatible structures, and more specifically the classical problem of integrability by quadratures, i.e. to study under what conditions you can determine the solutions by means of a finite number of algebraic operations (including inversion of functions) and quadratures of some functions.
We present a substantial generalisation of a classical result by Lie in this direction. Namely, we prove that all vector fields in a finite-dimensional transitive and solvable Lie algebra of vector fields on a manifold can be integrated by quadratures.
Gloria Marí Beffa (University of Wisconsin–Madison)
Title: Discrete geometry of polygons and Hamiltonian structures
Abstract: In this talk we will review how the discrete geometry of polygons in
some parabolic manifolds helps us de ne Hamiltonian structures for some
discrete evolutions, including well known integrable systems. We will use
projective polygons as running example to illustrate the results, and discuss
open problems. This is in part joint work with Jin Ping Wang (U of Kent at
Canterbury) and Anna Calini (College of Charleston).
Rui Pacheco (Universidade da Beira Interior, Covilhã)
Title: Harmonic maps and shift-invariant subspaces
Abstract: In the early 90s, G. Segal formulated the harmonicity equations for maps from surfaces into the unitary group in terms of the Grassmannian model of loop groups, in which harmonic maps (the non-linear sigma models of theoretical physics) correspond to certain families of shift-invariant subspaces of $L^2(S^1,\mathbb{C}^n)$. This point of view leads to a beautiful interplay between differential geometry and operator theory. We will present some new interesting results about harmonic maps from surfaces into Lie groups and their symmetric spaces that make explicit use of operator-theoretic methods. This is joint work in progress with Alexandru Aleman (University of Lund) and John C. Wood (University of Leeds).
Chiara Pagani (Università del Piemonte Orientale, Alessandria, Italy)
Title: Noncommutative gauge theories through twist deformation quantization
Abstract: In noncommutative geometry principal bundles consist of algebra extensions that satisfy the condition to be Hopf-Galois. In this algebraic setting, quantum groups and their (co)actions play a central role in the description of symmetries of noncommutative spaces.
In this seminar we describe a general theory of Drinfeld twist deformation quantization of Hopf-Galois extensions [Aschieri, Bieliavsky, Pagani, Schenkel, Commun. Math. Phys. 352 (2017)] and present recent results on the study of the group of gauge transformations of a noncommutative bundle [Aschieri, Landi, Pagani, arXiv:1806.01841]. We illustrate the general theory through examples, focusing on instantons on quantum spheres.
Elsa Prada (Universidad Autónoma de Madrid)
Title: Topological insulators and superconductors
Abstract: Traditionally, we understand phases of matter such as liquid/solid or ferromagnetic/paramagnetic within Landau's paradigm, which is based on order parameters of a local nature. With the advent of the topology paradigm we have been able to discover new phases of matter that are realized in the so-called topological materials. These are described by topological invariants instead of local order parameters. Invariants take on discrete sets of values that describe order of a global nature. In this talk I will review general concepts of topological band theory and the bulk-boundary correspondence. I will apply these concepts first to topological insulators in one-, two- and three-dimensions, and then to topological superconductors in one- and two-dimensions. I will show specific paradigmatic materials within each category and the most celebrated condensed-matter experiments where their existence has been demonstrated. I will finish with important applications of their topologically-protected boundary states.
Ana Ros Camacho (Mathematical Institute of Utrecht University)
Title: Orbifold equivalence for simple, unimodal and bimodal singularities
Abstract: In this talk I will introduce orbifold equivalence, an equivalence relation between polynomials satisfying certain conditions ("potentials") which describe Landau-Ginzburg models. We will review how it relates the potentials associated to simple, (exceptional) unimodal and bimodal singularities, reproducing classical results like strange duality from the classification of singularities from Arnold. In addition, we will see that these equivalences seem to be controlled by Galois groups. Based on ongoing work with T. Kluck and G. Cornelissen and on joint work with R. Newton, I. Runkel et al.
