## Q-Math Seminar

### Kurusch Ebrahimi-Fard, ICMAT

#### The algebra of the feedback loop in nonlinear control theory

##### Thursday 26th of June 2014, 10:00, ICMAT, Aula Gris 2

Fliess operators, also known as Chen-Fliess functional expansions, play a central role in the theory of nonlinear control systems. To represent systems made out of simpler subsystems, Fliess operators are combined in several ways through interconnections. The four basic system interconnections found in most control systems are the parallel, product, cascade and feedback connections. The latter, also known as a feedback loop, is one of the building blocks of modern control theory, roughly speaking, it can be described as feeding the output of one system into another system, the output of which is then fed back into the first one. Through the work of Gray and Duffaut Espinosa, it has become clear that a composition of Chen-Fliess functional expansions characterizes a feedback loop and can be considered a natural generalization of classical composition of functions. As a result, a Faà di Bruno type Hopf algebra has emerged. Foissy made a crucial contribution by proposing an appropriate grading of this Hopf algebra. In this talk, we present a combinatorial description of the algebra underlying the feedback loop by realizing its algebraic structures in terms of combinatorial operations on rooted circle trees. A Zimmermann type forest formula for the antipode is also given. This is of immediate interest with respect to applications such as system inversion, where the antipode needs to be computed explicitly to high order. Thus, combinatorially efficient methods are needed for its computation. This talk is based on joint work with W.S. Gray and L.A. Duffaut Espinosa.

### Juan Carlos Marrero, Universidad de la Laguna

#### The exact Lagrangian function for discrete mechanical systems with symmetries

##### Friday 30th of May 2014, 10:00, ICMAT, Aula Gris 2

In this talk, I will present a definition for the exact discrete Lagrangian function associated with a regular Lagrangian function L on an integrable Lie algebroid. For this purpose, I will use that L induces a regular Lagrangian function on the vertical bundle of the source of the corresponding Lie groupoid. Some applications of this construction to discrete Lagrangian functions with symmetries will be also presented.

### Marta Farré Puiggalí, ICMAT

#### Isotropy in the inverse problem of the calculus of variations

##### Friday 23rd of May 2014, 10:00, ICMAT, Aula Gris 2

The inverse problem of the calculus of variations consists in deter- mining if a given system of second order differential equations is equivalent to the Euler-Lagrange equations corresponding to some regular Lagrangian. We will introduce the problem from a per- spective that is easily extended to constrained systems, both in the autonomous and non-autonomous cases. This transition is given by passing from Lagrangian submanifolds to isotropic submanifolds.

### Julio de Vicente, UC3M

#### On nonlocality as a resource theory and nonlocality measures

##### Friday 9th of May 2014, 10:00, ICMAT, Aula Gris 2

Bell's theorem establishes the incompatibility of the predictions of quantum mechanics with those of any local realistic theory (i.e. a local hidden variable theory). This feature of the theory is nowadays referred to as quantum nonlocality. Besides its foundational interest, nonlocality has been lately an active field of research because of its application in quantum information science in what is known as device-independent quantum information processing. In this setting, quantum nonlocal states are regarded as a resource to implement various tasks such as cryptography and randomness generation. In a resource theory, one aims at providing means on how to characterize which elements constitute a resource and quantify how useful they are as well as at providing protocols on how to manipulate them. I will approach nonlocality from this perspective. In order to do so, I will analyze in full mathematical detail the operations that can be implemented in the device independent scenario. This provides a theoretical ground to systematically and rigorously study how to order and quantify nonlocal resources. Finally, I will review several nonlocality measures and discuss their validity from this point of view.

### Marco Zambon, ICMAT

#### Singular foliations and Lie groupoids

##### Friday 25th of April 2014, 10:00, ICMAT, Aula Gris 2

We start reviewing (regular) foliations, their holonomy, and a Lie groupoid associated to them (called holonomy groupoid). We then consider singular foliations, meant as a suitable submodule of vector fields on a manifold. We will review the ingenious construction by Androulidakis-Skandalis of a groupoid H associated to any singular foliation. H is only a topological groupoid, but we will show that the restriction of H to any leaf is a smooth Lie groupoid. Further, we will relate H to the holonomy transformations of the singular foliation, thus justifying the name "holonomy groupoid". This is based on joint work with Androulidakis.

### María Barbero Liñán, UC3M-ICMAT

#### Interconnection of Dirac systems for morse families

##### Friday 11th of April 2014, 10:00, ICMAT, Aula Gris 2

Dirac structures provide a framework to describe Hamiltonian and Lagrangian mechanics by means of Dirac systems. Such systems are given implicitly by a Lagrangian submanifold defined by the differential of a function. Here, we extend the notion of Dirac system to Lagrangian submanifolds given by a Morse family. Examples such as constrained variational calculus on Poisson manifolds, optimal control problems can be described intrinsically with this new approach. Moreover, the notion of backward and forward Dirac structures allow us to describe the interconnection of Dirac systems.

### Fernando Lledó, UC3M-ICMAT

#### Quantum symmetries, self-adjoint extension and reduction theory.

##### Friday 4th of April 2014, 10:00, ICMAT, Aula Gris 2

In this talk we will address the question of how does the process of self-adjoint extensions of symmetric operators intertwine with the notion of quantum symmetry. In particular, given a unitary representation of a Lie Group $G$ on a Hilbert space H, we develop the theory of $G$-invariant self-adjoint extensions of symmetric operators using the theory of quadratic forms. We will prove a $G$-equivariant version of Kato's representation theorem for quadratic forms. Moreover, we will apply von Neumann algebra techniques to analyze the relation between the reduction theory of the unitary representation and the reduction of the $G$-invariant unbounded operator.

### Kurusch Ebrahimi-Fard, ICMAT

#### On q-analogs of Multiple Zeta Values

##### Friday 28th of March 2014, 10:00, ICMAT, Aula Naranja

We will review recent progress made in the understanding of double shuffle relations for different q-analogs of multiple zeta values.

### Juan Manuel Pérez Pardo, UC3M-ICMAT

#### Integrability and chaos of infinite dimensional Hamiltonian systems: New ideas and perspectives.

##### Friday 21st of March 2014, 10:00, ICMAT, Aula Gris 1

The objective of this talk is to review the main notions of integrability of infinite dimensional Hamiltonian systems. For this purpose the Schroedinger equation will be presented as an infinite dimensional Hamiltonian system. The integrability of the system is mainly related with the structure of the spectrum of the corresponding Hamiltonian operator. We will propose a criterion for integrability based on the analytic properties of the algebra of unbounded operators generated by the latter. Namely, a system will be not integrable if the associated Gelfand triple does not define a domain of essential self-adjointness for the complete algebra. As the motivating example we will consider the quantum mechanical systems described by the Laplace-Beltrami operator defined on Riemannian manifolds with boundary.

### Alberto Ibort, UC3M-ICMAT

#### Convex bodies of states, majorization and Atiyah’s convexity theorem

##### Friday 14th of March 2014, 10:00, ICMAT, Aula Gris 1

We will try to understand the relation between majorization problems, Atiyah’s convexity theorem, the Jordan-Schwinger map and properties of convex bodies of quantum states.

### Alberto López Yela, UC3M

#### Group theory and Quantum Tomography

##### Friday 28th of February 2014, 10:00, ICMAT, Aula Gris 2

A quantum system can be described using a C* algebra approach where the observables of the system can be identified with the self-adjoint part of the algebra, and the states with normalized positive functionals on such algebra. A tomographic picture of Quantum Mechanics is a theory that describes how to reconstruct a state of a quantum system from sampling functions that parametrize the state. It is natural to use unitary representations of certain groups to obtain a tomographic picture of quantum systems because of Naimark's theorem and the GNS construction. Thus, given a state, we will show how to obtain sampling functions and probability distributions associated to them, called quantum tomograms, that under certain conditions can actually be measured in a laboratory.

### Mario Sigalotti, Centre de Recherche Saclay - Île-de-France

#### Approximately controllable finite-dimensional quantum systems are exactly controllable

##### Friday 21st of February 2014, 10:00, Universidad Carlos III de Madrid, room 2.3.A05

We discuss the controllability problem of a closed control-affine quantum system driven by two or more external fields. We provide a sufficient condition for controllability in terms of existence of conical intersections between eigenvalues of the Hamiltonian in dependence of the controls seen as parameters. The spectral condition appears naturally in the adiabatic control framework, which can be used to prove approximate controllability. It actually turns out that, under such condition, the system is Lie-bracket generating when lifted to the group of unitary transformations, and hence exactly controllable. We conclude by showing, from a representation theory viewpoint, that approximate and exact controllability are equivalent properties for general finite-dimensional quantum systems.

### Pedro Tradacete, UC3M

#### Spectral theory for positive operators on Banach lattices and domination.

##### Monday 27th of January 2014, 13:00, Universidad Carlos III de Madrid, room 2.2D08

I will present the theory of positive operators on Banach lattices together with several aspects that illustrate the close relation between topological and lattice structure in these spaces. The domination problem for positive operators will be introduced, with a particular emphasis on the class of Riesz operators and their spectral theory.

### Antonio García, UC3M

#### Stone's Theorem: an unexpected tool in Sampling Theory

##### Monday 20th of January 2014, 11:00, Universidad Carlos III de Madrid, room 2.2D08

The classical sampling theory in shift-invariant subspaces of $L^2(R)$ can be extended to $U$-invariant subspaces where $U$ denotes a unitary operator defined on a Hilbert space $H$. Roughly speaking, regular sampling is obtained by using the discrete group of unitary operators $(U^n)_{n\in Z}$. In dealing with irregular sampling, more specifically with the one associated with the time-jitter error, a continuous group of unitary operators $(U^t)_{t\in R}$ is needed, where $U^1=U$; in this case, Stone's Theorem plays a significant role.

### Julio de Vicente, UC3M

#### Bell's theorem and the modern view of quantum nonlocality

##### Monday 13th of January 2014, 11:00, Universidad Carlos III de Madrid, room 2.2D08

In this seminar talk I will review Bell's theorem which establishes the incompatibility of quantum mechanics with any local realistic theory (i.e. a local hidden variable theory). This feature of the theory is usually termed as quantum nonlocality. Besides its foundational interest, nonlocality is nowadays an active field of research because of its application in quantum information processing. I will introduce the mathematical framework which is used at present to address this theory, present some interesting mathematical problems that arise in this setting and discuss some of the problems I am currently working on in this field.

### Alberto Ibort, UC3M

#### Quantum holonomy control and quantum control at the boundary

##### Monday 9th of December 2013, 11:00, Universidad Carlos III de Madrid, room 2.2D08

We will discuss the adaptation of holonomy control to quantum systems. As an application, the possibility of controlling quantum systems by modifying the “boundary conditions” of the free dynamics will be discussed.

### Fernando Lledó, UC3M

#### Roe algebras and amenability

##### Monday 25th of November 2013, 11:00, Universidad Carlos III de Madrid, room 2.2D08

In the present talk I will present some basic material on coarse metric spaces, i.e., spaces whose topology reflect only the large scale structure of the space. Based on these spaces one can introduce uniform Roe algebras which are generated by operators with finite propagation speed. We will ask then the question if these algebras may be approximated in terms of matrices. We will also give examples in terms of amenable groups.

### Juan Manuel Pérez-Pardo, UC3M

#### Self-adjoint extensions on smooth manifolds with corners

##### Monday 18th of November 2013, 11:00, Universidad Carlos III de Madrid, room 2.2D08

We will study how do the corners at the boundary of a smooth manifold take part in the process of defining a self-adjoint extension of a symmetric differential operator. Von Neumann's theorem of s.a. extensions will be introduced briefly and we will work out a simple example of this situation. A scheme to handle this situation in greater generality will be proposed.

References:

A. Posilicano. On the many Dirichlet Laplacians on a non-convex polygon and their approximations by point interactions. J Funct. Anal. 265(3), 303-323 (2013)

### María Barbero, UC3M

#### Infinitesimal holonomy algebra and controllability

##### Monday 4th of November 2013, 11:00, Universidad Carlos III de Madrid, room 2.2D08

For a particular family of nonholonomic control systems on a principal bundle it has been proved that the infinitesimal holonomy algebra characterizes the controllability of the system [Radford,Burdick,Proceed. MTNS 98]. For a mechanical control system associated to an affine connection it is expected that the infinitesimal holonomy algebra might also play a role when trying to extend the controllability results [Lewis, Murray, 1997] at zero velocity to the nonzero case [Barbero, Proceed. CDC 2011].

### Leonardo Ferro, UC3M

#### Exceptional Jordan algebras, pre-Jordan algebras and Lie-Jordan algebras

##### Monday 28th of October 2013, 11:00, Universidad Carlos III de Madrid, room 2.2D08

Given an associative algebra, there is a natural splitting of the product in a symmetric and an antisymmetric part. The symmetric part defines a Jordan algebra, but not all Jordan algebras can be derived in this way from an associative product. Zelmanov proved that the only "exceptional" Jordan algebra is the Albert algebra of 3x3 matrices with octonionic entries. We will discuss the possibility of incorporating this exceptional Jordan algebra in the framework of pre-Jordan algebras. Then we will introduce Lie-Jordan algebras and discuss recent advances.

References:

H. Hanche-Olsen, E. Stormer "Jordan operator algebras"

www.math.ntnu.no/~hanche/joa/joa-h.pdf

F. Falceto, L. Ferro, A. Ibort and G. Marmo "Reduction of Lie-Jordan

Banach algebras and quantum states"

http://iopscience.iop.org/1751-8121/46/1/015201

### Kurusch Ebrahimi-Fard, ICMAT

#### From pre-A to A. The splitting of algebra products.

##### Monday 21st of October 2013, 11:00, Universidad Carlos III de Madrid, room 2.2D06B

We review recent developments in algebra, which focus on the notion of splitting of algebra products - into linear combination of binary compositions. The shuffle algebra plays a paradigmatic role in these developments. Particular emphasis will be put on associative, Lie and Jordan algebras.