2019-03-13
15:45 hrs.

Monika Anna Winklmeier. Universidad de los Andes
Estimates for eigenvalues in gaps of the essential spectrum
Sala 5
Abstract:
In this talk I will show how bounds for eigenvalues in gaps of the essential spectrum of a linear operator can be obtained. The main example will be a one-dimensional Dirac type operator.

2019-03-06
15:45 hrs.

Christian Jaekel. University of São Paulo
On reflection positivity, modular localisation and Connes cocycles
Sala 5
Abstract:
The unitary irreducible representations of the Lorentz group carry an intrinsic notion of localisation on de Sitter space, known as modular localisation. An extension of Araki’s perturbation theory of modular automorphisms can be used to define interacting representations of the Lorentz group, as well as the corresponding Haag-Kastler nets. The analyticity properties of the correlation functions allow us to extend these theories to “nets" of (non-abelian) von Neumann algebras on the sphere. Reflection positivity can be used to recover the interacting quantum (field) theories on the de Sitter space from the sphere. Explicit examples are scalar bosons with polynomial or exponential interactions in 1+1 space-time dimensions, but our aim is to classify all interacting quantum theories compatible with the space-time symmetries. The Minkowski space limit is the limit of space-time curvature to zero, which is well-behaved on the level of local von Neumann algebras.

2018-12-19
15:45 hrs.

Jean Bellissard. Georgia Institute of Technology
Viscosity of liquids: A simplistic but effective model
Sala 5
Abstract:
Using a new degree of freedom called "anankeon" we design a Markov process describing the competition with phonons. The viscosity can be computed analytically. It can be shown that the behavior in temperature follows experimental results.

2018-11-28
15:45 hrs.

Francisco Correa. Universidad Austral de Chile
PT-Deformation of Calogero-Sutherland Models
sala 5
Abstract:
In this talk we will discuss how Calogero-Sutherland models of identical particles on a circle are deformed away from hermiticity but retaining a symmetry, preserving the integrability structure. The interaction potential gets completely regularized, which adds to the energy spectrum an infinite tower of previously non-normalizable states. For integral values of the coupling, extra degeneracy occurs and a nonlinear conserved charge enlarges the ring of Liouville charges.

2018-11-07
15:45 hrs.

Olivier Bourget. Pontificia Universidad Católica de Chile
Kicked Random Quantum Systems Revisited
Sala 5
Abstract:
We explain how various localization results obtained for kicked random quantum systems can be recast in the framework of the fractional moment method and then generalized (joint work with G. Moreno).

2018-10-31
15:45 hrs.

Jorge Zanelli. Centro de Estudios Científicos, Valdivia
Parallelizable (pseudo) spheres in $3$ and $7$ dimensions
Sala 5
Abstract:

It is a classic result in geometry that $\mathbb S^1$, $\mathbb S^3$ and $\mathbb S^7$ are parallelizable: they admit a globally defined flat connection (Cartan & Schouten, 1926). Moreover, these are the only parallelizable spheres (Adams Theorem, 1959).

We explore the extension of these results for different spacetime signatures and give explicit formulas for the connections for $H^{2,1}$ and $H^{1,2}$ in three dimensions, and for $H^{4,3}$ and $H^{3,4}$ in dimension seven.

2018-10-24
15:45 hrs.

Enrique Reyes. Universidad de Santiago de Chile
El problema de Cauchy para la jerarquía de Kadomtsev-Petviashvili
Sala 5
Abstract:
Esta charla es sobre una solución al problema de Cauchy para la jerarquía de Kadomtsev-Petviashvili (KP) que se ha venido refinando en los últimos años. La jerarquía KP es un conjunto infinito de ecuaciones  diferenciales no-lineales en una "variable espacial" e infinitas "variables temporales", que contiene como casos particulares ecuaciones completamente integrables tales como la famosa ecuación de Korteweg-de Vries.

Es posible solucionar todas las ecuaciones de la jerarquía KP usando teoremas de factorización de grupos de Lie de dimensión infinita. En esta charla se mostrará este resultado en tres contextos distintos:

a) Algebraico: Los actores principales son grupos de Lie construidos en base a operadores pseudo-diferenciales formales; a su vez, estos operadores se definen usando álgebras equipadas con derivaciones y valuaciones no-arquimideanas. La solución del problema de Cauchy para la jerarquía KP es formal.

b) Geométrico: Los grupos de Lie de a) se equipan con estructuras de grupos de Frölicher. La solución del problema de Cauchy para la jerarquía KP es suave.

c) Analítico: La jerarquía KP misma se plantea como una ecuación no-lineal en un grupo de Frölicher construido con la ayuda de una clase de operadores pseudo-diferenciales introducida por Kontsevich y Vishik en 1994. La solución del problema de Cauchy para la jerarquía KP es suave.

2018-10-17
15:45 hrs.

Pablo Miranda. Universidad de Santiago de Chile
Resonances in deformed tubes: twisting and bending
Sala 5
Abstract:
In this talk we will consider an infinite straight tube and we will deform it by a periodic twisting and a local bending. On the deformed tube we will define the Laplacian and will study the existence of scattering resonances created by the deformations. We will show the existence of exactly one resonance or one eigenvalue near the bottom of the essential spectrum, depending on the strength of the twisting and the bending. We will also obtain the asymptotic behavior of the resonance/eigenvalue as a function of the bending and twisting.

2018-10-10
15:45 hrs.

Rafael Benguria. Pontificia Universidad Católica de Chile
A sharp estimate for Neumann eigenvalues of the Laplace-Beltrami operator for domains in a hemisphere
Sala 5
Abstract:

2018-10-03
15:45 hrs.

Christian Sadel. Pontificia Universidad Católica de Chile
Matrices de transfer y teoría espectral para operadores discretos
sala 5
Abstract:
Operadores discretos de Jacobi o block-Jacobi y su teoría espectral están muy estudiado con el método de matrices de transfer. Voy a hablar de estos conexiones y generalizaciones para operadores mas generales aun resumen de algunos resultados.

2018-09-26
15:45 hrs.

Hagop Tossounian. Universidad de Chile
Mark Kac's Model And The $d_2$ Metric
Sala 5
Abstract:
In 1956 Mark Kac introduced a stochastic model to derive a Boltzmann-like equation. His model is linear, based on $N\gg1$ particles that undergo binary-collisions. The rate of approach to equilibrium is an important question for the Kac model. In this talk I will introduce Kac's model and the $d_2$ metric, due to Gabetta-Toscani-Wennberg, in the context of Kac's model; demonstrate the almost intensivity property of $d_2$ in $N$, and show how $d_2$ lead to a class of initial states that do not show approach to equilibrium for time of order $1$.

2018-09-12
15:45 hrs.

Akito Suzuki. Shinshu University
Supersymmetric aspects of quantum walks
Sala 5
Abstract:
Chiral symmetric quantum walks exhibit supersymmetry. In this talk, we define an index for such quantum walks so that it agrees with the Witten index for a supersymmetric Hamiltonian. We also give several concrete models for which we can calculate the index.

2018-08-29
15:45 hrs.

Norbert Heuer. Pontificia Universidad Católica de Chile
Una formulación ultra-débil del modelo de Kirchhoff-Love y aplicaciones
Sala 5
Abstract:
Encontrar formulaciones variacionales bien planteadas es un punto central para el análisis numérico de problemas definidos por ecuaciones en derivadas parciales. Resulta que hay un método numérico, llamado DPG, donde este buen planteamiento basta para obtener sistemas discretos estables. Normalmente no es así, como en el caso de los elementos finitos. Dada la estabilidad automática del DPG, para un problema específico se pueden diseñar formulaciones variacionales con foco en las variables de interés, con la única condición lograr un buen planteamiento. Ilustramos esto para el caso de las ecuaciones de Kirchhoff-Love que son un modelo para la flexión de placas delgadas bajo tensión vertical. La dificultad del modelo consiste en la falta de regularidad estandard de incógnitas relevantes. Esta falta impide el uso de formulaciones sencillas y complica el diseño de métodos numéricos.

2018-08-22
15:45 hrs.

Emanuela Radici. Università Degli Studi Dell'aquila
Deterministic particle approximation for scalar aggregation-diffusion equations with nonlinear mobility
Sala 5
Abstract:
We aim to describe the one dimensional dynamic of a biological population influenced by the presence of a nonlocal attractive potential and a diffusive term, under the constraint that no over crowding can occur. It is well known that this setting can be expressed by a class of aggregation-diffusion PDE's with nonlinear mobility. We investigate the existence of weak type solutions obtained as large particle limit of a suitable nonlocal version of the follow-the-leader scheme, which is interpreted as the discrete Lagrangian approximation of the target continuity equation. We restrict the analysis to nonnegative bounded initial with finite total variation, away from vacuum and supported in a closed interval with zero-velocity boundary conditions. The main novelties of this work concern the presence of a nonlinear mobility term and the non strict monotonicity of the diffusion function, thus, our result applies also to strongly degenerate diffusion equations. We also address the pure attractive regime, where we are able to achieve a stronger notion of solution. Indeed, in this case our scheme converges towards the unique entropy solution to the target PDE as the number of particles tends to infinity. This is a joint work with Marco Di Francesco and Simone Fagioli.

2018-06-27
15:45 hrs.

Dominique Spehner. Universidad de Concepción
Interacting bosons in a double-well potential: localization regime
Sala 5
Abstract:
We study the ground state of a large bosonic system trapped in a symmetric double-well potential, letting the distance between the two wells increase to infinity with the number of particles. In this context, one expects an interaction-driven transition between a delocalized state (the particles are independent and live in both wells) and a localized state (half of the particles live in each well). We start from the full many-body Schrödinger Hamiltonian in a large-filling situation where the on-site interactions and kinetic energies are comparable. When tunneling is negligible against the interaction energy, we prove a localization estimate showing that the particle number fluctuations in each well are strongly reduced and that the particles are strongly correlated. The modes in which the particles condense are minimizers of nonlinear-Schrödinger-type functionals. This is a joint work with Nicolas Rougerie.

2018-06-20
15:45 hrs.

Esteban Castillo. Pontificia Universidad Católica de Chile
Non Commutative Geometry for Particle Physics
Sala 5
Abstract:

The Standard Model of Particle Physics remains the most successful theory in science as a hole. As such its interpretation, derivation and inner workings are of fundamental importance. Non commutative geometry allows for a different point of view, giving central importance to Algebraic rather than Geometric information. In this talk the basic ideas of non commutative geometry and the fundamental results that bridge the gap to Quantum Field Theory are presented. These are applied in detail to Quantum Electro Dynamics and other theories, including the Standard model and beyond. Finally, the phenomenological aspects of the Standard Model are analyzed.

2018-06-13
15:45 hrs.

Rolando Rebolledo. Pontificia Universidad Católica de Chile
The generalised quantum Harmonic oscillator and its decoherence-free sub-algebra
Sala 5
Abstract:
We consider the definition of the generalised Quantum Harmonic Oscillator (QHO) introduced by Bath and Parthasarathy in [2]. It is well-known (see [1]) that all QMS suffers decoherence in its evolution. Loosely speaking, one says that "the quantum evolution becomes classical", this means that after some time, the dynamics concentrates on a commutative sub-algebra of observables. In [3] we characterized decoherence-free subalgebras where the evolution preserves its quantum structure. In general, these sub-algebras are trivial, nevertheless some physical systems do contain non trivial decoherence sub-algebras. More precisely, the decoherence-free subalgebra is the biggest (non commutative) where the semigroup act as a group of endomorphisms. The conference will show that the generalised QHO has a non trivial decoherence-free subalgebra.

[1] Ph. Blanchard and R. Olkiewicz, Decoherence induced transition from quantum to classical dynamics. Rev. Math. Phys. 15 (2003), no. 3, 217-243.

[2] B.V.R. Bhat and K.R. Parthasarathy, Generalized harmonic oscillators in quantum probability, Sem. Probab. XXV (1991) 39-51.

[3] A. Dhahri, F. Fagnola and R. Rebolledo, The Decoherence-free Subalgebra of a Quantum Markov Semigroup with Unbounded Generator. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (2010), no. 3, 413-433.

[4] F. Fagnola and R. Rebolledo, Entropy Production for Quantum Markov Semigroups, Commun. Math. Phys. 335 (2015), 547-570.

2018-06-06
15:45 hrs.

Edgardo Stockmeyer. Pontificia Universidad Católica de Chile
Asymptotic dynamics for certain 2D magnetic quantum systems
Sala 5
Abstract:

In this talk I will present new results concerning the long time localisation in space (dynamical localisation) of certain two-dimensional magnetic quantum systems. The underlying Hamiltonian may have the form $H=H_0+W$, where $H_0$ has dense point spectrum and rotational symmetry and $W$ is a perturbation that breaks the symmetry. (Joint with: I. Anapolitanos, E. Cárdenas, D. Hundertmark, and S. Wugalter)

2018-05-30
15:45 hrs.

Mircea Petrache. Pontificia Universidad Católica de Chile
Crystallization principles for the simplest many body systems in 2 dimensions
Sala 5
Abstract:
Consider the following fundamental crystallization question: if a point configuration (x_1,..,x_N) is a ground state for the energy E_f given as the sum of pairwise energies f(|x_i-x_j|), what principles force the configuration to approximate a particular lattice in 2-dimensions, asymptotically for very large N? The two possible settings are (a) the one in which we fix the particle density as a constraint for the minimization, or (b) the one in which we fix a natural scale via the shape of f itself, i.e. we look at one-well potentials f. In both cases, we want to find the relevant and tractable properties of the potential f which allow to treat the minimization amongst lattice and some non-lattice configurations. I will recall important known results for the two situations (a), (b), and present some new results and counterexamples, obtained in collaboration with L. Betermin.

2018-05-23
15:45 hrs.

Mauro Spera. Università Cattolica del Sacro Cuore
Remarks on Landau levels, braid groups and Laughlin wave functions
Sala 5
Abstract:
In this talk, after discussing ([1]) the holomorphic geometric quantization of a charged particle on a plane subject to a constant magnetic field perpendicular to the latter (see also [2]), we shall outline the geometric approach to unitary Riemann surface braid group representations via stable holomorphic bundles on Jacobians developed in [3] and the ensuing construction of generalized Laughlin wave functions. 
[1] A. Galasso and M. Spera: Remarks on the geometric quantization of Landau levels, Int. J. Geom. Meth. Mod. Phys. 13 (10) (2016), 1650122 (19 pages).
[2] J. Klauder and E. Onofri: Landau levels and geometric quantization, Int. J. Modern Phys. A 4 (1989) 3939–3949.
[3] M. Spera: On the geometry of some unitary Riemann surface braid groups representations and Laughlin-type wave functions, J. Geom. Phys. 94 (2015), 120-140.