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.

2018-05-16
15:45 hrs.

Kevin Morand. Universidad Técnica Federico Santa María Valparaíso
Three approaches to non-Riemannian geometries
Sala 5
Abstract:
Recent years have seen a surge of interest regarding non-Riemannian geometries, the most prominent examples thereof being Newton-Cartan and Carroll geometries. We review these two dual geometries and related aspects from three perspectives:

1) Intrinsic à la Cartan

2) Ambient à la Eisenhart

3) Algebraic à la Cartan (again)

2018-05-09
15:45 hrs.

Martin Chuaqui. Pontificia Universidad Católica de Chile
La derivada Schwarziana de Ahlfors y ceros de soluciones de ecuaciones lineales
Sala 5
Abstract:
Expondremos la definición y propiedades de la derivada Schwarziana de Ahlfors, junto a un resumen de resultados conocidos y problemas abiertos. Se discutira el concepto de parametrizacion de Möbius de curvas de largo infinito, para terminar con un teorema sobre los ceros de soluciones de ecuaciones lineales en la recta real.

2018-05-02
15:45 hrs.

María Isabel Cortez. Universidad de Santiago de Chile
Propiedades espectrales de los sistemas minimales de Cantor
Sala 5
Abstract:
Ejemplos de sistemas minimales de Cantor son los subshifts de substitución y algunas extensiones simbólicas de las rotaciones sobre el círculo. En esta charla veremos algunas propiedades que caracterizan a los valores propios de estos sistemas, así como su relación con el grupo de dimensión, invariante algebraico asociado a estos sistemas. Algunas de estas propiedades se pueden extender a los sistemas dinámicos generados por las traslaciones de un tiling en R^d.

2018-04-24
15:45 hrs.

Richard Froese. University of British Columbia
Resonances lost and found
Sala 5
Abstract:
We compute the large $L$ asymptotics of the resonances for one dimensional Schrödinger operators $H_L=V_1(x)+\mu(L)V_2(x−L)$, where $V_1$ and $V_2$ are compactly supported and $\mu(L)\sim e^{-cL}$ for $c\ge0$. These are compared to the Schrödinger dynamics of $H_L$. This is joint work with Ira Herbst.

2018-04-18
15:45 hrs.

Amal Taarabt. Pontificia Universidad Católica de Chile
On the current-current correlation measure for random Schrödinger operators
Sala 5
Abstract:
We review various properties of random Schrödinger operators and recall formulations of conductivity and current-current correlation measure. In this talk we will present a panoramic view and recent results on localized regime. We will focus in particular on the diagonal behaviour problem of the ccc-measure and explain how it is related to the localization length. This is a work in progress with J. Bellissard and G. De Nittis.

2018-04-11
15:45 hrs.

Carlos Román. Leipzig University & Max Planck Institute for Mathematics in The Sciences
On the 3D Ginzburg-Landau model of superconductivity
Sala 5
Abstract:
The Ginzburg-Landau model is a phenomenological description of superconductivity. A crucial feature is the presence of vortices (similar to those in fluid mechanics, but quantized), which appear above a certain value of the applied magnetic field called the first critical field. We are interested in the regime of small ε, where ε>0 is the inverse of the Ginzburg-Landau parameter (a material constant). In this regime, the vortices are at main order codimension 2 topological singularities.
In this talk I will present a quantitative 3D vortex approximation construction for the Ginzburg-Landau energy, which provides an approximation of vortex lines coupled to a lower bound for the energy, optimal to leading order, analogous to the 2D ones, and valid for the first time at the ε-level. I will then apply these results to describe the behavior of global minimizers for the 3D Ginzburg-Landau functional below and near the first critical field. I will also provide an ε-quantitative product-type lower bound for the study of Ginzburg-Landau dynamics.

2018-04-04
15:45 hrs.

Rafael Tiedra de Aldecoa. Pontificia Universidad Católica de Chile
Quantum time delay for unitary propagators
Sala 5
Abstract:
We give the definition of quantum time delay in terms of sojourn times for unitary propagators in a two-Hilbert spaces setting. We prove that this time delay defined in terms of sojourn times (time-dependent definition) exists and coincides with the expectation value of a unitary analogue of the Eisenbud-Wigner time delay operator (time-independent definition). Our proofs rely on a new summation formula relating localisation operators to time operators and on various tools from functional analysis such as Mackey's imprimititvity theorem, Trotter-Kato Formula and commutator methods for unitary operators. Joint work with Diomba Sambou (PUC).
https://doi.org/10.1142/S0129055X19500181

2018-03-28
15:45 hrs.

Diomba Sambou. Pontificia Universidad Católica de Chile
Complex eigenvalues for a non-self-adjoint Dirac operator
Sala 5
Abstract:
We will consider a 2d Dirac operator with constant magnetic field perturbed by non-self-adjoint potentials. It is well known that when it is perturbed by certain self-adjoint potentials, then, there is creation and accumulation of real eigenvalues near every point of its essential spectrum given by a set of degenerate isolated eigenvalues called the Landau levels. Recently, similar results have been proved for Schrödinger operators perturbed by non-self-adjoint perturbations showing the existence of complex-valued potentials generating infinitely many non-real eigenvalues accumulating at every point of [0,+∞). We will present a similar result for the 2d Dirac operator above, showing the existence of non-self-adjoint perturbations generating infinitely many non-real eigenvalues accumulating at every Landau level.

2018-03-21
15:45 hrs.

Nicolas Raymond. University of Rennes 1
Survey on the semiclassical magnetic Laplacian
Sala 5
Abstract:
In the first part of the talk, I will describe recent advances about the description of the discrete spectrum of the magnetic Laplacian, in the semiclassical limit. In the second part, I will describe two results in two dimensions, essentially in the case of a non-degenerate magnetic well: the Birkhoff normal forms and their applications (collaboration with S. Vu Ngoc) and the magnetic WKB constructions (collaboration with Y. Bonthonneau).

2018-03-14
15:45 hrs.

Timo Weidl. University of Stuttgart
The edge resonance in elastic media with zero Poisson coefficient
Sala 5
Abstract:
A two-dimensional elastic semistrip with stress-free boundary conditions and zero Poisson coefficients has an embedded eigenvalue on top of the continuous spectrum. This effect is known as edge resonance. For an infinite plate of finite thickness with a drilling hole $(R^2\setminus\Omega)\times I$ actually infinitely many edge resonances will occur. This is related to the spectral problem of perturbations of symbols with strongly degenerated minima, which also appear in BCS theory. I give an overview on some of our results in this area, which still poses a number of mathematical challenges.