2019-08-28
15:45 hrs.

Daniel J. Pons. Universidad Andrés Bello
Métricas no-canónicas en Diff(S1)
Sala 5
Abstract:
Re-visitamos ideas de V. I. Arnold sobre grupos de difeomorfismos de variedades. Cuando la variedad subyacente es el círculo, estudiamos la geometría de tal grupo dotado con algunas métricas.

2019-08-21
15:45 hrs.

Marouane Assal. Pontificia Universidad Católica de Chile
A double well problem for a system of Schrödinger operators with energy-level crossing
Sala 5
Abstract:
We study the existence and the asymptotic distribution of the eigenvalues of a 2*2 semiclassical system of coupled Schrödinger operators, in the case where the two electronic levels (potentials) cross at some real point and each of them admits a simple well. Considering energy levels above that of the crossing, we give the asymptotics of the eigenvalues close to such energies. In the case of symmetric wells, eigenvalues splitting occurs and we give a precise estimate of it.

This is a joint work with Setsuro Fujiie (Ritsumeikan University, Kyoto, Japan).

2019-08-14
15:45 hrs.

Julien Royer. Universidad de Toulouse
Local energy decay for the periodic damped wave equation
Sala 5
Abstract:
In this talk, we will discuss the local (or global) energy decay for the wave equation with damping at infinity. We are in particular interested in the case of a periodic (or asymptotically periodic) setting. We will mainly describe the contribution of low frequencies and observe that it behaves like the solution of some heat equation. We will see how this emerges from the spectral analysis of the damped wave equation.

2019-07-24
15:45 hrs.

Fabian Belmonte. Universidad Católica del Norte
Canonical Quantization of Constants of Motion
Sala 5
Abstract:
It is well known that Weyl quantization does not intertwine the Poisson bracket of two functions with the commutator of the corresponding operators (Groenewold- van Hove’s no go theorem). The latter suggest that Weyl quantization does not preserve the constants of motion of every given Hamiltonian, however, there are very important examples where it does so. In this talk we are going to approach the following problems:
a) Is it possible to determine the Hamiltonians for which a given canonical quantization preserves its constants of motion? We will give an interesting criteria partially answering this question in terms of the Wigner transform. We will give some important examples as well.
b) Conversely, is it possible to construct a canonical quantization preserving the constants of motion of a prescribed Hamiltonian? Under certain conditions, we will show a construction of such quantization based in the structural analogy between the description of classical and quantum constants of motion.

2019-06-05
15:45 hrs.

Horia Cornean. Aalborg University
A Beals criterion for magnetic pseudo-differential operators proved with magnetic Gabor frames
Sala 5
Abstract:
First, we give a new proof for the Beals commutator criterion for non-magnetic Weyl pseudo-differential operators based on classical Gabor tight frames. Second, by introducing a modified ‘magnetic’ Gabor tight frame, we naturally derive the magnetic analogue of the Beals criterion originally considered by Iftimie-Mantoiu-Purice. This is joint work with Bernard Helffer (Nantes) and Radu Purice (Bucharest). https://doi.org/10.1080/03605302.2018.1499777

2019-05-29
15:45 hrs.

Andrés Fernando Reyes Lega. Universidad de los Andes (Colombia)
Emergent gauge symmetries, quantum operations and anomalies
Sala 5
Abstract:
The Gelfand-Naimark-Segal (GNS) construction is a fundamental tool for the study of the representation theory of operator algebras. It also plays a prominent role in the algebraic approach to quantum field theory. In this talk I will discuss some examples of applications of the algebraic approach to quantum physics to systems with a finite number of degrees of freedom. I will illustrate how the GNS construction naturally leads to interesting connections between gauge symmetries, anomalies and quantum-information concepts like entanglement entropy and quantum operations.

2019-05-15
15:45 hrs.

Massimo Moscolari. Sapienza University of Rome
Beyond Diophantine Wannier diagrams: gap labelling for Bloch-Landau Hamiltonians
Sala 5
Abstract:
In 1978 Wannier discovered a Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. I will show how to extend this relation to a gap labelling theorem for any 2D Bloch-Landau Hamiltonian operator and to certain non-covariant systems having slowly varying magnetic fields. The integer slope will be interpreted as the Chern character of the projection onto the space of occupied states. The talk is based on a joint work with H. Cornean and D. Monaco.

2019-04-24
15:45 hrs.

Svetlana Jitomirskaya. University of California, Irvine
Cantor spectrum of a model of graphene in magnetic field
Sala 5
Abstract:
We consider a quantum graph as a model of graphene in magnetic fi elds and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero and Hausdorff dimension bounded by 1/2.

Based on joint works with S. Becker, R. Han, and also I. Krasovsky.

2019-04-17
15:45 hrs.

Walter de Siqueira Pedra. University of São Paulo
Thermodynamical Stability and Dynamics of Lattice Fermions with Mean-Field Interactions
Sala 5
Abstract:
For lattice fermions we study the thermodynamic limit of the time evolution of observables when the corresponding finite-volume Hamiltonians contain mean-field terms (like, e.g., the BCS model). It is well-known that, in general, this limit does not exist in the sense of the norm of observables, but may exist in the strong operator topology associated to a well-chosen representation of the algebra of observables. We proved that this is always the case for any cyclic representation associated to an invariant minimizer of the free energy density, if the Hamiltonians are invariant under translations. Our proof uses previous results on the structure of states minimizing the free energy density of mean-field models along with Lieb-Robinson bounds for the corresponding families of finite-volume time evolutions. This is a joint work with Jean-Bernard Bru, Sébastien Breteaux and Rafael Miada.

2019-04-03
15:45 hrs.

Jorge Antezana. National University of la Plata
Quasicrystals and Fourier analysis
Sala 5
Abstract:
Quasicrystals are non-periodic structures discovered by Shechtman in 1984 (see [Sh]). Nowadays, one of the best mathematical descriptions quasicrystals are the so called "model sets". These sets were introduced by Meyer in [M], many years before the discovery of Shechtman. In that moment, one of the aims of Meyer was to study approximation of algebraic characters by continuous ones in locally compact abelian groups (see also [L]).

Recently, important applications of quasicrystals to Fourier Analysis have been found (see [MM], [GL], [LO], [AACM] ). In this talk we will discuss some of these applications, making focus in those related with problems of sampling and interpolation in Paley Wiener spaces.

[AACM]  E. Agora, J. Antezana, C. Cabrelli, Existence of quasicrystals and universal stable sampling and interpolation in LCA groups, to appear in Trans. Amer. Math. Soc.

[GL] S. Grepstad, N. Lev,  Multi-tiling and Riesz bases. Adv. Math. 252 (2014), 1-6.

[L] J. C. Lagarias, Mathematical quasicrystals and the problem of diffraction. Directions in mathematical quasicrystals,  CRM Monogr. Ser., 13, Amer. Math. Soc., Providence (2000) 61-93.

[LO] N. Lev, A. Olevskii, Quasicrystals and Poisson's summation formula, Invent. math. 200 (2015), 585-606.

[MM] B. Matei, Y. Meyer, Simple quasicrystals are sets of stable sampling, Complex Var. Elliptic Equ. 55 (2010), 947-964.

[M] Y. Meyer, Algebraic Numbers and Harmonic Analysis, (1970) North Holland.

[Sh] D. Shechtman, I. Blech, D. Gratias, J.W. Cahn,  Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53 (1984) 1951-1953.

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.