# Seminario FisMat

El objetivo de este seminario es de reunir, de la manera la mas amplia posible, investigadores y estudiantes de la comunidad chilena e internacional alrededor de las diversas temáticas de física matemática. Profesores, investigadores jóvenes, así como estudiantes, son los bienvenidos como expositores.

Los miércoles, a las 15:45 hrs, sala 5 de la Facultad de Matemáticas.

Organización: Olivier Bourget, Giuseppe De Nittis, Christian Sadel, Edgardo Stockmeyer, Rafael Tiedra de Aldecoa.
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:
In this talk I will present a proof  an isoperimetric inequality for the harmonic  mean of the first $N-1$ non-trivial Neumann eigenvalues of the Laplace-Beltrami operator for domains contained in a hemisphere of $\mathbb{S}^N$. I will also present an overview of isoperimetric inequalities for Neumann Laplacians. This is joint work with Barbara Brabdolini and Francesco Chiacchio (U. Degli Studi Federico II, Napoli).
2018-10-03
15:45 hrs.
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.
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.