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FisMat Days Afunalhue 2024, 16.12.-19.12.

From 16th to 18th of December 2024 we will have our first FisMat days in the Ruka in Afunalhue, Campus Villarica of the Pontifical Catholic University of Chile. Poster

Tentative schedule

The following schedule is only tentative and will be adjusted. The idea is to give all participants some space to talk if they want to.

TimeMon. 16.12.Tue. 17.12.Wed. 18.12.Thu. 19.12.
8:00 - 9:00---BreakfastBreakfastBreakfast
9:00-9:40Javier LorcaDominique Spehnerfinal discussion
9:45-10:25Fabian BelmonteDaniel Parra
10:30-11:00 Coffee Coffee departure
11:00-11:40Arias CesarHanne van den Bosch
11:45-12:25Per SundellPablo Miranda
12:30-14:00 Lunch Lunch
14:00-14:40Julian CalderonChristian Sadel
14:45-15:25Santiago GonzalezMarouane Assal
15:30-16:00 Coffee Coffee
16:00-16:45ArrivalJorge Acuña---
17:00-18:00 Welcome DiscussionDiscussion
18:30-20:00 Dinner Dinner Dinner

Participants

NameUniversity
Acuña, JorgePUC Chile
Arias, CesarPUC Chile
Assal, MarouaneUSACH
Baeza, AugustinPUC Chile
Belmonte, FabianU. Católica del Norte
Calderon, JulianPUC Chile
De Nittis, GiuseppePUC Chile
Figueroa, CristianUSACH
Gonzalez, SantiagoPUC Chile
Lorca, JavierU. Frontera
Miranda, PabloUSACH
Moreno, GregorioPUC Chile
Núñez, LuisPUC Chile
Paraguez, IgnacioPUC Chile
Parra, DanielU. Frontera
Sadel, ChristianPUC Chile
Saenz, CamilaPUC Chile
Spehner, DominiqueUDEC
Sundell, PerU. Andres Bello
Taarabt, AmalPUC Chile
Van Den Bosch, HanneCMM, U. Chile

Talks

Javier Lorca, Higher Abelian Quantum Double Models: Introduction to the Characterization and Classification of the Ground State Subspace
Higher dimensional abelian quantum double models have been shown to be well defined in any finite dimension and exhibit the characteristic behavior of SPT phases models. In this talk, we will introduce the formalism of these models in a pedagogical manner, focusing on the characterization of the topological ground state subspace and briefly presenting its classification scheme. We will discuss the connection of these models with pressing problems in condensed matter physics and quantum computation.

Fabian Belmonte, TBA

Arias Cesar, What geometry can teach us about holography?
One of the most fundamental open problems in theoretical physics is the unification of the four elementary interactions into a single theory of quantum gravity. To this end, the holographic principle—the idea that the degrees of freedom that characterize a certain physical system are encoded at the boundary of spacetime—may arguably provide some key insights into our microscopic understanding of gravitational phenomena. In the context of string theory, the holographic principle crystallizes into the celebrated AdS/CFT correspondence, which conjectures that quantum gravity on a negatively curved spacetime (known as anti-de Sitter space, or AdS) is equivalent to a non- gravitational theory (referred to as a conformal field theory, or CFT) which has support on the boundary of spacetime. In this picture, spacetime may be thought of as emerging from the boundary data of the dual theory.
In this talk, we propose an holographic scheme in which dual theories do not necessarily have support on the boundary of spacetime, but they can also be localized on special interior subregions called defects. These are distinguished hypersurfaces that can be thought of as spacetime impurities, and that arguably exhibit all the relevant geometric textures that we find at the boundary of spacetime—including the capability to encapsulate degrees of freedom which can be independently described by means of some quantum field theory. In this proposed framework, spacetime can not only be reconstructed from a boundary theory, but also from dual defect theories that are located in the deep interior of spacetime itself.

Per Sundell, AKSZ quantization of unfolded fields
Quantum fields on spacetime emerges naturally within operator algebras arising from solutions to the Batalin-Vilkovisky master equation formulated in spaces of maps between Q manifolds due to Alexandrov, Kontsevich, Schwarz and Zaboronsky. We describe how the formalism applies to various theories containing gravity in their unfolded form, including higher spin gravity, and its impact on our understanding of holography

Julian Calderon, Sistemas cuánticos a temperatura finita T > 0: Fases de Berry y Uhlmann
En esta charla exploraremos la transición desde la caracterización geométrica del modelo de Ising XY a temperatura T = 0 donde un índice Z_2 se relaciona con una clase de Chern del haz generado por el estado base, hacia el análisis a T > 0. En este régimen, los estados de Gibbs, que son estados quasilibres, permiten asociar operadores positivos invertibles en el espacio de una partícula. Estos operadores generan una holonomía en la fibración de Uhlmann, que, mediante una noción de purificación, puede reinterpretarse geométricamente en términos de fases similares a las de Berry. Veremos cómo, bajo ciertas condiciones, los espectros de las respectivas holonomías coinciden.

Santiago Gonzalez, Topological classification of translation-invariant states.

Jorge Acuña, Generalizaciones del Modelo de Kitaev.

Dominique Spehner, TBA

Daniel Parra, TBA

Hanne van den Bosch, Edge modes at soft walls.
The spectrum of a periodic Hamiltonian, such as the Wallace model for graphene or the SSH chain in dimension 1, consists of absolutely continuous bands.
When such a Hamiltonian is confined to a halfspace, additional points can appear in the spectrum. They are exponentially decaying away from the interface and thus called edge modes.
The edge states correspond to eigenvalues of the Bloch Hamiltonian. One way to show their existence is through spectral flow methods.
In this talk, I will introduce these methods for the case of a discrete hamiltonian where the confinement is due to a bounded potential diverging to plus infinity on one side. This is based on joint work with David Gontier and Camilo Gomez.

Pablo Miranda, Asintótica de valores propios para operadores de Dirac magnéticos en dos dimensiones
En esta charla, presentamos resultados sobre la distribución de valores propios para operadores de Dirac magnéticos bidimensionales. Consideramos operadores con campo magnético constante, con perturbaciones eléctricas y magnéticas que tienen soporte compacto. Derivamos fórmulas asintóticas de tercer orden que incorporan una propiedad geométrica del soporte de la perturbación. Notablemente, nuestro enfoque nos permite considerar algunas perturbaciones que no necesariamente tienen signo fijo, lo cual constituye una de las principales novedades de nuestro trabajo.

Christian Sadel, Transfermatrix methods for one-channel unitary operators.
Transfermatrix methods have been extremely useful in one-dimensional systems like one-dimensional discrete Schrödinger operators or one-dimensional quantum walks. In the Hermitian case such methods have been generalized first to a certain ‘one-channel’ scheme and later to any locally finite hopping Hermitian operator (including higher dimensional discrete Schrödinger operators) giving a formula for the spectral measure at some designated root of the graph. This also implies criteria for delocalization. In the unitary world we recently developed the analogue of the one-channel scheme.

Marouane Assal, Crossings of classical trajectories and quantum resonances for Schrödinger systems
In physics, the notion of quantum resonances goes back to the very beginnings of quantum mechanics. It was proposed to explain results in scattering experiments, and it is associated to the notion of pseudo-particles with finite lifetime. To some extent, it is reasonable to think of resonances as complex numbers which are generalized eigenvalues of quantum observable. The imaginary part of a resonance is interpreted as the inverse of the lifetime of the pseudo-particle it is associated with. Thus, the closer they are to the real axis, the more meaningful they should be.
I will present some recent results on the semiclassical asymptotic distribution of resonances for systems of Schrödinger operators with crossings of the underlying classical trajectories.

Organizers

Christian Sadel and Giuseppe De Nittis.

Sponsors

The event is sponsored by the Mathematics Faculty of the Pontifical University of Chile

conference, Events, meeting

2nd Chile-Japan Workshop on Mathematical Physics and Partial Differential Equations

The Second Chile-Japan Workshop on Mathematical Physics and Partial Differential Equations will take place at the

Mathematics and Computer Science Department, University of Santiago of Chile, Sept. 25-28, 2023.

More information is available on the conference website.

The first workshop took place in the Graduate School of Mathematical Sciences, University of Tokyo, on September 2018. 

seminar FisMat

FisMat Seminar, Diciembre 7, 2022, 14:30

Speaker: Tobias Ried. Max Planck Institute

Title: Cwikel’s bound reloaded

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 1

Abstract:
The Cwikel-Lieb-Rozenblum (CLR) inequality is a semi-classical bound on the number of bound states for Schrödinger operators. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel’s proof is said to give a constant which is at least about 2 orders of magnitude off the truth.
In this talk I will give a brief overview of the CLR inequality and present a substantial refinement of Cwikel’s original approach which leads to an astonishingly good bound for the constant in the CLR inequality. Our proof is quite flexible and leads to rather precise bounds for a large class of Schrödinger-type operators with generalized kinetic energies. Moreover, it highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis. (joint work with D. Hundertmark, P. Kunstmann, and S. Vugalter)

seminar FisMat

FisMat Seminar, Noviembre 23, 2022, 14:30

Speaker: Danilo Polo, PUC

Title: Topological quantization of interface currents in magnetic quarter-plane systems

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 1

Abstract:
A magnetic interface is a thin region of the space that separates the material in two parts subjected to perpendicular uniform magnetic fields of distinct intensity. It is well-known that interfaces may lead to surface modes which can carry edge currents. The aim of this talk is to show (in the tight-binding approximation) the topological quantization of such currents for magnetic quarter-plane systems, and by using K-theory, we derive bulk-interface dualities. If time allows, I will state the necessary conditions to produce topological non-trivial corner states in these systems.Joint work with: Giuseppe De Nittis

seminar FisMat

FisMat Seminar, Noviembre 16, 2022, 14:30

Speaker: Avelio Sepúlveda, U de Chile

Title: Relaciones entre el campo libre Gaussiano y el modelo de spins O(N)

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 1

Abstract:
En esta presentación, se discutirá la correlación de dos puntos del modelo de spin O(3).  Siguiendo las ideas de Patrascioiu y Seiler, expresamos la correlación como una constante por la probabilidad que ambos puntos pertenezcan a una componente conexa de un modelo de percolación. Hacemos esto para mostrar que uno de los argumentos de estos autores puede ser contradecido construyendo un modelo XYen un ambiente aleatorio que satisface lo siguiente: El modelo tiene baja temperatura excepto en un pequeño conjunto que no percola; y la función a dos puntos decrece exponencialmente. Es instrumental para estos resultados comprender la relación entre el campo libre Gaussiano y los modelos de spin O(N).Trabajo en conjunto con Juhan Aru y Christophe Garban. 

seminar FisMat

FisMat Seminar, Octubre 26, 2022, 14:30

Speaker: Giuseppe de Nittis, PUC

Title: The Magnetic Spectral Triple: Applications and Open Questions

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 1

Abstract:
Since the early works by Bellissard, non-commutative geometry (NCG) has proved to be an excellent tool for the analysis of the quantum Hall effect (QHE), and more in general for the study of the topological phases of matter. The central object of the Bellissard’s NCG for the QHE is a spectral triple designed to deal with tight-binding operators. In this talk we will present a new spectral triple suitable to treat  continuous magnetic operators. We will show how the QHE in the continuous can be described inside this new NCG. Certain possible new applications, along with some related open questions, will be also presented. Joint work with: F. Belmonte & M. Sandoval

seminar FisMat

FisMat Seminar, Septiembre 7, 2022, 15:30

Speaker: Bruno de Mendonça Braga, Puc-Rio

Title: Embeddings of von Neumann algebras into uniform Roe algebras

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 1

Abstract:
Given a uniformly locally finite metric space X, its uniform Roe algebra, denoted by Cu(X), is a C-algebra of bounded operators on the Hilbert space ℓ2(X) which captures the large scale geometry of X. This algebra was introduced by John Roe in 1988 and it has since become a topic of interest to researchers in many different fields such as operator algebras, geometric group theory, and mathematical physics. As for the latter, uniform Roe algebras have recently started to be used as a framework in mathematical physics to study the classification of topological phases. In this talk, we will discuss some recent developments about the structure of Cu(X). More precisely, we discuss which von Neumann algebras can be found inside Cu(X). 

seminar FisMat

FisMat Seminar, Agosto 31, 2022, 15:30

Speaker: Pablo Miranda, USACH

Title: Resonancias cerca de umbrales para operadores de Schrödinger discretos

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 1

Abstract:
En esta charla consideramos versiones generalizadas de operadores Schrödinger definidos en Z⊗CN. Cerca de los umbrales del espectro estudiamos la distribución de las resonancias de nuestros operadores. La cantidad de resonancias cerca de cada umbral es finito y nuestro primer resultado es obtener el número exacto de estas. Además, obtenemos una descripción precisa de la localización de las resonancias, en términos de cúmulos cerca de ciertos puntos en el plano complejo. Estos resultados son bastante naturales pero no aparecen en la literatura físico-matemática.En la segunda parte, consideramos operadores definidos en Z⊗Z pero con restricciones que inducen a las partículas a moverse de manera “horizontal”. Estos operadores tienen una estructura similar a los de la primera parte, pero presentan un fenómeno de acumulación de resonancias que nosotros describimos a través del comportamiento asintótico de la función de conteo de estas.Parte de nuestro resultados son válidos para operadores no auto-adjuntos. Este es parte de un trabajo conjunto con Marouane Assal, Olivier Bourget y Diomba Sambou.

seminar FisMat

FisMat Seminar, Julio 28, 2022, 15:30

Speaker: Mircea Petrache, PUC

Title: Building examples of graphs that allow infinitely many sharp isoperimetric shapes

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 1

Abstract:
Discrete isoperimetric shapes are configurations that reach equality in the edge-isoperimetric inequality among subsets of a fixed graph. Equivalently, the question is to find the shape of the best crystal grain, given the crystal structure of a material. Equality in the discrete edge-isoperimetric inequality is hard to achieve, and the ambient graphs which have infinite families of discrete isoperimetric shapes are rare: Our goal is to build a large class of examples. We start from the “macroscopic” or “continuum” isoperimetric problem, with two approaches, one via PDE and one via Optimal Transport. We build a new discrete strategy which combines the two approaches. Our strategy poses several nice new challenges, and it highlights the close link between semidiscrete optimal transport and convexity. In this introductory talk, I describe what new classes of examples we find, and also some mysterious directions still to be explored.

seminar FisMat

FisMat Seminar, Junio 22, 2022, 11:30

Speaker: Walter Alberto de Siqueira Pedra

Title: Equilibrium States of Many-Body (Fermion) Systems with Long-Range Interactions

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 1

Abstract:
We present our results on infinite volume equilibrium states of many-fermion systems on the lattice with mean-field interactions (“Non-cooperative Equilibria of Fermi Systems with Long-Range Interactions”, Memoirs of the AMS, 2013) in relation to our recent contributions on Kac limits for equilibrium states (“From Short-Range to Mean-Field Models in Quantum Lattices”, arXiv:2203.01021): We recently proved that under very general conditions such long-range limits of equilibria of short-range models are equilibria of some naturally associated mean-field model. This is reminiscent of well-known works of Penrose and Lebowitz (1966, classical case), and Lieb (1971, quantum), on the Kac limit of the pressure in the thermodynamic limit.