conference, doctoral schools, meeting

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. 

doctoral schools

Spring School in Analysis and Mathematical Physics, October 14-22, 2019

From October 14 to 22, we will have the Spring School in Analysis and Mathematical Physics in Santiago at the Pontificia Universidad Católica de Chile. The school is directed to undergraduate and graduate students.

Guide:
Here is the schedule ,
Here is a list of titles and abstracts ,
Here are some course files to download ,
Here is a list of participants

– Oficial website of the Doctoral School in english –
– Página oficial de la Escuela Doctoral en español –

Schedule

Weak 1, October 14-18

Mo
14.10.
Tu
15.10.
We
16.10.
Th
17.10.
Fr
18.10.
9:00-9:30Welcome
9:30-10:00Course KrejcirikCourse KrejcirikCourse RivièreCourse Richard
10:00-11:00 Talk P. Herreros
11:00-11:30coffeecoffeecoffeecoffeecoffee
11:30-1:00Course RivièreCourse Rivièretutorial C. Krejcirik
(Sadel, Krejcirik)
Course RichardCourse Sirakov
1:00-2:00lunchlunchlunchlunchlunch
2:00-3:00Course Krejcirik Talk R. Benguria Talk O. Bourget Talk C. Roman Course Sirakov
3:00-3:30coffeecoffeecoffee
3:30-4:00Presentation
graduate program
tutorial C. Riviere
(Petrache)
Course Richard
tutorial  C. Richard
(Richard, Tiedra)
coffee
4:00-5:00coffeeClub de Matematica
Mariel Saez

Weak 2, October 21-22

Mo
21.10.
Tu
22.10.
9:30-10:00Tutorial C. Sirakov
(Kamburov)
10:00-11:00 Talk A. Taarabt
11:00-11:30coffeecoffee
11:30-12:30Course Sirakov Talk M. Chuaqui
12:30-1:00TOUR
1:00-2:00lunchBBQ
2:00-3:00interviews
3:00-3:30coffee
3:30-6:00interviews

Program, Speakers and Abstracts

Courses

  • Geometrical aspects of spectral theory, by David Krejcirik (Czech Technical University).
    Abstract: Spectral theory is an extremely rich field which has found its application in many areas of physics and mathematics. One of the reason which makes it so attractive on the formal level is that it provides a unifying framework for problems in various branches of mathematics, for example partial differential equations, calculus of variations, geometry, stochastic analysis, etc.
    The goal of the lecture is to acquaint the students with spectral methods in the theory of linear differential operators coming both from modern as well as classical physics, with a special emphasis put on geometrically induced spectral properties. We give an overview of both classical results and recent developments in the field, and we wish to always do it by providing a physical interpretation of the mathematical theorems.
     
  • Operator algebras: what are they good for ?, by Serge Richard (Nagoya University).
    Abstract: Operator algebras naturally appear when dealing with families of operators having common properties. From this point of view, they can be considered as a generalization of spectral theory of a single operator. Depending on the purpose and on the context, operator algebras can be of very different types. For the study of operators on Hilbert spaces, C*-algebras and von Neumann algebras are of the greatest interest.
    Our aim for this mini-course is to introduce the concept of C*-algebras and to show how they can be effectively used for the study of operators on Hilbert space. In the first lecture we shall present the main definitions and deal with bounded operators only. During the second lecture, C*-algebraic technics will be used for the study of unbounded operators. In the final lecture, the key concepts of K-theory for C*-algebras will be introduced, and an application to Toeplitz operators will conclude this mini-course.
     
  • Minmax methods in the calculus of variations of curves and surfaces, by Tristan Riviere (ETH Zürich).
    Abstract: The study of the variations of curvature functionals takes its origins in the works of Euler and Bernouilli from the eighteenth century on the Elastica. Since these very early times, special curves and surfaces such as geodesics, minimal sur- faces, elastica, Willmore surfaces, etc. have become central objects in mathematics much beyond the field of geometry stricto sensu with applications in analysis, in applied mathematics, in theoretical physics and natural sciences in general. Despite its venerable age the calculus of variations of length, area or curvature function- als for curves and surfaces is still a very active field of research with important developments that took place in the last decades. In this mini-course we shall con- centrate on the various minmax constructions of these critical curves and surfaces in euclidian space or closed manifolds. We will start by recalling the origins of minmax methods for the length functional and present in particular the “curve shortening process” of Birkhoff. We will explain the generalization of Birkhoff’s approach to surfaces and the ”harmonic map replacement” method by Colding and Minicozzi. We will then recall some fundamental notions of Palais Smale deformation theory in infinite dimensional spaces and apply it to the construction of closed geodesics and Elastica. In the second part of the mini-course we will present a new method based on smoothing arguments combined with Palais Smale deformation theory for performing successful minmax procedures for surfaces. We will present various applications of this so called “viscosity method” such as the problem of computing the cost of the sphere eversion in 3-dimensional Euclidian space.
     
  • Variational approach to boundary value problems, by Boyan Sirakov (Pontificia Universidade Católica do Rio de Janeiro)
    Abstract: Assuming some knowledge of functional analysis and Lebesgue spaces, we start by making a brief introduction to Sobolev spaces. We develop the variational formulation of boundary value problems for ordinary and partial differential equations, and use basic results from functional analsis to show their solvability in an appropriate weak sense. If time permits, we will also discuss regularity of weak solutions.

Talks

  • Can one hear the shape of a drum? by Rafael Benguria (Pontificia Universidad Católica de Chile)
    Abstract: In 1966, Mark Kac wrote a,  now classical, paper for the American Mathematical Monthly posing that appealing question. For that manuscript Mark Kac got the 1968  Chauvenet Prize for “mathematics expository writing” of the Mathematical Association of America. In this talk I will present a review of that problem which was eventually answered in the negative by C. Gordon, D. Webb and S. Wolpert in 1992.
       
  • On Discrete Time Dependent Quantum Systems, by Olivier Francois Bourget (Pontificia Universidad Católica de Chile).
    Abstract: After a review of various concepts involved in the study of quantum dynamical systems, we will illustrate them with models displaying spectral and dynamical transitions.
     
  • Analysis of singularities in elliptic equations, by Carlos Román Parra (Pontificia Universidad Católica de Chile).
    Abstract: In this talk I will discuss problems arising in mathematical physics and conformal geometry, whose common point is about the analysis of singularities in nonlinear elliptic partial differential equations. I will begin by analyzing the occurrence of quantized vortices in the Ginzburg-Landau model of superconductivity. Then, I will discuss the occurrence of “bubbling” in the classical problem in geometric analysis of prescribing the Gaussian or scalar curvature of a manifold.
     
  • Boundary Rigidity problems in Geometry, by Pilar Herreros (Pontificia Universidad Católica de Chile).
    Abstract: In general, boundary rigidity refers to the question of whether the metric on a manifold is determined by some data on the boundary. The classical problem is boundary distance rigidity, where the distance between boundary points is known. When this fails, we can study the scattering data of the region; for each point and inward direction on the boundary, it associates the exit point and direction of the corresponding unit speed geodesic. We will discus conditions on spaces where we have distance rigidity, scattering rigidity or lens rigidity, where the boundary data considered is the scattering data plus the length of each geodesic.
     
  • Transport and localization properties in disordered models, by Amal Taarabt (Pontificia Universidad Católica de Chile)
    Abstract: Electronic transport is a central object in the study of condensed matter. In this talk we discuss how random Schrödinger operators appear naturally as auxiliary models in the study of wave propagation in such materials, and its consequences on the transport and localization properties.
     
  • Invariantes geometricos de curvas en el espacio, by Martin Chuaqui (Pontificia Universidad Carolina de Chile)
    Abstract: Discutimos invariantes geometricos de curvas en el espacio con respecto al grupo de isometrias y al grupo de transformaciones conformes.
     
  • Club de Matemática, El problema de la reina Dido, by Mariel Saez (Pontificia Universidad Católica de Chile)
    Abstract: En la Eneida se relata que la reina Dido quizo comprar tierras al rey de los Berber y éste le ofreció tanto territorio como el que ella pudiese encerrar con una piel de buey. Dido encontró la mejor solución posible al problema y fundó el reino de Cartago, el cual fue uno de los mas importantes del Mediterraneo en la antigüedad.
    ¿Cuál fue la solución que propuso Dido? ¿Cuál es la formulación matemática de este problema y como se justifica la elección de Dido?

Course materials

Organizers

Participants

Alejandro Quiroga Triviño
Alfredo Soliz Gamboa
Alisson Cordeiro Alves Tezzin
Angela Andrea Flores Concha
Angela Patricia Vargas Mancipe
Artur Henrique de Oliveira Andrade
Boris Bermudez Cardenas
Carlos Alberto Santana Rosas
Cristóbal Ignacio Vallejos Benavides
Danilo Jose Polo Ojito
Danko Aldunate bascuñan 
Fabian Domingo Caro Perez
Felipe Ignacio Flores Llarena
Fernando Rodriguez Avellaneda
Jaime Andrés Gómez Ortiz
Jarvin Javier Mestra Páez.
Javier Rolando Corregidor Maldoando
José David Beltrán Lizarazo
Juan Felipe Lopez Restrepo
Juan Manuel González Brantes
Katharina Klioba
Luciano Sciaraffia Rubio
Lukas Daniel Palma Torres
Lya Alejandra Hurtado Guzmán
Marcos Andrés Hermosilla Cartes
Mauro Javier Mendizábal Pico
Nora Christine Doll
Rubén Reinaldo Ariel Tobar Quiroz
Sandra Katherine Moreno Lemos
Sebastián Alajandro Muñoz Thon
Sebastian Camilo Puerto Galindo
Sebastian Cuellar
Simeona Quispe Monterola
Victor Eloy Ramos Marrugo
Wankar Fulguera Condori
Yikang Li
Yina Ospino Buelvas
Zamir David Beleño Rodriguez
Announcement, conference

2nd JNMP Conference on Nonlinear Mathematical Physics: 2019

The 2nd JNMP Conference on Nonlinear Mathematical Physics: 2019 conference is held from May 26 till June 4, 2019 at the University of Santiago de Chile.
USACH – Centro de estudios de postgrado y educación continua 
(Center for postgraduate studies and continuing education) 
Piso 3, Av. Apoquindo 4499, Las Condes, Región Metropolitana, Santiago, Chile

Description

This conference is being organized for the Journal of Nonlinear Mathematical Physics (JNMP) community. We aim to bring together experts and young scientists in the area of Mathematical Physics that concern Nonlinear Problems in Physics and Mathematics. The main topic of the conference is centered around the scope of JNMP: continuous and discrete integrable systems including ultradiscrete systems, nonlinear differential- and difference equations, applications of Lie transformation groups and Lie algebras, nonlocal transformations and symmetries, differential-geometric aspects of integrable systems, classical and quantum groups, super geometry and super integrable systems.