Posts/News

Seminar Spectral Theory and PDE

Seminar Spectral Theory and PDE, October 24, 17:00

Speaker: Nora Doll. Friedrich-Alexander University of Erlangen

Title: Connecting Real index pairings to spectral flows

Place: PUC Chile, Campus San Joaquin, Fac. Matematicas, Sala 2

Abstract:
Object of this talk are index pairings of a projection and an unitary where both, the projection and the unitary fulfill Real symmetry relations. For a given combination of  symmetries the Noether index of the pairing vanishes, but there may be a Z_2-index given by the dimension of the kernel, modulo 2. Aim of this talk is to give an interpretation of these Z_2-indices as spectral flows. Two applications are given, one concerning topological insulators, the other concerning the Spectral Localizer.

seminar FisMat

FisMat Seminar, October 23, 15:45

Speaker: Max Lein. Tohoku University

Title: On the Topological Nature of Electromagnetic Surface Modes at Metal-Dielectric Interfaces

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 5

Abstract:
Phenomena that can be linked to the “topology of the system” have become quite ubiquitous and popular in physics, as they tend to be very robust to perturbations. Bulk-boundary correspondences make this mathematically precise. In a recent publication we have identified the presence of electromagnetic surface waves at metal-dielectric interfaces of a given polarization as a topological phenomenon. More precisely, we have shown that we can predict the existence of these surface modes by a hitherto unknown type of bulk-boundary correspondences that has no quantum analog. From a mathematical point of view, these bulk-boundary correspondences should still be regarded as a conjecture, since our work rests on explicit solutions to Maxwell’s equations for planar interfaces.

In this talk, I will explain how we have arrived at these conjectures despite the absence of a complete mathematical theory explaining the underlying mechanism. And I will propose a mathematical framework in which this bulk-boundary correspondence might be made rigorous. Three aspects are particularly intriguing: from the conceptual point of view, this is to our knowledge the first topological effect that is explained not by the analog of the hamiltonian, but by the topology of another conserved quantity — helicity. Secondly, a mathematical explanation involves Krein-selfadjoint operators on Krein spaces (as opposed to selfadjoint operators on Hilbert spaces). Roughly speaking, Krein spaces are Hilbert spaces equipped with a second, indeterminate inner product. And lastly, even the new bulk classification of non-hermitian topological insulators obtained independently by Zhou and Lee as well as Kawabata et al. does not correctly predict the topology of the system.

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
seminar FisMat

FisMat Seminar, October 9, 15:45

Speaker: Marcone Corrêa Pereira. Universidad de Sao Paulo

Title: A nonlocal approach to spatial spread in thin structures

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 5

Abstract:
In this talk we discuss an approach to considerer spatial spread in N-dimensional thin structures.  We introduce equations with nonlocal dispersal and defined in tight domains contrasting it with its corresponding local diffusion equation with Neumann and Dirichlet boundary conditions. Here the thin structure effect is modeled by an ϵ-parameter family of open sets which squeezes to a lower dimension open set as ϵ→0. The asymptotic behavior of the solutions is analyzed and the results are compared with classical situations to elliptic equations in thin domains.

conference, meeting

Mathematical Physics at the Eclipse, Afunalhue, 13.12.-16.12. 2020

From 13 to 16 of December 2020 we will have a Chilean Mathematical Physics meeting in the Ruka of the Pontificia Universidad Católica de Chile in Afunalhue close to Villarica. The idea is that this will be start of a regular Mathematical Physics meeting in Afunalhue.

On Monday, December 14 there will be a total solar eclipse visible in the lunch break!

Organizers: Christian Sadel and Giuseppe De Nittis

Preliminary Schedule

Sunday, 13.12.2020 is the arrival day, and there is a dinner in the evening.

TimeMon. 14.12.Tue. 15.12.Wed. 16.12.
8:00 - 9:30BreakfastBreakfastBreakfast
9:30-10:00Registration / WelcomeTalk 3Talk 6
10:00-10:30 Talk 1
10:30-11:00CoffeeCoffee
11:00-12:00Early LunchTalk 4Talk 7
12:00-14:30Lunch / Eclipse
(Maximum ~13:13)
LunchLunch
14:30-15:30Talk 2Talk 5Discussion Session
15:30-16:30CoffeeCoffeeCoffee
16:30-17:30Discussion SessionDiscussion SessionDeparture
18:30-20:00DinnerDinner

Participants

Rafael BenguriaPUC Chile
Olivier BourgetPUC Chile
Giuseppe De NittisPUC Chile
Christian SadelPUC Chile
Amal TaarabtPUC Chile

Sponsors

seminar FisMat

FisMat Seminar, September 11, 15:45

Speaker: Juan Felipe Lopez Restrepo. Universidad de los Andes, Colombia

Title: Edge States and Selfadjoint Extensions in the Kitaev Chain

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 5

Abstract:
In this seminar, finite discrete Kitaev chain shall be presented with its topological phase transition. A continuum limit model is derived and the diagonalization problem of the resulting bilinear hamiltonian is translated in choice of selfad-joint extensions for a one dimensional Dirac operator, which is given by a set of possible boundary conditions. It is shown that the previous result coincides with the application of the Araki’s self dual formalism and its connection with edge sates is discussed. Joint work with A. F. Reyes.

seminar FisMat

FisMat Seminar, September 4, 15:45

Speaker: Rafael Tiedra de Aldecoa. Pontificia Universidad Católica de Chile

Title: Ruled Strips With Asymptotically Diverging Twisting

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 5

Abstract:
We consider the Dirichlet Laplacian in a 2-dimensional strip composed of segments translated along a straight line with respect to a rotation angle with velocity diverging at infinity. We show that this model exhibits a “raise of dimension” at infinity leading to an essential spectrum determined by an asymptotic 3-dimensional tube of annular cross section. If the cross section of the asymptotic tube is a disc, we also prove the existence of discrete eigenvalues below the essential spectrum. Joint work with David Krejcirik (Prague).

seminar FisMat

FisMat Seminar, August 21, 15:45

Speaker: Marouane Assal. Pontificia Universidad Católica de Chile

Title: A double well problem for a system of Schrödinger operators with energy-level crossing

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

seminar FisMat

FisMat Seminar, August 14, 15:45

Speaker: Julien Royer. Universidad de Toulouse

Title: Local energy decay for the periodic damped wave equation

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