Author: Christian Sadel

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)

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

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. 

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

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 C^∗_u(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 C^∗_u(X). More precisely, we discuss which von Neumann algebras can be found inside C^∗_u(X). 

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.

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.

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.