Seminarios

Futuros Eventos

2019-04-17
15:45 hrs.
Seminario Fismat
Walter de Siqueira Pedra. University of São Paulo
Tba
Sala 5
2019-04-03
15:45 hrs.
Seminario Fismat
Jorge Antezana. National University of la Plata
Tba
Sala 5
2019-03-27
17:00hrs.
Seminario de Modelamiento Matemático
Claire Delplancke. Center of Mathematical Modeling in Santiago, Universidad de Chile
Bayesian Modeling for Inverse Problems? The Example of a Scalable Algorithm for Passive Seismic Tomography in Mining.
Sala 1, Edificio Rolando Chuaqui, Campus San Joaquín, Avda. Vicuña Mackenna 4860, Macul, Chile.
2019-03-26
14:00hrs.
Seminario de Geometría Algebraica
Pedro Montero. Utfsm
Introducción al Minimal Model Program y Acotamiento
sala 2
Abstract:
Uno de los objetivos de la geometría algebraica es la clasificación de variedades proyectivas. En dimension 1, existe una clara distinción entre los tipos de curvas de género 0, 1 y mayor o igual a 2, tanto de un punto de vista topológico (revestimiento universal), de la curvatura y, más algebraicamente, de acuerdo a la negatividad, trivialidad o positividad de su clase canónica. En esta charla, discutiremos sobre cómo el Minimal Model Program (MMP) permite conjeturalmente asociar a toda variedad algebraica un "modelo minimal" y reducir el problema de clasificación a estudiar tres grandes familias: variedades de Fano (canónico negativo), Calabi-Yau (canónico trivial) y canónicamente polarizadas (canónico positivo). En este marco, la existencia de una cantidad finita de familias (acotamiento) es crucial para poder esperar obtener clasificaciones explícitas, y unos de los primeros grandes pasos en esta dirección son los resultados de acotamiento de variedades de Fano y canónicamente polarizadas obtenidos por Alexeev en el caso de superficies (medianamente singulares). Dichos resultados han sido generalizados recientemente a dimensiones superiores por Birkar en el caso de variedades de Fano (gracias a lo cual ha obtenido la medalla Fields) y por Hacon-McKernan-Xu en el caso canónicamente polarizado. El objetivo de este seminario será entender el lenguaje y técnicas del MMP en el caso de superficies singulares así como el rol de la "teoría de complementos", introducida por Shokurov y utilizada por Birkar para extender los resultados de Alexeev.
 
El plan preliminar de las charlas podría ser entonces el siguiente (también, modificar a gusto):
1. Introducción al Minimal Model Program y acotamiento (Pedro)
2. MMP para superficies suaves (Pedro)
3. Log MMP para superficies.
4. Singularidades ADE
5. Propiedades de pares logarítmicos
6. Clasificación de pares log canónicos en dimensión 2
7. Adjunción y Lema de Conectividad
2019-03-25
16:30 - 17:30hrs.
Seminario de Sistemas Dinámicos
Michele Triestino. U. Bourgogne
Cantor Dynamics and Simple Left-Orderable Groups
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central)
Abstract:
I will present a construction of simple groups of homeomorphisms of the real line.

Given a homeomorphism of a Cantor set $\sigma: X \to X$, consider the suspension $Y=X \times [0,1] / (x,1) \sim (\sigma(x),0)$, and look at the group $H_0(Y)$ of homeomorphisms of $Y$, isotopic to the identity. If $\sigma$ is minimal, then $H_0(Y)$ is simple [Aliste-Prieto - Petite], and I will describe countable subgroups $T(Y)$ which are also simple. These are reminiscent of the classical Thompson groups, and feature several nice properties. For instance, when \sigma is a minimal subshift, $T(Y)$ is finitely generated.

Joint work with Nicolás Matte Bon.
2019-03-22
16:00hrs.
Club de Matemática
Jan Kiwi. PUC
¿ Cómo Ubicarse? Matemáticas y Gps
Ninoslav Bralic
Abstract:
Uno de los inventos relativamente recientes que nos ha cambiado la vida es el GPS (Global Positioning System). En esta charla veremos algunos aspectos matemáticos involucrados en el funcionamiento de este sistema de posicionamiento.
http://www.clubdematematica.cl/
2019-03-22
12:00 hrs.
Seminario de Estadística
Debajyoti Sinha. Florida State University
Semiparametric Bayesian Latent Variable Regression for Skewed Multivariate Data
Sala 5, Facultad de Matemáticas, Edificio Rolando Chuaqui, Campus San Joaquin, Pontificia Universidad Católica de Chile
Abstract:
For many real-life studies with skewed multivariate responses, the level of skewness and association structure assumptions are essential for evaluating the covariate eff ects on the response and its predictive distribution. We present a novel semiparametric multivariate model and associated Bayesian analysis for multivariate skewed responses. Similar to multivariate Gaussian, this multivariate model is closed under marginalization, allows a wide class of multivariate associations, and has meaningful physical interpretations of skewness levels and covariate eff ects on the marginal density. Other desirable properties of our model include the Markov Chain Monte Carlo computation through available statistical software, and the assurance of consistent Bayesian estimates of the parameters and the nonparametric error density under a set of plausible prior assumptions. We illustrate the practical advantages of our methods over existing alternatives via simulation studies, the analysis of a clinical study on periodontal disease and extensions to Bayesian regression trees.

This is a joint work with Drs. A.Bhingare, S.Lipsitz, D.Bandopadyay and A.Linero.

Seminario organizado por el Centro para el Descubrimiento de Estructuras en Datos Complejos - MiDaS.
http://midas.mat.uc.cl
2019-03-22
14:00hrs.
Seminario de Teoría de Números
Natalia García. Pontificia Universidad Católica de Chile
Alturas en Teoría de Números
Sala 2
Abstract:
En esta charla vamos a definir algunas funciones de altura, las cuales son esencialmente una manera de medir complejidad aritmética, y constituyen una herramienta fundamental. Veremos también algunas de sus propiedades aplicadas a diversos temas en teoría de números.
http://www.mat.uc.cl/~natalia.garcia/stn.html
2019-03-20
14:00hrs.
Seminario de Ingeniería Matemática y Computacional
Ignacio Labarca. Alumno Magíster, Instituto de Ingeniería Matemática y Computacional
Convolution Quadrature Methods for Time-Domain Scattering From Unbounded Penetrable Interfaces
Auditorio San Agustín Campus San Joaquin
Abstract:
We present a class of boundary integral equal on methods for the numerical solution of acoustic and electromagnetic time-domain scatering problems in the presence of unbounded penetrable interfaces in two-spatial dimensions. The proposed methodology relies on Convolution Quadrature (CQ) methods in conjunction with the recently introduced Windowed Green Function (WGF) method. As in standard time-domain scatering from bounded obstacles, a CQ method of the user’s choice is utilized to transform the problem into a finite number of (complex) frequency-domain
problems posed on the domains involving penetrable unbounded interfaces. Each one of the frequency-domain transmission problems is then formulated as a second-kind integral equation that is effectively reduced to a bounded interface by means of the WGF method—which introduces errors that decrease super-algebraically fast as the window size increases. The resulting windowed integral equations can then be solved by means of any (accelerated or unaccelerated) off-the-shelf Helmholtz boundary integral equation solver capable of handling complex wavenumbers with large imaginary part. A high-order Nystrom method based on Alpert quadrature rules is utilized here. A variety of numerical examples including wave propagation in open waveguides as well as scatering from multiply layered media, demonstrate the capabilities of the proposed approach.

Eventos Pasados

2019-03-19
16:30hrs.
Santiago Number Theory and Algebra Seminar (Santas)
José Ignacio Burgos Gil. Instituto de Ciencias Matemáticas, Madrid
Height pairing between arithmetic cycles
Abstract:
The linking number between two circles is the number of windings of one circle around the other. This is a topological invariant and is a first example of a secondary characteristic class. Analogues of the linking number can be defined in many situations. For instance the height pairing between algebraic cycles is a generalization of the cross ratio between four points in the projective line and can be seen as a "linking number" that has a very nice Hodge theoretical interpretation. 
Higher Chow groups have been introduced by Bloch as a concrete way to represent motivic cohomology. In this talk I will explain how to define a height pairing between higher cycles. This is joint work with S. Goswami and G. Pearlstein.

Sala de seminarios (4to piso, lado norte de) Dpto de Matemática y Ciencia de la Computación USACH
2019-03-19
16:00hrs.
Seminario de Análisis y Geometría
Azahara de la Torre Pedraza. University of Freiburg
On higher dimensional singularities for the fractional Yamabe problem
Abstract:
We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, for which we establish the classical gluing method of Mazzeo and Pacard for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of a fractional order ODE,
and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. Note, however, that no traditional phase-plane analysis is available here. Instead, first, we provide a rigorous construction of radial fast-decaying solutions by a blow-up argument and a bifurcation method. Second, we use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be. Third, for the linear theory, we use complex analysis and some non-Euclidean harmonic analysis to  examine a fractional Schrödinger equation with a Hardy type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of  Frobenius method. Surprisingly enough, a fractional linear ODE may still have a two-dimensional kernel as in the second order case.
Sala 2
2019-03-18
16:30-17:30hrs.
Seminario de Sistemas Dinámicos
Yiwei Zhang. Center for Mathematical Sciences, Huazhong University of Science and Technology, China
Understanding physical mixing processes via transfer operator approach
Abstract:
Industrial and chemical mixing processes of various kinds occur throughout nature and are vital in many technological applications.In the context of discrete dynamical systems, the transfer operator approach has been shown as a powerful tools from both theoretic and numerical viewpoint.

In this talk, I will use a toy model (i.e., the one dimensional stretch and fold map) as an example to provide a brief introductionon the relationships between the spectral properties of the associated transfer operator and the estimations of the optimal mixing rate of the mixing process. Moreover, I will address how the optimal mixing rate varies according to the stretch and fold map has "cutting and shuffling'' behaviour (i.e., composing with a permutation). 

If time permits, I will also talk about how to interpret this problem to the eigenvalue estimations for the Random bi-stochastic matrices (free probability theory) and the locations of poles of the dynamical zeta function.
Sala 1, PUC, Facultad de Matemáticas, Av. Vicuña Mackenna 4860, Macul, La Florida
2019-03-15
14:00hrs.
Seminario de Teoría de Números
Ricardo Menares. Pontificia Universidad Católica de Chile
La constante de Hermite
Abstract:
Un empaquetamiento de discos en el plano es una familia de discos del mismo radio y cuyos interiores no se intersectan. No es posible cubrir el plano con un empaquetamiento de discos, por lo que la pregunta natural es encontrar uno que recubra la mayor área posible. La respuesta involucra un cierto reticulado del plano.
 
En esta charla consideraremos empaquetamientos de bolas en un espacio euclidiano de dimensión arbitraria n. Nos restringiremos a empaquetamientos provenientes de reticulados. La n-ésima constante de Hermite mide la eficiencia, en términos de volumen recubierto, que puede alcanzar un empaquetamiento de bolas en tal espacio. 
 
La constante de Hermite es conocida solo para valores pequeños de n. También se dispone de información sobre su valor asintótico. Explicaremos algunos de estos resultados, así como las preguntas abiertas. No asumiremos conocimientos especializados de Teoría de Números.

Sala 2
2019-03-15
16:00hrs.
Coloquio de Matemática UC
Aníbal Medina. University of Notre Dame
Conmutatividad en topología, homotopía y análisis de datos
Abstract:
En esta charla hablaremos sobre el uso de la conmutatividad en la clasificación de espacios topológicos. Recordaremos la construcción del invariante "cohomología" con su estructura de álgebra conmutativa y cómo ésta se levanta, módulo homotopias coherentes, a un modelo de cocadenas. Dicha manifestación homotópica de la conmutatividad nos entrega aún más información topológica que, gracias al desarrollo de nuevos algoritmos, puede ser efectivamente incorporada en los actuales protocolos para el análisis topológico de datos.
auditorio Ninoslav Bralic
2019-03-14
16:30 -- 17:30hrs.
Seminario de Sistemas Dinámicos
Neil Dobbs. University of Geneva
TBA
Sala 5
2019-03-13
15:45 hrs.
Seminario Fismat
Monika Anna Winklmeier. Universidad de los Andes
Estimates for eigenvalues in gaps of the essential spectrum
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.
Sala 5
2019-03-12
16:30 -- 18:00hrs.
Seminario de Sistemas Dinámicos
Hamal Hubbard. Cornell University
Construccion de aplicaciones pseudo-Anosov
Sala 1
2019-03-12
16:30hrs.
Santiago Number Theory and Algebra Seminar (Santas)
Sebastián Herrero. Instituto de Matemáticas, Pucv
Equidistribución p-ádica de puntos enteros en hipersuperficies cuadráticas
Abstract:
En la primera parte de esta charla repasaremos resultados de Pommerenke, Linnik, Duke y Schulze-Pillot, entre otros, sobre la equidistribución de puntos enteros en hipersuperficies cuadráticas en espacios euclideanos. 
En la segunda parte presentaremos resultados análogos en el mundo p-ádico, cuya demostración hace uso de la teoría de formas modulares y cotas para sus coeficientes de Fourier.
Los resultados que serán presentados forman parte de un proyecto en colaboración con R. Menares y J. Rivera-Letelier.

Sala de seminarios (4to piso, lado norte de) Dpto de Matemática y Ciencia de la Computación USACH
2019-03-12
16:00hrs.
Seminario de Análisis y Geometría
Andrés Larraín-Hubach. University of Dayton, Ohio
Conexiones auto-duales sobre espacios Taub-NUT
Abstract:
Las ecuaciones de Yang-Mills son un sistema de ecuaciones en derivadas parciales, definidas sobre variedades suaves en cuatro dimensiones, con un profundo significado geométrico. Las propiedades de las soluciones de estas ecuaciones, sobre variedades compactas, han sido  analizadas desde los años sesenta y han arrojado resultados importantes tanto en matemáticas como en física. Las soluciones sobre  variedades no compactas no han sido estudiadas tan ampliamente y aún hay muchas preguntas importantes sin respuesta. En esta charla, basada en resultados obtenidos en colaboración con Sergey Cherkis y Mark Stern, explicaré diversas propiedades de ciertas  soluciones a las ecuaciones de Yang-Mills, definidas sobre unas variedades abiertas especiales llamadas Espacios Taub-NUT. En particular, explicaré dos argumentos distintos para probar un teorema de índice necesario en la construcción.
 

Sala 2
2019-03-11
16:30 -- 18:00hrs.
Seminario de Sistemas Dinámicos
Hamal Hubbard. Cornell University
Construccion de aplicaciones pseudo-Anosov
Sala 1
2019-03-08
15:00hrs.
Coloquio de Matemática UC
Mark Spivakovsky. Institut de Mathématiques de Toulouse, Université Paul Sabatier
Introduction to the problem of resolution of singularities in algebraic geometry
Abstract:
The subject of this talk is the problem of resolution of singularities in algebraic geometry, but it is intended for a general mathematical audience. The problem of resolution of singularities asks whether, given an algebraic variety X over a field k, there exists a non-singular algebraic variety X' and a proper map X' -> X which is one-to-one over the non-singular locus of X. If we cover X' by affine charts, the problem becomes one of parametrizing pieces of X by small pieces of the Euclidean space k^n.

All the basic notions such as algebraic variety, singularity, birational map, etc., will be defiend from scratch. We will describe an algorithm for resolving the singularities of plane curves. We will explain how to generalize this algorithm to higher dimensions, thereby giving a brief sketch of the proof of Hironaka's celebrated theorem on resolution of singularities of varieties over fields of characteristic zero.

Time permitting, we will briefly discuss the difficulties that arise in trying to generalize Hironaka's result to fields of positive characteristic.
Auditorio Ninoslav Bralic
2019-03-08
16:00hrs.
Coloquio de Matemática UC
Heisuke Hironaka. Japan Association for Mathematical Sciences / Harvard University
Embedded Resolution of Singularities in Algebraic Geometry
Abstract:
Algebraic geometry underwent phenomenal transform from *geometric* to *algebraic* in the manner of concepts and proof techniques. Resolution of singularity is typical example among many other classical and/or new problems.

Personally, I had been strongly influenced and much indebted by foundational contributions of my teachers: Oscar Zariski, Masayoshi Nagata, Alexander Grothendieck and many others.

Here I want to present my own additions and implementations strictly focusing my attention on problems about embedded resolution of singularities. Technically new concepts and techniques in my own contributions will be explained by the following technical terms:

1) "Idealistic exponent" and its application to "MOIE"
2) Singularity set "S" and algebraic translation "P"
3) "Q" smoothing of arithmetic singularity
4) "Escalator-elevator"  imagery transformation of "Q" smooth
Auditorio Ninoslav Bralic
2019-03-06
15:45 hrs.
Seminario Fismat
Christian Jaekel. University of São Paulo
On reflection positivity, modular localisation and Connes cocycles
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.
Sala 5
2019-03-05
16:30hrs.
Santiago Number Theory and Algebra Seminar (Santas)
Stefan Gille. University of Alberta
Residue maps for hermitian forms of central simple algebras
Abstract:
Residue maps for quadratic forms over fields of characteristic not 2 are well known and a useful tool to study these objects. One can construct these maps also using derived Witt groups, and this approach can be generalized to hermitian Witt groups of central simple algebras with involutions. In case of involutions of the first can there is also a direct definition possible, and one can prove an analog of Springer's exact sequence. The latter holds for certain more general involutions as well.

Sala de seminarios (4to piso, lado norte de) Dpto de Matemática y Ciencia de la Computación USACH
2019-01-25
14:30hrs.
Santiago Number Theory and Algebra Seminar (Santas)
Diego Izquierdo. Mpim Bonn
Dimensión de los cuerpos, K-teoría y aritmética de los espacios homogéneos
Abstract:
En 1986, Kato y Kuzumaki hicieron varias conjecturas con las que esperaban dar una caracterización diofántica de la dimensión cohomológica de los cuerpos en términos de K-teoría de Milnor y de puntos racionales sobre hipersuperficies proyectivas de pequeño grado. Hoy en día, sabemos que estas conjecturas son falsas en general. En esta charla, veremos que si uno sustituye las hipersuperficies proyectivas de pequeño grado por los espacios homogéneos, la conjectura de Kato y Kuzumaki se vuelve cierta. Se trata de un trabajo en colaboración con Giancarlo Lucchini Arteche.
Sala 2
2019-01-14
16:30hrs.
Seminario de Sistemas Dinámicos
Alberto Pinto. Faculty of Sciences, University of Porto
Piecewice chaotic maps
Abstract:
We will consider the class of Cr unidimensional piecewise maps with a transitive attractor. These maps can have simultaneously discontinuities, criticalities and singularities. We will show that topological chaos is equivalent to metric chaos. We recall that this result is known in several classes strictly contained in the general class that we are presenting.
Sala 2
2019-01-11
12:00 hrs.
Seminario de Estadística
David Dahl. Department of Statistics, Brigham Young University
Summarizing Distributions of Latent Structure
Abstract:
In a typical Bayesian analysis, consider effort is placed on "fitting the model" (e.g., obtaining samples from the posterior distribution) but this is only half of the inference problem.  Meaningful inference usually requires summarizing the posterior distribution of the parameters of interest.  Posterior summaries can be especially important in communicating the results and conclusions from a Bayesian analysis to a diverse audience.  If the parameters of interest live in R^n, common posterior summaries are means, medians, and modes.  Summarizing posterior distributions of parameters with complicated structure is a more difficult problem.  For example, the "average" network in the posterior distribution on a network is not easily defined. This paper reviews methods for summarizing distributions of latent structure and then proposes a novel search algorithm for posterior summaries.  We apply our method to distributions on variable selection indicators, partitions, feature allocations, and networks.  We illustrate our approach in a variety of models for both simulated and real datasets.

Seminario organizado por el Centro para el Descubrimiento de Estructuras en Datos Complejos - MiDaS.
Sala 5, Facultad de Matemáticas, Edificio Rolando Chuaqui, Campus San Joaquin, Pontificia Universidad Católica de Chilehttp://midas.mat.uc.cl
2019-01-10
12:00 hrs.
Seminario de Estadística
Dae-Jin Lee. Bcam - Basque Center for Applied Mathematics
Hierarchical modelling of patient-reported outcomes data based on the beta-binomial distribution
Abstract:
The beta-binomial distribution does not belong to the exponential family and, hence classical regression techniques cannot be used when dealing with outcomes following the mentioned distribution. In this talk, we propose and develop regression models based on the beta-binomial distribution for the analysis of U, J or inverse J-shaped discrete and bounded outcomes. In fact, although this work is focused on the analysis of patient-reported outcomes (PROs), which usually follow the mentioned distributional shapes, proposed models can also be extended to several fields. First of all, we make a review and comparison of existing beta-binomial regression approaches in independent data context, concluding that the marginal approach is the most adequate. However, PRO studies are usually carried out in a longitudinal framework, where patients' responses are measured over time. This leads to a multilevel or correlated data structure and consequently, we extend the marginal beta-binomial regression approach to the inclusion of random effects to accommodate the hierarchical structure of the data. We develop the estimation and inference procedure for the model proposal. Furthermore, we compare the performance of our proposal with similar approaches in the literature, showing that it gets better results in terms of reducing the bias of the estimates. We apply the model to a longitudinal Chronic Obstructive Pulmonary Disease study carried out at Galdakao Hospital in Biscay, Spain, reaching clinically and statistically relevant results about the evolution of the patients over time. PROs are usually obtained using rating scale questionnaires consisting of questions or items, grouped into one or more subscales, often called dimensions or domains. Therefore, we also propose a multivariate regression model based on the beta-binomial distribution for the joint analysis of all the longitudinal dimensions provided by different questionnaires. Finally, it is worth mentioning that we have implemented all the proposed regression models in the PROreg R- package which is available at CRAN.
Sala 1, Facultad de Matemáticas, Edificio Rolando Chuaqui, Campus San Joaquin, Pontificia Universidad Católica de Chile
2019-01-10
16:30hrs.
Seminario de Sistemas Dinámicos
Alberto Pinto. Faculty of Sciences, University of Porto
Pseudo-Anosov diffeomorphisms
Abstract:
We will introduce pseudo Cr smooth structures on surfaces that will have the following property; the Pseudo-Anosov diffeomorphisms are uniformly Cr hyperbolic. We will conjecture that the Bochi-Mane theorem will extend to such pesudo C1 smooth structures recovering the duality of the result for all surfaces.
Sala 2
2019-01-04
14:30hrs.
Santiago Number Theory and Algebra Seminar (Santas)
Yuri Bilu. Université de Bordeaux
Singular units do not exist
Abstract:
In the first part, I will revise the classical theory of complex multiplication of elliptic curves (or C-lattices); in particular, I will define the notion of a singular modulus, the j-invariant of an elliptic curve (or a C-lattice) with complex multiplication.  According to the old result of Weber, a singular modulus is an algebraic integer. In the second part, I will briefly describe the recent work of Habegger, Kühne and myself proving that a singular modulus cannot be algebraic unit.
Sala 2
2019-01-03
16:00hrs.
Seminario de Análisis y Geometría
Armin Schikorra . University of Pittsburgh
Self-repulsive curvature energies for curves and surfaces: regularity theory and relation to harmonic maps
Abstract:
I will talk about a class of curvature energies for curves, the O'Hara energies, that are nonlocal in nature. In particular, I will present an approach for regularity theory of minimizers and critical points for these curves which is based on a relation to (fractional) harmonic maps. Then I will present some results towards attempts of generalizing this idea to surfaces.
Sala 2