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# On a fully nonlinear version of the Min-Oo Conjecture

## José M. Espinar Instituto Nacional de Matemática Pura e Aplicada

In this talk, we prove that the Min-Oo's conjecture holds if we consider a compact connected locally conformally flat manifold with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality, and the mean curvature of the boundary is controled bellow by the mean curvature of a geodesic ball in the standard unit-sphere. This is a joint work with E. Barbosa and M.P. Cavalcante.

Seminario 1ª planta, IEMath

# Defensa de Tesis Doctoral: Translating soliton of the mean curvature flow

Director: Francisco Martín.

Salón de Conferencias, IEMath

# Boundaries in Non-positive Curvature IV

## Maciej Czarnecki Uniwersytet Łódzki

Manifolds and spaces of non-positive curvature
Ideal boundary
Contracting boundary
Applications to foliations and laminations.

Seminario, 1ª Planta, IEMath

# Boundaries in Non-positive Curvature III

## Maciej Czarnecki Uniwersytet Łódzki

Manifolds and spaces of non-positive curvature
Ideal boundary
Contracting boundary
Applications to foliations and laminations.

Seminario, 1ª Planta, IEMath

# Boundaries in Non-positive Curvature II

## Maciej Czarnecki Uniwersytet Łódzki

Manifolds and spaces of non-positive curvature
Ideal boundary
Contracting boundary
Applications to foliations and laminations.

Seminario, 1ª Planta, IEMath

# Boundaries in Non-positive Curvature I

## Maciej Czarnecki Uniwersytet Łódzki

Manifolds and spaces of non-positive curvature
Ideal boundary
Contracting boundary
Applications to foliations and laminations.

Seminario, 1ª Planta, IEMath

# Algunos problemas clásicos de los solitones de traslación del FCM

Seminario 1ª planta, IEMath

# Stable minimal surfaces in semidirect products

We will study geometric properties of compact stable minimal surfaces with boundary in homogeneous 3-manifolds $X$ that can be expressed as a semidirect product of $\mathbb{R}^2$ with $\mathbb{R}$ endowed with a left invariant metric. For any such compact minimal surface $M$, we provide an a priori radius estimate which depends only on the maximum distance of points of the boundary $\partial M$ to a vertical geodesic of $X$. In particular, there are no complete stable minimal surfaces inside solid metric cylinders around vertical geodesics in $X$ .​ We also give a generalization of the classical Rado's Theorem to the context of compact minimal surfaces with graphical boundary over a convex horizontal domain in $X$ , and we study the geometry, existence and uniqueness of this type of Plateau​ problem.​ Joint work with Bill Meeks and Pablo Mira.​

Seminario 1ª planta, IEMath

# A complex associated family for isothermic surfaces

## Katrin Leschke University of Leicester

In my talk I explain how Quaternionic Holomorphic Geometry can be used to discuss the associated family of isothermic surfaces. Isothermic surfaces are surfaces which have a conformal curvature line parametrisation, and cylinder, minimal surfaces and constant mean curvature surfaces are examples. It is known that any isothermic surface has a family of flat real connections such that parallel sections give new isothermic surfaces. In this talk, I will introduce a complex family which specialises to the known complex families for minimal surfaces and constant mean curvature surfaces.

Seminario 1ª planta, IEMath

# Static & Stationary Spacetimes: Causal Curiosities and the Fundamental Cocyle

## Steven G. Harris Saint Louis University

This talk presents an algebraic invariant that measures the degree to which a (non-standard) static or stationary spacetime can exhibit violations of naive expectations on causal structure; for a static spacetime, this invariant ("fundamental cocycle") is a de Rham cohomology class, but for stationary spacetimes it is something of looser structure.  Using this cocycle provides a simple measure for determining whether two events are timelike related, so it leads to classification of a spacetime along the "causal ladder" (globally hyperbolic, strongly causal, etc.).  It has excellent properties with group actions, leading to conclusions about when spacetimes and their covers or quotients have similar causality properties.  In particular, it can be used to create (by means of quotient) spacetimes of desired causality structure, such as cosmic strings on non-flat backgrounds.

# Geodesic Connectedness in Spacetimes with Lightlike Killing Vector Fields

## Anna María Candela Università degli Studi di Bari

In the last years a great deal of e ort has been invested in order to investigate the existence of geodesics connecting two given points in Lorentzian manifolds as a statement similar to the Hopf-Rinow Theorem is not known for spacetimes.
In particular, some geometric sufficient conditions have been introduced so that stationary spacetimes, i.e. spacetimes equipped with a timelike Killing vector eld, are geodesically connected. Unluckily, a similar global result is not true for space- times equipped with a lightlike Killing vector eld. Anyway, points connected by geodesics can be characterized.
Joint works with Rossella Bartolo and José Luis Flores.

# Magnetic maps - V

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

Abstract: We present some basic notions on magnetic curves on Riemannian manifolds and give several examples in dimension 3, emphasizing the case of Killing magnetic curves. We present some results on magnetic curves in almost contact metric geometry in arbitrary dimension. Later on we introduce the notion of magnetic map between Riemannian manifolds. Magnetic maps are generalizations of both magnetic curves and harmonic maps. We provide some fundamental examples of them. Further on we describe the problem in almost contact metric geometry. Then we produce examples of magnetic maps, having as either source or target manifold the tangent bundle of a Riemannian manifold equipped with several Riemannian metrics. In particular we study when the canonical projection, a vector field and the tangent map are, respectively, magnetic maps.

Temas:

• Magnetic curves on Riemannian manifolds
• Geodesics and harmonic maps
• Magnetic maps: definition and first examples
• Magnetic maps in almost contact metric geometry
• Magnetic maps and tangent bundle of a Riemannian manifold

# Effective index estimates via Euclidean isometric embeddings

## Alessandro Carlotto ETH Zurich

I will present a general method (pioneered by Ros and Savo) to obtain universal and effective index estimates for minimal hypersurfaces inside a Riemannian manifold, given an isometric embedding of the latter in some (possibly high-dimensional) Euclidean space. This approach can be applied, on the one hand, to tackle a conjecture by Schoen and Marques-Neves asserting that the Morse index of a closed minimal hypersurface in a manifold of positive Ricci curvature is bounded from below by a linear function of its first Betti number, which we settle for a large class of ambient spaces. On the other hand, these methods turn out to be very powerful in studying free-boundary minimal hypersurfaces in Euclidean domains: among other things, we prove a lower bound for the index of a free boundary minimal surface which is linear both with respect to the genus and the number of boundary components. Applications to compactness theorems, to the explicit analysis of known examples (due to Fraser-Schoen and to Folha-Pecard-Zolotareva) and to novel classification theorems will also be mentioned. This is joint work with Lucas Ambrozio and Benjamin Sharp.

Seminario 1ª planta, IEMath

# Magnetic maps - IV

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

Abstract: We present some basic notions on magnetic curves on Riemannian manifolds and give several examples in dimension 3, emphasizing the case of Killing magnetic curves. We present some results on magnetic curves in almost contact metric geometry in arbitrary dimension. Later on we introduce the notion of magnetic map between Riemannian manifolds. Magnetic maps are generalizations of both magnetic curves and harmonic maps. We provide some fundamental examples of them. Further on we describe the problem in almost contact metric geometry. Then we produce examples of magnetic maps, having as either source or target manifold the tangent bundle of a Riemannian manifold equipped with several Riemannian metrics. In particular we study when the canonical projection, a vector field and the tangent map are, respectively, magnetic maps.

Temas:

• Magnetic curves on Riemannian manifolds
• Geodesics and harmonic maps
• Magnetic maps: definition and first examples
• Magnetic maps in almost contact metric geometry
• Magnetic maps and tangent bundle of a Riemannian manifold

# Magnetic maps - III

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

Abstract: We present some basic notions on magnetic curves on Riemannian manifolds and give several examples in dimension 3, emphasizing the case of Killing magnetic curves. We present some results on magnetic curves in almost contact metric geometry in arbitrary dimension. Later on we introduce the notion of magnetic map between Riemannian manifolds. Magnetic maps are generalizations of both magnetic curves and harmonic maps. We provide some fundamental examples of them. Further on we describe the problem in almost contact metric geometry. Then we produce examples of magnetic maps, having as either source or target manifold the tangent bundle of a Riemannian manifold equipped with several Riemannian metrics. In particular we study when the canonical projection, a vector field and the tangent map are, respectively, magnetic maps.

Temas:

• Magnetic curves on Riemannian manifolds
• Geodesics and harmonic maps
• Magnetic maps: definition and first examples
• Magnetic maps in almost contact metric geometry
• Magnetic maps and tangent bundle of a Riemannian manifold

# Magnetic maps - II

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

Abstract: We present some basic notions on magnetic curves on Riemannian manifolds and give several examples in dimension 3, emphasizing the case of Killing magnetic curves. We present some results on magnetic curves in almost contact metric geometry in arbitrary dimension. Later on we introduce the notion of magnetic map between Riemannian manifolds. Magnetic maps are generalizations of both magnetic curves and harmonic maps. We provide some fundamental examples of them. Further on we describe the problem in almost contact metric geometry. Then we produce examples of magnetic maps, having as either source or target manifold the tangent bundle of a Riemannian manifold equipped with several Riemannian metrics. In particular we study when the canonical projection, a vector field and the tangent map are, respectively, magnetic maps.

Temas:

• Magnetic curves on Riemannian manifolds
• Geodesics and harmonic maps
• Magnetic maps: definition and first examples
• Magnetic maps in almost contact metric geometry
• Magnetic maps and tangent bundle of a Riemannian manifold

# Magnetic maps - I

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

Abstract: We present some basic notions on magnetic curves on Riemannian manifolds and give several examples in dimension 3, emphasizing the case of Killing magnetic curves. We present some results on magnetic curves in almost contact metric geometry in arbitrary dimension. Later on we introduce the notion of magnetic map between Riemannian manifolds. Magnetic maps are generalizations of both magnetic curves and harmonic maps. We provide some fundamental examples of them. Further on we describe the problem in almost contact metric geometry. Then we produce examples of magnetic maps, having as either source or target manifold the tangent bundle of a Riemannian manifold equipped with several Riemannian metrics. In particular we study when the canonical projection, a vector field and the tangent map are, respectively, magnetic maps.

Temas:

• Magnetic curves on Riemannian manifolds
• Geodesics and harmonic maps
• Magnetic maps: definition and first examples
• Magnetic maps in almost contact metric geometry
• Magnetic maps and tangent bundle of a Riemannian manifold

# Compact stable surfaces with constant mean curvature in Killing submersions

## José M. Manzano King's College London

A Killing submersion is a Riemannian submersion from an orientable 3-manifold to an orientable surface, such that the fibres of the submersion are the integral curves of a Killing vector field without zeroes. The interest of this family of structures is the fact that it represents a common framework for a vast family of 3-manifolds, including the simply-connected homogeneous ones and the warped products with 1-dimensional fibres, among others. In the first part of this talk we will discuss existence and uniqueness of Killing submersions in terms of some geometric functions defined on the base surface, namely the Killing length and the bundle curvature. We will show how these two functions, together with the metric in the base, encode the geometry and topology of the total space of the submersion. In the second part, we will prove that if the base is compact and the submersion admits a global section, then it also admits a global minimal section. This gives a complete solution to the Bernstein problem (i.e., the classification of entire graphs with constant mean curvature) when the base surface is assumed compact. Finally we will talk about some results on compact orientable stable surfaces with constant mean curvature immersed in the total space of a Killing submersion. In particular, if they exist, then either (a) the base is compact and it is one of the above global minimal sections, or (b) the fibres are compact and the surface is a constant mean curvature torus.

Seminario 1ª Planta, IEMath-GR

# An inclusive immersion into a quaternion manifold and its invariants

## Kazuyuki Hasegawa Kanazawa University

We introduce a quaternion invariant for an inclusive immersion in a quaternion manifold, which is a quaternion object corresponding to the Willmore functional. The lower bound of this invariant is given by topological one and the equality case can be characterized in terms of the natural twistor lift. When the ambient manifold is the quaternion projective space and the natural twistor lift is holomorphic, we obtain a relation between the quaternion invariant and the degree of the image of the natural twistor lift as an algebraic curve. Moreover the first variation formula for the invariant is obtained. As an application of the formula, if the natural twistor lift is a harmonic section, then the surface is a stationary point of the invariant under any variations such that the induced complex structures do not vary.

Seminario 1ª Planta, IEMath-Gr

# Una caracterización del volumen mediante la desigualdad de Brunn-Minkowski

## Jesús Yepes ICMAT

The Brunn-Minkowski inequality can be summarized by stating that the volume, i.e., the Lebesgue measure in $\mathbb {R}^n$, is ($1/n$)-concave. More precisely, we have $$(1)\quad vol\big((1-\lambda)A+\lambda B\big)^{1/n}\geq(1-\lambda)vol(A)^{1/n}+\lambda vol(B)^{1/n}$$ for all measurable sets $A, B$ so that $(1-\lambda) A+\lambda B$ is also measurable. It is easy to see that (1) is also true if we exchange $1/n$ by an arbitrary $p\leq 1/n$. When working with absolutely continuous measures $d\mu(x)=f(x)dx$ associated to densities $f$ with some convexity assumptions, one can also obtain the following Brunn-Minkowski inequality $$(2)\quad \mu((1-\lambda) A+\lambda B)^{p}\geq(1-\lambda)\mu(A)^p+\lambda\mu(B)^p$$ for any pair of measurable sets $A, B$ with $\mu(A)\mu(B)>0$ and such that $(1-\lambda) A+\lambda B$ is also measurable, where $p\leq 1/n$ is associated to the type of convexity'' of $f$. The convexity conditions of such density functions f allow us to understand whether the volume should be the sole measure satisfying the latter inequality or not. Thus, in this talk we will discuss whether, for a given measure on $\mathbb {R}^n$ (not necessarily absolutely continuous), having an inequality like (2) for a certain (small') subfamily of sets in $\mathbb {R}^n$ implies that the measure is (up to a constant) the volume itself. To this respect, we will point out that the constraints $1/n\geq p>0$, when the support of the measure is the whole $\mathbb {R}^n$, and $p=1/n$, when it is an arbitrary open convex set, are both necessary in order to get such a characterization.

Seminario 1ª Planta, IEMath-Gr

# El espacio de esferas orientadas como puente entre $\mathbb{H}^3$ y $\mathbb{R}^3$

Es conocido que las superficies mínimas en $\mathbb{R}^3$ y las llanas de $\mathbb{H}^3$ admiten representaciones homomorfas y comparten varios e interesantes aspectos tanto de tipo geométrico como tipológico. A pesar de esto, no se ha dado hasta ahora ninguna descripción geométrica que conecte estos dos tipos de superficies inmersas en diferentes espacios ambiente. Nuestro objetivo es mostrar una construcción geométrica que asocia toda superficie llana de $\mathbb{H}^3$ (con singularidades admisibles) a un par de mínimas en $\mathbb{R}^3$ relacionadas por una transformación de Ribaucour. La construcción es un caso particular de una conexión geométrica general entre superficies del espacio hiperbólico y envolventes a una congruencia de esferas del espacio euclídeo.

Seminario 1ª Planta, IEMath-GR

# Laminaciones mínimas en $\mathbb{R}^3$ y la conjetura de Hoffman-Meeks

La conjetura de Hoffman-Meeks afirma que si M es una superficie mínima con curvatura total finita en ​​$\mathbb{R}^3$​ con género $g$ y $k$ finales, entonces $k\leq g+2$. Este problema abierto motiva estudiar los posibles límites de una sucesión de superficies mínimas embebidas $M_n\subset \mathbb{R}^3$​ con género fijo y curvatura total finita. Normalmente, los objetos que aparecen en el límite son laminaciones mínimas con singularidades. Mediante el uso de la teoría de Colding-Minicozzi, daremos un resultado de convergencia para una parcial de la sucesión $M_n$ anterior, si asumimos una cota uniforme del radio de inyectividad de las superficies $M_n$ fuera de un cerrado numerable de $\mathbb{R}^3$. Usaremos este resultado de convergencia para obtener una cota (no explícita) $k\leq C(g)$ del número de finales en la conjetura de Hoffman-Meeks, que sólo depende del género. Esta cota del número de finales produce una cota para el índice de estabilidad de una superficie mínima de curvatura total finita, también en función sólo de su género.​

Seminario 1ª planta, IEMath

# The generalized Tanaka-Webster Ricci tensor of real hypersurfces in symmetric spaces

## Konstantina Panagiotidou Hellenic Army Academy

Seminario 1ª planta, IEMath

# A study of hypersurfaces in terms of $*$-Ricci tensor

## Georgios Kaimakamis Hellenic Army Academy

Seminario 1ª planta, IEMath

# Unicidad de esferas en 3-variedades. Demostración de una conjetura de Alexandrov.

## Pablo Mira Universidad Politécnica de Cartagena

Un famoso teorema de Hopf establece que toda esfera de curvatura media constante en $\mathbb{R}^3$ es una esfera redonda. En esta charla generalizaremos el teorema a clases de superficies gobernadas por EDPs elípticas sobre cada plano tangente en 3-variedades arbitrarias, con la única hipótesis de la existencia de una familia transitiva de superficies candidatas. Así, probamos que toda esfera en estas condiciones es una esfera de la familia de candidatos. Como aplicación, demostraremos una conjetura de 1956 de A.D. Alexandrov sobre la unicidad de esferas de curvaturas predeterminadas en $\mathbb{R}^3$ , y completaremos la caracterización de las esferas redondas como las únicas esferas de Weingarten elípticas en $\mathbb{R}^3$. Este es un trabajo con José A. Gálvez.

Seminario 1ª planta, IEMath

# The parametric h-principle for minimal surfaces in $\mathbb{R}^n$ and null curves in $\mathbb{C}^n$

## Franc Forstneric Univerza v Ljubljani

Let $M$ be an open Riemann surface. It was proved by Alarcón and Forstneric that every conformal minimal immersion $M\to\mathbb R^3$ is isotopic to the real part of a holomorphic null curve $M\to\mathbb C^3$. We prove the following substantially stronger result in this direction: for any $n\ge 3$, the inclusion of the space of real parts of non flat null holomorphic immersions $M\to\mathbb C^n$ into the space of non flat conformal minimal immersions $M\to \mathbb R^n$ satisfies the parametric h-principle with approximation; in particular, it is a weak homotopy equivalence. Analogous results hold for several other related maps. For an open Riemann surface $M$ of finite topological type, we obtain optimal results by showing that the above inclusion and several related maps are inclusions of strong deformation retracts; in particular, they are homotopy equivalences. (Joint work with Finnur Lárusson.)

Seminario 1ª planta, IEMath

# Minimal Graphs in the Heisenberg Space

## Barbara Nelli Università degli Studi di L'Aquila

I will talk about the existence and the non existence of minimal graphs in the Heisenberg space, with a given boundary. Moreover height and area estimates for such graphs are obtained.

Seminario 1ª planta, IEMath

# New conformal methods in Riemannian and Lorentzian geometry

## Olaf Müller Universität Regensburg

The talk gives an overview over some recently developed methods in global analysis and geometry that involve conformal factors. First we review a global existence result, obtained with Nicolas Ginoux, for Dirac-Higgs-Yang-Mills systems under the assumption that the underlying spacetime has a conformal extension, which ist the case for solutions to the Einstein equations for initial values in a weighted neighborhood of the standard ones. Then we switch to Riemannian geometry and show, using the novel ‚flatzoomer’ method, that every conformal class contains a metric of bounded geometry. Finally we sketch the consequences of the result for the Yamabe flow on noncompact manifolds and a related result for Cheeger-Gromov convergence of some relevance in the context of positive scalar curvature on compact manifolds.

Seminario 1ª planta, IEMath

# Type changes of zero mean curvature surfaces of Riemann-type in the Lorentz-Minkowski 3-space

## Shintaro Akanime Kyushu University

A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann-type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann-type according to their causal characters, and we give new examples of ZMC surfaces containing lightlike lines and a zero mean curvature entire graph of mixed type.

# Study of real hypersurfaces in complex hyperbolic two-planeGrassmannians with Ricci tensors

## Changhwa Woo Kyoonpook National University

In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field $T$, that is, $R_{\xi}\phi T=TR_{\xi}\phi$, where $T=A$ or $T=S$ for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. By using simultaneous diagonalzation for commuting symmetric operators, We give a complete classification of real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting condition respectively.

Seminario 2ª Planta, IEMath-GR

# Recent Progress on Complex Quadric in Hermitian Symmetric Spaces

## Young Jin Suh Kyungpook National University

Seminario 2ª Planta, IEMath-GR

# Finsler-Lagrange geometry and its applications - a brief review

## Nicoleta Voicu Universitatea Transilvania Brasov

Finsler-Lagrange geometries are a generalization of Riemannian geometry, obtained by allowing the metric tensor to depend not only on the points of the manifold under discussion, but also on a tangent vector at each of these points. We present here in brief the specific features of these geometries together with some of their applications - with a special focus on classical field theories - and the surrounding open questions.

Seminario 1ª planta, IEMath

# Constant mean curvature surfaces in discrete geometry

## Pascal Romon Université Paris-Est Marne-la-Vallée

Defining a relevant notion of constant mean curvature and constant mean curvature surfaces is difficult in discrete geometry. In order to make sense of these definitions, we will first recall what properties are shared by the smooth CMC surfaces, e.g. criticality, associated family, integrable system PDE, that we would like to hold also in the discrete case. Then we will describe how such notions can or cannot be carried out in the discrete case. (Joint work with Sasha Bobenko)

Seminario 1ª planta, IEMath

# Superficies Mínimas con Borde

## José M. Espinar Instituto Nacional de Matemática Pura e Aplicada

Introducción y conceptos básicos. Primera fórmula de variación y aplicaciones. Teorema de Nitsche. Aplicaciones conformes. Teorema de Fraser-Schoen. Segunda fórmula de variación y aplicaciones. Teorema de Shiffman.

Seminario 2ª Planta, IEMath-GR

# Discrete Differentil Geometry, I

## Pascal Romon Université Paris-Est Marne-la-Vallée

Topics: Introduction and basic concepts Discrete curves in the plane. Aproximation properties Discrete surfaces. Mean curvature flow. Discrete Gauss-Bonnet theorem

Seminario 2ª Planta, IEMath-GR

# Superficies Mínimas con Borde

## José M. Espinar Instituto Nacional de Matemática Pura e Aplicada

Introducción y conceptos básicos. Primera fórmula de variación y aplicaciones. Teorema de Nitsche. Aplicaciones conformes. Teorema de Fraser-Schoen. Segunda fórmula de variación y aplicaciones. Teorema de Shiffman.

Seminario 2ª Planta, IEMath-GR

# Regularity and singularity of area-minimizing surfaces

## Camillo De Lellis Universität Zürich

The Plateau's problem, named after the Belgian physicist J. Plateau, is a classic in the calculus of variations and regards minimizing the area among all surfaces spanning a given contour. Although Plateau's original concern were 2-dimensional surfaces in the 3-dimensional space, generations of mathematicians have considered such problem in its generality. A successful existence theory, that of integral currents, was developed by De Giorgi in the case of hypersurfaces in the fifties and by Federer and Fleming in the general case in the sixties. When dealing with hypersurfaces, the minimizers found in this way are rather regular: the corresponding regularity theory has been the achievement of several mathematicians in the 60es, 70es and 80es (De Giorgi, Fleming, Almgren, Simons, Bombieri, Giusti, Simon among others). In codimension higher than one, a phenomenon which is absent for hypersurfaces, namely that of branching, causes very serious problems: a famous theorem of Wirtinger and Federer shows that any holomorphic subvariety in $\mathbb C^n$ is indeed an area-minimizing current. A celebrated monograph of Almgren solved the issue at the beginning of the 80es, proving that the singular set of a general area-minimizing (integral) current has (real) codimension at least 2. However, his original (typewritten) manuscript was more than 1700 pages long. In a recent series of works with Emanuele Spadaro we have given a substantially shorter and simpler version of Almgren's theory, building upon large portions of his program but also bringing some new ideas from partial differential equations, metric analysis and metric geometry. In this talk I will try to give a feeling for the difficulties in the proof and how they can be overcome. Moreover I will touch some recent developments which go beyond Almgren's result.

Sala de Conferencias, IEMath-GR

# Eventos

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Lugar: Seminario 1ª planta IE-Math
Horario:
9:30 - 10:30 Vicente Miquel, de la UV, On the mean curvature flow associated to a density
10:30 - 11:00 Descanso café
11:00 - 12:00 Mariel Sáez de la PUC (Chile), On the Evolution by Fractional Mean Curvature

Almuerzo
16:00 - 17:00 Magdalena Rodríguez de la UGR, Harmonic diffeomorpphisms between surfaces
17:00 - 17:30 Descanso café
17:30 - 18:30 Francisco Martín (UGR), Translating graphs of the mean curvature flow.

# 2016-17 Warwick EPSRC Symposium: Geometric PDEs

## Warwick (UK)

»

Organisers: P. Topping (Warwick), A. Neves (Imperial), N. Wickramasekera (Cambridge), M. Dafermos (Cambridge), C. Warnick (Warwick)

Scientific summary: Geometric PDE has been flourishing around the world over recent years, and particularly in the UK. This workshop is aimed at accelerating this development, and making the progress accessible to UK PhD students and postdocs. We plan to emphasise areas of exceptional UK expertise such as geometric flows (e.g. Ricci flow, mean curvature flow etc.), minimal surfaces (including min-max methods and regularity theory), and mathematical relativity (e.g. black hole stability problems). All these fields have enjoyed spectacular progress over recent years. The field of geometric flows has developed remarkably and its impact on neighbouring areas through the solution of famous problems in topology and geometry has even reached a lay audience. Minimal surface theory has seen a resurgence of interest in min-max methods leading to the solution of the Willmore conjecture amongst others. Recent breakthroughs in the field of mathematical relativity include the resolution of the L2-curvature conjecture, and significant advances towards addressing the major open problems of black hole stability and cosmic censorship. All these fields promise much more to come, both in terms of their development and in terms of their success in solving famous open problems in other fields. We will be aiming to provide a forum for many of the new UK appointments in this general area to speak alongside international experts. The large number of UK postdocs and students in the area will have an excellent opportunity to hear leaders of their fields. In addition, we plan to offer short speaking opportunities to a substantial number of UK postdocs in the spirit of the well-established annual ”Junior Warwick-Imperial-Cambridge Geometric Analysis Seminar" which will be incorporated into this event for 2016.

# Encuentro Red Española de Análisis Geométrico

»
El encuentro de investigadores de la REAG es una de las actividades programadas de la Red Española de Análisis Geométrico. Sus objetivos principales son favorecer el intercambio de ideas y fomentar la cooperación entre investigadores, tanto miembros de la Red como de grupos afines cuya investigación se desarrolla en el campo del Análisis Geométrico.

#### Participantes

• Miguel A. Javaloyes (Universidad de Murcia): Estructuras de Finsler con viento: de la navegación de Zermelo a la causalidad de espaciotiempos
• Pablo Mira (Universidad Politécnica de Cartagena): Clasificación de esferas inmersas modeladas por EDPs elípticas
• Joan Porti (Universitat Autònoma de Barcelona): Espinas de longitud mínima y polígonos hiperbólicos

# Taller de jóvenes investigadores de la REAG

»
El Taller de Jóvenes Investigadores de la Red Española de Análisis Geométrico es una actividad que se ha llevado a cabo durante los últimos años y que surgió con el objetivo de compartir los avances entre distintos grupos de investigación españoles. Está dirigido a investigadores que estén realizando los estudios de doctorado o bien que hayan defendido su tesis recientemente.

#### Participantes

• Teresa García (Universitat Autònoma de Barcelona): Compactificación de una acción diagonal en el producto de espacios CAT(-1)
• Vicent Gimeno (Universitat Jaume I): Subvariedades con segunda forma fundamental domada
• Irene Ortiz (Universidad de Murcia): El primer valor propio del operador de estabilidad para superficies compactas CMC
• Jaime Santos (Universidad Autónoma de Madrid): Isometrías en espacios métricos con medidas y curvatura de Ricci acotada inferiormente
• Jesús Yepes (ICMAT): Desigualdad de Brunn-Minkowski bajo hipótesis sobre proyecciones y secciones

# 8th International Meeting on Lorentzian Geometry

## Malaga (Spain)

»

Welcome to the 8th International Meeting on Lorentzian Geometry. After seven successful meetings in Benalmádena (Málaga), Murcia, Castelldefels, Santiago de Compostela, Martina Franca (Italy), Granada, and São Paulo (Brazil), the next edition will take place in Málaga (Spain), from the 20th to the 23rd, September 2016. Also a special edition on "Lorentzian and conformal Geometry" was celebrated in 2014 in honour to Prof. Helga Baum in Greifswald (Germany).

Topics on pure and applied Lorentzian geometry include, but are not limited to, geodesics, submanifolds, causality, black holes, Einstein equations, geometry of spacetimes or AdS-CFT correspondence.

# Modern Advances in Geometry and Topology

## Kharkiv (Ukraine)

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The international conference MODERN ADVANCES IN GEOMETRY AND TOPOLOGY will be organized in honor of professor Alexander A. Borisenko to emphasize his diverse contribution to geometry and to celebrate his 70th birthday.

The scientific program of the conference will include 40 minutes invited talks, along with 15 minutes contributed talks and posters in one of the following conference topics:

• Geometry of submanifolds
• Global differential and Riemannian geometry
• Curvature flows
• Foliations
• Pseudo-Riemannian, Finsler and Kahler geometries
• Lie groups and homogeneous spaces
• Topology

Please visit the conference web-site http://magt.karazin.ua/ for further details on registration, abstracts submission, accommodation, traveling, cultural program, etc.

# XXV International Fall Workshop on Geometry and Physics

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The XXV edition of the International Fall Workshop on Geometry and Physics (IFWGP) will take place at the main campus of the Spanish National Research Council (CSIC) in Madrid, as it did 25 years ago, from August 29th to September 2nd, 2016. This series of international workshops, held at Spanish and Portuguese universities and research centers, covers topics in the fields of Differential Geometry, Applied Mathematics and Physics, with a particular emphasis on geometric aspects of Mechanics and Field Theory (both in their classical and quantum incarnations).

# XIX Geometrical Seminar

## Zlatibor (Serbia)

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This is the 19th meeting of the Geometrical Seminar which started its activities during the eighties in the last century, under the name Yugoslav Geometrical Seminar. At the beginning, the seminar was of the national level. With great efforts meetings were organized even in the difficult period of the nineties, but, nevertheless, the level of meetings has raised, the 17th and 18th Geometrical Seminar, which took place in Zlatibor 2012 and Vrnjacka Banja 2014, had over 100 participants from more than 30 countries. The aim of these meetings is to bring together mathematicians and physicists interested in geometry and its applications, to give lectures on new results, exchange ideas, problems and conjectures.

### Conference topics

Differential Geometry, Topology, Lie Groups, Mathematical Physics, Discrete Geometry, Integrable Systems, Visualization, as well as other subjects related to the main themes are welcome.

### Preliminary list of Speakers

*-to be confirmed

Dmitri Alekseevsky (Brno, Czech Republic), David Blair (East Lansing, USA), Victor Buchstaber (Moscow, Russia), Bang-Yen Chen, (East Lansing, USA) Branko Dragovich (Belgrade, Serbia), Boris Dubrovin (Moscow, Russia), Anatoly T. Fomenko (Moscow, Russia), Peter B. Gilkey (Eugene, USA), Graham Hall (Aberdeen, UK), Oldrich Kowalski (Prague, Czech Republic), Miodrag Mateljević (Belgrade, Serbia), Svetozar Minčić (Niš, Serbia), Emil Molnar (Budapest, Hungary), Masafumi Okumura (Saitama, Japan), Mileva Prvanović (Belgrade, Serbia), Udo Simon (Berlin, Germany), Hellmuth Stachel (Vienna, Austria), Leopold Verstraelen (Leuven, Belgium), Luc Vrancken (Vallenciennes, France)

### Registration

All participants are kindly asked to fill in the registration form, or submit it as an e-mail to the geometricalseminar2016@gmail.com`, until April 15, 2016.

# Alterman Conference on Geometric Algebra and Summer School on Kähler Calculus

## Brasov (Romania)

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The Alterman Conference on Geometric Algebra and first Summer School on Kähler Calculus are sponsored by Eric Alterman and will be held at the Transilvania University of Brasov (Romania) from August 1st to 9th. The main objectives of the Summer School and the Conference are:

• Teaching Clifford Algebra and Kähler Calculus to the non initiated.
• Enhancing the applications to physics, engineering, computer science and image processing.
• Promoting other areas of the Grassmann legacy connected with Clifford Algebra.

The structure of both the Alterman Conference and the Summer School is modular with the aim that each attendant can choose the components of the event in which he wishes to participate.

# Curso de Verano “Espacio y Tiempo” 2016

## Casa de la Cultura de Almuñécar (Granada)

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Los conceptos de “espacio” y “tiempo” son, a priori, muy intuitivos y familiares. De hecho, aprendemos a ubicarnos espacial y temporalmente desde edades muy tempranas y nos acompañan constantemente en nuestra vida diaria. Sin embargo, al profundizar en ellos descubrimos que, bajo ciertos prismas, estos conceptos esconden interpretaciones abstractas, misteriosas, a veces difíciles de captar. La Física moderna, por ejemplo, apoyándose en elaborados formalismos matemáticos los trata como un objeto único e inseparable para describir la Naturaleza. Aunque el espacio y el tiempo son fundamentales en el desarrollo de las ciencias exactas, especialmente en Física y Matemáticas, en los Programas de Estudios de Grado apenas se discuten estos conceptos de manera conjunta y, cuando se hace, se ofrece una visión muy polarizada. En este curso trataremos la idea del “espacio y tiempo” desde diferentes ángulos y a diferentes escalas, ofreciendo al alumno una formación específica a la vez que transversal. Abordaremos las teorías matemáticas y físicas más modernas sobre la estructura de nuestro Universo, expondremos resultados experimentales recientes en la detección de radiación cósmica, observaciones astronómicas y astrobiología. A una escala más pequeña, analizaremos la evolución espacio-temporal de nuestro planeta, los mecanismos biológicos de envejecimiento, la lógica escondida en los sistemas de medición del tiempo y los secretos de la geometría fractal. Trataremos la realidad del espacio-tiempo desde el pensamiento filosófico y sus aplicaciones en la música y la danza. El curso se orienta, pues, hacia estudiantes de ramas tanto científicas como no científicas, interesados en ampliar sus conocimientos sobre el concepto del “espacio y tiempo”. Los contenidos tratados en el Curso aportan una formación complementaria a los actuales Grados universitarios, no solapando en ningún caso con los contenidos de los Planes de Estudios.

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# Modern Topics in Nonlinear PDE and Geometric Analysis

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The school will cover important recent developments in three central areas of mathematical analysis, namely Calculus of Variations, Geometric nonlinear PDE and Mathematical Physics. For each of these research areas there will be a course consisting of two mini-courses.

# Coloquio de Geometría del Sur+Este

## Murcia (España)

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Durante los próximos días 20 y 21 de junio vamos a celebrar en Murcia el "I Coloquio de Geometría en el Sur + Este", que esperamos que sea el primero de una exitosa serie de encuentros que sirva para reforzar y revitalizar la amistosa relación existente entre los diferentes grupos de Geometría de esta zona de la geografía española.

# Achievements and Perspectives in Nonlinear Analysis. A tribute to Donato Fortunato

## Bari (Italy)

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We are organizing the International Workshop Achievements and Perspectives in Nonlinear Analysis. A tribute to Donato Fortunato during which we will take the opportunity to celebrate our friend Dino Fortunato’s 70th birthday.

Dino has been a full professor of Mathematical Analysis at the Department of Mathematics of the Università degli Studi di Bari “Aldo Moro” since 1980.

He has authored or co-authored more than one hundred papers published in international journals and the Springer monograph “Variational Methods in Nonlinear Fields Equations”. Over the years he has been principal investigator of several research projects.

Both his teaching and his passion for research have contributed to the training of many students who have then pursued an academic career in Mathematics.

The workshop will address Dino’s main research interests: variational and topological methods applied to the study of nonlinear differential equations that arise in Differential Geometry and in Mathematical Physics (Hamiltonian Systems, General Relativity, classical and quantum Field Theory, and more).

The meeting will take place in Bari, at the Department of Mathematics of the Università degli Studi di Bari “Aldo Moro”.

# Aspects of Membrane Dynamics

## Stockholm (Sweden)

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Despite of the first study of relativistic extended objects in physics (Dirac, 55 years ago) having been aimed at membranes, and mathematicians (e.g. Eisenhart) having considered higher-dimensional minimal hypersurfaces to be of great interest more than one century ago, progress in this fascinating field has been slow, because of its intrinsic non-linearity and immense complexness. It therefore needs a joint review of the state of the art, as well as communication between leading experts of various aspects of the problem.

# Second joint Conference of the Belgian, Royal Spanish and Luxembourg Mathematical Societies

## Logroño (La Rioja, Spain)

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### Plenary speakers

• Sara Arias de Reyna
• María Jesús Carro (Valdivia Lecture)
• Raf Cluckers
• Sergei Merkulov
• Johannes Nicaise
• Jesús María Sanz Serna
• Anton Thalmaier

### Special sessions

Session 1
Functional Analysis. Organizers: Françoise Bastin, José Bonet, Catherine Finet, Domingo García, Jasson Vindas
Session 2
Model Theory and Applications. Organizers: Elías Baro, Raf Cluckers, Françoise Point.
Session 3
Algebra and Number Theory. Organizers: Stef Caenepeel, Consuelo Martínez, Antonio Rojas, Gabor Wiese
Session 4
Partial Differential Equations. Organizers: Denis Bonheure, Salvador Villegas
Session 5
Algebraic Geometry and Singularities. Organizers: Nero Budur, Antonio Campillo, Francisco Monserrat, Wim Veys.
Session 6
Dynamical Systems and ODE. Organizers: Peter De Maesschalck, Freddy Dumortie, Santiago Ibáñez, Jesús Palacián.
Session 7
Probability and Statistics. Organizers: Alfonso Gordaliza, Christophe Ley.
Session 8
Orthogonal Polynomials and Special Functions. Organizers: Mirta Castro, Judit Mínguez, Walter Van Assche.
Session 9
Combinatorial and Computational Geometry. Organizers: Philippe Cara, Jan De Beule, David Orden
Session 10
Geometric Analysis, Differential Geometry and Quantization. Organizers: Simone Gutt, Magdalena Rodríguez, Martin Schlichenmaier.

# International Workshop on Theory of Submanifolds

## Istanbul (Turkey)

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This workshop will take place at Istanbul Technical University, Turkey between June 02-04, 2016. The previous conference “International Workshop on Finite Type Submanifolds” in 2014, which took place at Istanbul Technical University, had several participants from different countries. The aim of this meeting is to bring together mathematicians in differential geometry and its applications, to give talks or posters on new results. We would be pleased to see you in Istanbul.

# Perspectives on integral geometry

## University of Georgia, Athens, Georgia (USA)

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In view of the intense activity in integral geometry over the past few years, it seems an auspicious moment for the mathematical sciences community to collect our thoughts on the subject. Our hope is to air the major viewpoints that have been brought to bear on the subject, both in its internal development and in its applications, and in this way stimulate further progress.

# The 1st International Conference on Differential Geometry

## Fez (Morocco)

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The International Conference on Differential Geometry 2016 will be held on 11-15 April, 2016 at Fez, Morocco, the country which constitutes a rare exception of political stability and sustainable development in the region of Middle East and North Africa. This stability, the hospitality of its citizens, its experience as one of the most important touristic countries in the world and also its old tradition in organizing great events are guarantees of success of any conference.

The international conference in Differential Geometry of Fez will provide a forum and an excellent venue for researchers, academic faculty and students to present -and eventually publish- their research results and approaches.The ICDG-Fez’2016 conference seeks original and high quality contributions in the fields chosen as topics for the three sections of this conference.

Areas of interest

Original contributions in ICDG are solicited in, but not limited to, the following directions:

• Riemannian and pseudo-Riemannian manifolds and Submanifolds.
• Geometric Structures and Representation Theory.
• Geometric properties of functions and vector fields.

# Un siglo de Relatividad General y aun en la vanguardia

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7 de marzo "Historia de la Relatividad General", por Eduardo Bataner

8 de marzo "Las ideas físicas de la Relatividad General", por Bert Janssen

9 de marzo "Las geometrías del espaciotiempo", por Miguel Sánchez

Todas las charlas tendrán lugar en principio en el Salón de Grados "Florentino García Santos" de la Facultad de Ciencias, a las 19.00 horas.

# Geometric Flows (and related topics)

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## Confirmed speakers (main conference)

• Esther Cabezas-Rivas (Frankfurt)
• Pau Figueras (QMUL)
• Panagiotis Gianniotis (UCL)
• Robert Haslhofer (Toronto)
• Tobias Lamm (Karlsruhe)
• Jason Lotay (UCL)
• Sylvain Maillot (Montpellier)
• Andre Neves (Imperial)
• Huy Nguyen (Queensland)
• Melanie Rupflin (Oxford)
• Ben Sharp (Pisa)
• Giuseppe Tinaglia (KCL)
• Peter Topping (Warwick)

## Speakers of Junior WIC seminar (shorter talks):

• Otis Chodosh (Cambridge)
• Cecile Huneau (Cambridge, TBC)
• Hassan Jolany (Lille)
• Dan Ketover (Imperial)
• Kai Zheng (Warwick)