# Activities

One of the purpose of the network is to promote activities to favor the investigation exchange between its members and other mathematicians working on the geometric analysis topic. In this page you find all the information about the activities that may interest to the members of the network.

# School on the geometry and physics of moduli spaces

## Madrid (Spain),May 19th– 23rd, 2014.

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### Aim

The main goal of this school is to introduce graduate students, postdocs and senior researchers to several active fields of current research in the theory of moduli spaces and their interplay with geometry, topology and theoretical physics. The school is organized within the ICMAT Research Term on the Geometry and Physics of Moduli Spaces.

### Programme

The school will consist of 4 introductory courses:

• Sergei Gukov (Caltech): From Higgs bundles to knots and 3-manifolds
• Nigel Hitchin (Oxford): Spectral curves
• François Labourie (Orsay): Cyclic surfaces and Hitchin components
• Peter Topping (Warwick): The Teichmueller harmonic map flow and minimal immersions

### Venue

The school will take place in Miraflores de la Sierra, a small village located 49 Km North of Madrid, Spain.

# Meeting of the GDRE

## Roscoff (Bretagne, France),June 16th – 20th, 2014

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An international conference entitled "Geometric analysis in Roscoff" will be organised at the Station Biologique de Roscoff (CNRS), within the scientific program of Groupement de Recherche Européen Geometric Analysis. The meeting will be held  under the auspices of the Centre National de la Recherche Scientifique (CNRS), of the ANR France-Brésil "Surfaces", of the Labex Lebesgues, the Université de Bretagne Occidentale, the Université Paris Est Créteil , the Conseil Général du Finistère and of the Région Bretagne.

Geometric analysis lies on the interface between the study of partial differential equations and Riemannian geometry, influenced by fundamental problems from mathematical physics.  The domain has come to the fore over the last thirty years producing some spectacular results, for example the resolution of the Poincaré conjecture and the Willmore conjecture. The aim of this conference is to bring together experts in the domains of conformal geometry, geometric flows, minimal surfaces and related fields, in order to exchange ideas and to present some of the latest advances.  Doctoral students and young researchers are particularly encouraged to attend, and we hope that all participants will gain inspiration from this meeting.

### List of speakers

• B.Andrews, (ANU Camberra, Australia): TBA
• G.Catino, (Politecnico Di Milano, Italy): Rigidity of criticals metrics for quadratic functionals
• B.Coskunuzer (Koc Univ., Turkey):Minimal Surfaces with Arbitrary Topology in $\mathbb{H}^2\times\mathbb{R} • B.Daniel (Univ. Nancy, France): Minimal isometric immersions into$\mathbb{S}^2\times \mathbb{R}$and$\mathbb{H}^2\times\mathbb{R}\$
• O.Druet (ENS Lyon, France): Stability of the Einstein-Lichnerowicz constraint system
• A.Enciso (ICMAT, Spain): Geometric structures in steady solutions to the Euler equation
• R.Graham (Univ. of Washington, United States): TBA
• R. Haslhofer (Courant Institute, United States): Mean curvature flow of mean convex hypersurfaces
• E.Humbert (Univ. Tours) Some recent results on positive mass conjecture
• L. Hauswirth (Univ. Marne La Vallée, France): TBA
• A. Jimenez (Universidade Federal Fluminense, Brazil):Isolated singularities of some types of geometric PDE's
• A. Juhl (Univ. Berlin, Germany): TBA
• S. Montiel (Univ.Granada, Spain): The Cheeger constant of an asymptotic hyperbolic manifold and the Yamabe type of its conformal infinity
• A. Neves (Imperial College, United Kingdom): Minimal surfaces on metrics with positive Ricci curvature
• T.Rivière (ETH Zurich, Switzerland): Some recent results on the calculus of variations of surfaces
• M.Rupflin (Univ.of Leipzig, Germany) Teichmüller harmonic map flow
• A.Ros (Univ. Granada, Spain) TBA
• K.Smoczyk (Leibniz Univ., Germany): TBA
• M.Traizet (Univ.Tours): TBA
• G.Wang (Univ.Freiburg, Germany): Geometric inequalities in the hyperbolic space and applications
• T. Zolorateva(Ecole Polytechnique, France): Higher codimension isoperimetric problems

# School in Geometric Analysis Geometric Analysis on Riemannian and singular metric spaces

## Como, ItalySeptember 30– October 5, 2013

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The school aims at introducing Ph.D. students and young researchers to some important aspects and recent developements of Geometric Analysis both in the Riemannian and in the singular setting. The school will consist of 4 or 5 courses, ranging from 3 to 5 hours each, which will be held by world leading experts in the field. Lectures will be at the Ph.D level. Satellite seminars could be devoted to more advanced topics. Some of the students and young researchers attending the school will have the opportunity to give a 30 minutes talk. A poster section will be also available.

### Scientific Committee

• Gérard Besson (Université de Grenoble)
• Stefano Pigola (Università dell'Insubria)
• Alberto G. Setti (Università dell'Insubria)
• Marc Troyanov (EPFL)

### Organizing Committee

• Debora Impera (Università di Milano Bicocca)
• Stefano Pigola (Università dell'Insubria)
• Alberto G. Setti (Università dell'Insubria)
• Michele Rimoldi (Università dell'Insubria)
• Daniele Valtorta (Università degli Studi di Milano)
• Giona Veronelli (Université Paris 13)

# Conference on Qualitative and Geometric Aspects of Elliptic PDE's

## Centre de Recerca Matemàtica (Bellaterra, Spain) September 2nd– 6th, 2013

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### Objetives

In the last decades a great interest has arisen in elliptic PDEs originating from geometric problems and, conversely, important geometric objects often reach a deeper comprehension through the analytical techniques developed in elliptic PDEs. One of the aims of the conference is to bring together experts from both fields. Another topic which has attracted attention and has seen important developements in the last years concerns nonlinear elliptic and parabolic PDEs with fractional diffusion.

Special emphasis will be put on:

• Existence and symmetry properties of solutions to the Allen-Cahn and related equations, including free boundary problems.
• Minimal surfaces and isoperimetric inequalities.
• Fractional diffusion and front propagation

# Conference: Geometric Analysis

## Centre de Recerca Matemàtica (Bellaterra, Spain) July 1st– 5th, 2013

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### Objetives

Conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space. While in two dimensions, this is precisely the geometry of Riemann surfaces, in dimensions three and above the answer opens up many new different subjects, leading to the very wide field that is conformal geometry.

The first question is to find conformal invariants, or more specifically, conformally covariant operators, that is, operators which satisfy some invariant property under conformal change of metrics on a manifold, and its associated curvature. The model example is the Laplace-Beltrami operator, in relation to the Yamabe problem. The Yamabe equation is a second order, semilinear PDE; we would like to understand higher order or fully non-linear generalizations, such as the Paneitz operators together with Q-curvature, or the \sigma_k equation. As a consequence, new interesting directions in PDEs have been opened up, where existence or regularity theory is not developed as much.

Lately, there has been a lot of interest in the study of non-local, conformally covariant operators of fractional order constructed from Poincaré-Einstein metrics. While they are natural objects in other areas as probability, their geometrical meaning is not yet well understood. Particularly, the study of Poincaré-Einstein metrics has been and continues to be a rich source of activity relating conformal and Riemannian geometry. These are complete Einstein metrics which are asymptotically hyperbolic at infinity. Their boundary at infinity invariantly inherits a conformal structure. The asymptotic behavior of the metric encodes a great deal of information about the conformal structure at infinity, and this has led to new constructions and progress in conformal geometry. On the other hand, there are many analytic problems concerning the existence, uniqueness and regularity of Poincaré-Einstein metrics with a given conformal infinity and many open problems. This topic is stimulated by its role in the AdS/CFT correspondence in Physics.

In CR geometry there are formal similarities with conformal geometry. For example, there are conformally covariant operators analogous to the conformal Laplacian and the Paneitz operators. While these operators also come with associated Q-curvature quantities, their geometric/analytic meaning is quite different from conformal geometry. The analysis of these operators is closely connected with the geometry of the pseudoconvex manifolds which they may bound, hence of interest in several complex variables.​

# Advanced course: Topics in Conformal Geometry and Geometry Analysis

## Centre de Recerca Matemàtica (Bellaterra, Spain) June 25th– 28th, 2013

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This week of minicourses is aimed at presenting to students (and to anyone interested) the main topics of the research program Conformal geometry and geometric PDEs, and serves as an introduction to the following conference. Conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space. While in two dimensions, this is precisely the geometry of Riemann surfaces, in dimensions three and above the answer opens up many new different subjects (in PDEs, Physics, CR geometry and many others), leading to the very wide field that is conformal geometry. Geometric Analysis constitutes a central branch in Partial Differential Equations, that includes the study of non-linear or non-local equations.

### List of speakers

• Williams Meeks, University of Massachusets at Amherst
• Alice Chang, Princeton University
• Charles Fefferman, Princeton University​

# Workshop on Algebra and Geometry

## Badajoz (Spain) June 20th– 21st, 2013

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This activity aims to promote scientific discussion and debate on recent trends, future perspectives on algebra and geometry in a wide sense.

This event will be held at the Departament of Mathematics (Universidad de Extremadura) in Badajoz (Spain) in June 20-21, 2013.

### List of main speakers

• Pedro L. García (University of Salamanca, Spain).
• Antonio M. Naveira (University of Valencia, Spain).
• Juan B. Sancho (University of Extremadura, Spain).

### List of invited speakers

• Rui Albuquerque (University of Évora, Portugal).
• Pablo M. Chacón (University of Salamanca, Spain).
• José Navarro (University of Extremadura, Spain).
• Javier Seoane Bascoy (University of Santiago de Compostela, Spain).

# Variational problemas and Geometric PDE's

## Granada (Spain),June 17th – 21st, 2013

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This Conference is intended to cover some of the more recent progresses in the field of Geometric Analysis. It corresponds to the annual meeting of the French-Spanish GDRE Geometric Analysis and it will be one of the first activities of the Math Institute of Granada University. The conference is also related to the program Conformal Geometry and Geometric PDE's in CRM (Barcelona) from May 2013 to July 2013.

# Conformal Geometry and Geometric PDE's

## Centre de Recerca Matemàtica (Bellaterra, Barcelona, Spain),May 01– July 31, 2013

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### Coordinators

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• Sun-Yung Alice Chang (Princeton University)
• Maria del Mar González (Universitat Politècnica de Catalunya)
• Robin Graham (University of Washington)
• Paul Yang (Princeton University)
Clay grants support available for doctoral students and post-docs

# Conformal and Kähler Geometry

## Institut Henri Poincaré (Paris, France), September 10–December 14, 2012

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### Mini-courses

• Olivier Druet (ENS-Lyon): An introduction to conformal geometry.
• Simon Brendle (Stanford University): The Sphere Theorem.
• Fernando Marquez (IMPA): The Yamabe problem: compactness and noncompactness questions.
• Olivier Biquard (UPMC): Asymptotic geometry of Einstein metrics.
• Claude LeBrun (Stony Brook): Curvature functionals, Kähler metrics, and the geometry of 4-manifolds.
• Sean Paul (University of Wisconsin): An introduction to the Tian-Yau-Donalsdon conjecture.
• Piotr Chruściel (University of Vienna): Introduction to the Cauchy problem in general relativity.

### Conferences

• September 17 – 21: Conformal and Kähler Geometry - IHP.
• October 08 – 12: Journées de Physique Mathématiques - IHP.
• October 22 – 26: Recent Developments in Conformal Geometry - Université de Nantes.
• November 05 – 09: Geometric PDEs - IHP.
• December 10 – 14: Recent Developments in Kähler Geometry - IHP.

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# International Conference on Geometric Analysis

## Domaine de Seillac (France), September 4–7, 2012

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The International Conference on Geometric Analysis intend to bring together specialists in the field of Geometric Analysis for a week. We hope that the pleasant environment of Domaine de Seillac will enhance communication and collaboration between participants.

The topics covered include: Variational problems in submanifold theory (minimal surfaces, constant mean curvature surfaces, etc.), Geometric flows, applications to General Relativity.

Students and young researchers are encouraged to apply. Some financial support will be available.

# Spanish Relativity Meeting in Portugal

## Guimaraes (Portugal), September 3–7, 2012

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The Spanish Relativity Meeting (ERE) is an international conference devoted to Relativity and Gravitation which is organized every year by one of the Spanish or Portuguese groups working in this area. It is supported by the Spanish Society of Gravitation and Relativity (SEGRE).

## Leuven (Belgium), August 27–30, 2012

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### Topics

Minimal submanifolds, Lagrangian submanifolds and related topics, Affine differential geometry, Lorentzian geometry.

### Invited Lecturers

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• Ildefonso Castro (Universidad de Jaen)
• Bang-Yen Chen (Michigan State University)
• Benoît Daniel (Université Nancy 1) (tbc)
• An-Min Li (Sichuan University)
• Jason Lotay (UCL, London)
• Udo Simon (T.U. Berlin)
• Rabah Souam (Université Paris 7)

# ICTP-ESF School and Conference on Geometric Analysis

## Trieste (Italy), June 11–29, 2012

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The Abdus Salam International Centre for Theoretical Physics (ICTP) is organizing a School and Conference on Geometric Analysis within the framework of the ICTP-ESF Series of High Level Conferences on Geometry and Dynamics. The activity will be held from 11 to 29 June 2012, in Trieste, Italy. It will be directed by C. Arezzo (ICTP, Trieste, Italy), F. Pacard (Ecole Polytechnique, Palaiseau, France) and R. Schoen (Stanford University, USA).

### Main topics

• Evolution equations on real and complex manifolds
• Riemannian Geometry and General Relativity
• Extremal Kähler metrics

The first two weeks (11 - 22 June) will be an Advanced School: six mini-courses of high level on the main topics above. There will be seminars and training activities for the participants. Some leading and famous mathematicians will present the mini-courses.

Invited Lecturers: S. Brendle (Stanford), F. Pacard (Ecole Polytechnique), T. Riviere (ETH), R. Schoen (Stanford), G. Tian (Princeton and Beijing), P. Topping (Warwick).

The third week (25 - 29 June) will be an International Conference on Geometric Analysis. Its primary aim is to promote interaction among the leaders in Geometric Analysis and to introduce and discuss front-line research with young participants from all over the world.

The tentative list of speakers of the third week's Conference (24-29 June), includes: R. Berman, Jingyi Chen, J. Corvino, J. Fine, V. Guedj, J. Isenberg, G. La Nave, C. Le Brun, Y.-I. Lee, Chi Li, Jiayu Li, A. Malchiodi, F. Marques, Song Sun, V. Tosatti, J. Viaclovsky, B. Wilking*, Xiaohua Zhu.
* to be confirmed

### Participation

Mathematicians from all countries which are members of the United Nations, UNESCO or IAEA may attend the activity.

# Differential Geometry

## Bedlewo (Poland), June 3–8, 2012

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The conference is planned as the fifth in the series (previous conferences in the years: 2000, 2003, 2005, 2008) of meetings organized in the framework of the BC activities by geometers from the Jagiellonian University in Cracow and Technische Universität Berlin. Since the meeting of the group of researchers from Europe, Japan, China and USA in Oberwolfach in 1986 , they have organized and attended many conferences (in various places of the world) dealing with topics connecting Riemannian (or pseudo-Riemannian) geometry, affine differential geometry (leading the topic far beyond the classical affine differential geometry), PDEs and their geometric aspects. The former conferences essentially enlarged the cooperation between mathematicians representing the above fields as well as attracted young researchers. A detailed list (with references, information about non-published partial results) of open problems was formulated during each of the above mentioned conferences. This time we are going to formulate such a list as well.

A proposed list of topics of the planned meeting contains, in particular, the following:

• Manifolds and PDEs (in particular Ricci solitons, geometric solution of the Monge-Ampère equations, solutions for certain types of PDEs via the geometric context in which they arise, nonlinear forth order PDEs which appear when studying curvature problems in affine hypersurface theory)
• affine submanifolds (after some stagnation there have appeared new important papers in the field, for instance two spectacular papers of Z. Hu, C. Lee, H. Li, L. Vrancken, namely complete classification of all Blaschke hypersurfaces with parallel (Levi-Civita) cubic form - in case locally strongly convex (J. Diff. Geom. to appear)- in case of Lorentz metrics (RM to appear),
• relations between submanifolds in the Riemannian and affine settings (parallel submanifolds in the sense of Hicks, Backlund related submanifolds etc.),
• statistical and Hessian manifolds in relation with affine differential geometry,
• submanifolds of product spaces (submanifolds with extrinsic symmetry properties, submanifolds solving some variational problems etc.),
• affine geometry on abstract manifolds, in particular the theory of homogeneous affine connections, affine manifolds (in the sense of Auslander),
• Weyl geometries,
• manifolds with special structures (Sasakian, Kaehler-Norden, contact etc.),
• curvature conditions (in particular, decomposition of curvature tensor),
• Lagrangian and CR submanifolds (in Riemannian and affine setting) conditions in Riemannian and pseudo-Riemannian geometry)
• Lagrangian and CR submanifolds (metric and affine)

# Workshop on Geometric Analysis

## Goethe-Universität Frankfurt, March 27 –30, 2012

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The aim of the workshop is to bring together people working in Geometric Analysis.

### Speakers

• Spyros Alexakis (University of Toronto)
• Christine Breiner (MIT)
• Esther Cabezas-Rivas (Universität Münster)
• Jingyi Chen (University of British Columbia)
• Camillo De Lellis (Universität Zürich)
• Ailana Fraser (University of British Columbia)
• Gerhard Huisken (MPI für Gravitationsphysik Golm)
• Yuxiang Li (Tsinghua University Beijing)
• Andrea Malchiodi (SISSA)
• Jan Metzger (Universität Potsdam)
• William P. Minicozzi II (Johns Hopkins University)
• André Neves (Imperial College London)
• Tristan Rivière (ETH Zürich)
• Felix Schulze (Freie Universität Berlin)
• Jeffrey Streets (University of California, Irvine)
• Peter Topping (University of Warwick)
• Brian White (Stanford University)
• Neshan Wickramasekera (University of Cambridge)

# Research Programme on Geometry and Quantization of Moduli Spaces

## CRM Bellaterra (Barcelona), March–June, 2012

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### Scientific description

This Research Program will be centered on the study of the geometry of algebraic moduli spaces, mostly (but not exclusively) associated to compact Riemann surfaces. The scope is intended to be wide, ranging from questions on the topology (cohomology, stable homotopy, etc.) of moduli spaces to problems on their quantization (Verlinde algebra, Hitchin connection, etc.), including also problems on dynamics (e.g. action of the mapping class group of the surface), quantum cohomology (Atiyah–Floer conjecture, Geometric Langlands program) and other areas. Most of the moduli spaces to be studied in the Research Program will be either instances or related to moduli spaces of flat connections and Higgs bundles on Riemann surfaces.

### Perspectives of the Programme

The central aims of the programme are to bring together experts in various aspects on the geometry and quantization of moduli spaces and related areas, to advance these topics, and to introduce research students and post-docs to the wealth of ideas and problems in them. As stated above, the interdependence of the topics we have identified is crucial to the development of the theory, and a major goal is to develop these ideas further. The programme will include an advanced course, two workshops, a final conference, as well as a regular seminar.

# Workshop on Geometry of Interfaces and Capillarity

## Granada (Spain), June 25–27, 2012.

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### Description

The objective of the workshop is to bring together researchers from mathematics and physics working on geometric problems involving interfaces, capillarity, surfaces with constant mean curvature and wetting phenomena.

### Invited speakers

• Martin Brinkmann (MPI for Dynamics and Self-Organization, Göttingen)
• Rustum Choksi (McGill University)
• Robert Finn (Stanford University)
• Reinhard Lipowsky (MPI of Colloids and Interfaces, Potsdam)
• John McCuan (Georgia Institute of Technology)
• Bennett Palmer (Idaho State University)
• Ralf Seemann (Universität des Saarlandes)
• Henry Wente (University of Toledo)

### Organizers

• Robert Finn, Stanford
• Reinhard Lipowsky, MPIKG

# Workshop on the Willmore conjecture

## Granada (Spain), May 16–18, 2012.

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### Goal of the workshop

Fernando C. Marques (IMPA - Rio de Janeiro, Brazil) and André Neves (Imperial College - London, UK) have recently given a proof of the Willmore conjecture (arXiv:1202.6036v1 [math.DG]). In this workshop these authors will give a detailed survey on their proof.

Additionally, related aspects of the Willmore conjecture will be covered in the meeting.

### Speakers

• Ernst Kuwert (Albert-Ludwigs-Universität Freiburg, Germany)
• Fernando Codá Marques (IMPA, Brasil)
• André Neves (Imperial College, UK)
• Franz Pedit (University of Massachussets, Amherst, USA)

# 2nd European Young and Mobile Workshop: Geometric analysis and Partial Differential Equations

## Granada, November 28th to 30th, 2011.

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Since 2010, the European Research Councils supports an annual workshop on varying outstanding topics in Mathematics, where a limited number of young researchers across the European Union can lecture about their progresses in problems related to the topic to which the workshop is devoted. This year the selected topic is Geometric Analysis. These young researchers will be selected by their own countries. Additionally, the organization will invite a number of senior experts in the field to give plenary lectures.

### Main speakers

• Rafe Mazzeo
• William P. Minicozzi II
• Vicente Miquel
• Peter Topping
• Marco Rigoli
• Antonio Ros

# VI International Meeting on Lorentzian Geometry

## Granada, September 6th to 9th, 2011.

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This coference is the sixth one of a series of biennial meetings about Lorentzian Geometry which started in 2001. Topics on pure and applied Lorentzian Geometry such as geodesics, submanifolds, causality, black holes, Einstein equations, geometry of spacetimes or AdS-CFT correspondence, will be covered.

### Main speakers

• Luis Alias (Univ. Murcia, Spain)
• Jose Carlos Diaz-Ramos (Univ. Santiago de Compostela, Spain)
• Jonatan Herrera (Univ. Malaga, Spain)
• Erasmo Caponio (Politecnico di Bari, Italy)
• Ines Kath (Univ. Greifswald, Germany)
• Vladimir Matveev (Friedrich-Schiller-Universität Jena, Germany)
• Kotaro Yamada (Tokyo Institute of Technology, Japan)
• Vladimir Chernov (Univ. Dartmouth, USA)
• Paolo Piccione (Univ. Sao Paulo, Brasil)

# XX International Fall Workshop on Geometry and Physics

## ICMAT (Madrid), August 31st to September 3rd, 2011

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The Fall Workshops on Geometry and Physics have been held yearly since 1992, and bring together Spanish and Portuguese geometers and physicists, along with an ever increasing number of participants from outside the Iberian peninsula. The meetings aim to provide a forum for the exchange of ideas between researchers of different fields in Differential Geometry, Applied Mathematics and Physics, and always include a substantial number of enthusiastic young researchers amongst the participants.

### Invited speakers

• Mauro Carfora (U. Pavia)
• Piotr T. Chruściel (U. Vienna)
• Marisa Fernández (U. País Vasco)
• Gary Gibbons (U. Cambridge)
• Janusz Grabowski (Polish Academy of Sciences)
• Renate Loll (Utrecht U.)
• Álvaro Pelayo (Institute for Advanced Study, Princeton)
• Patrizia Vitale (Naples U.) To be confirmed.

### Minicourses

• Alberto Ibort (U. Carlos III)
• Franco Magri (U. Milano Bicocca)

# Geometric Analysis Seminar

## Valencia, June 16th to 18th, 2011.

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### Aim and scope

The workshop will be divided in two types of activities. During the mornings there will be talks given by young researchers in Geometric Analysis where they will present their current research interests.In the afternoons there will be informal talks as well as working or problem sessions suggested by the participants. The meeting will close with a lecture given by the senior speaker.

### Invited speakers

• Esther Cabezas-Rivas, Universidad de Münster
• Antonio Cañete, Universidad de Sevilla
• Pablo Miguel Chacón Martín, Universidad de Salamanca
• Fernando Galaz, Universidad de Münster
• Pablo Mira, Universidad Politécnica de Cartagena

### Senior invited speakers

• Steen Markvorsen, DTU Matematik, Denmark.

# Doctoral course: Submanifold theory and applications

## Sevilla, March 28th to April 8th, 2011.

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This is an advanced school that is part of the doc-course (introductory research international school) organized by the Research Institute of Mathematics of the University of Seville (IMUS) for the year 2011. The aim is to give an introduction to some aspects in surface and submanifold theory, specially for PhD students and young researchers.

It will consist of 4 courses of 7 hours each, as well as an international conference of surface theory in 3-manifolds, with the participation of internationally renowned experts in the field.

# International Conference on Surface Theory

## Sevilla, April 5th to 8th, 2011.

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This conference will have an emphasis in the most important facts of the global theory of minimal and constant mean curvature surfaces, as well as other specially relevant geometric PDEs. It will also be dealing with other types of curvature conditions (such as constant curvature surfaces) and other natural geometric conditions (as for example, local convexity).