Speaker: Paul Baird (Université de Bretagne Occidentale, France)
Title 1: An introduction to biharmonic submersions via the space of circles in Euclidean space
Abstract 1: The origins of biharmonic maps can be traced back to 1744 and Euler's elastica.  These curves are important in elasticity theory, and arise as extrema of the total squared curvature with the constraint that the length of the curve be fixed. More generally, for a mapping between Riemannian manifolds, the bi-energy functional evaluates the total square norm of the tension field and extrema are called biharmonic maps.  The corresponding Euler-Lagrange equations are a fourth order elliptic system.  In this first talk, I will discuss Euler's elastica and their two-dimensional generalization (elastic surfaces), essentially extrema of the Willmore energy.  In harmonic map theory one finds a duality between minimal surfaces and a class of harmonic submersions with minimal fibres, leading to integral formulae which express complex-valued harmonic mappings on Euclidean space.  Given the strong connection between the Willmore energy and constant mean curvature surfaces, one may seek a similar duality, leading to integral formulae discovered by my past doctoral student M. Wehbe, for complex-valued biharmonic maps via a class of submersions which have circles as fibres.

Title 2: Biharmonic submersions through conformal geometry
Abstract 2: The fourth order Euler-Lagrange equations associated to the bi-energy functional are in general difficult to solve and in spite of several attempts, no satisfactory general theory has emerged.  Any harmonic map is biharmonic so one is interested in non-harmonic biharmonic maps, so-called proper biharmonic maps.  A nice example arising from the construction of my first talk, is the Hopf fibration viewed as a map from Euclidean 3-space to the plane by pre-composing and post-composing with stereographic projections.  This motivates the approach in this talk developed with my past doctoral student D. Kamissoko, where we begin with a harmonic map (such as the Hopf fibration) and conformally deform the metrics to arrive at a proper biharmonic map.  By this method, the fourth order system has become two second order systems.  On an Einstein manifold, isoparametric functions occur as the parameter in the conformal deformation, again highlighting the interplay between CMC hypersurfaces and biharmonic submersions.  A tool which is particularly suitable in the study of submersive mappings, is the associated stress-energy tensor, from which growth formulae can be deduced leading to Liouville-type theorems.

Speaker: Balázs Csikós (Eötvös Loránd University, Hungary)
• Joint work with Márton Horváth.
ˆ• The research was supported by the grants OTKA K72537 and OTKA K112703 of the Hungarian National Science and Research Foundation.

Title 1: A characterization of harmonic spaces
Abstract 1: We prove that in a complete, connected, and simply connected Riemannian manifold, the volume of the intersection of two small geodesic balls of equal radii depends only on the distance between the centers and the common value of the radii if and only if the space is harmonic.
References
[1] B. Csikós, M. Horváth, On the volume of the intersection of two geodesic balls, Dierential Geometry and its Applications 29:(4) pp. 567576. (2011)
[2] B. Csikós, M. Horváth, A characterization of harmonic spaces, Journal of Dierential Geometry 90: pp. 383-389. (2012)

Title 2: Harmonic Manifolds and the Volume of Tubes about Curves
Abstract 2: H. Hotelling proved that in the  $n$-dimensional Euclidean or spherical space, the volume of a tube of small radius about a curve depends only on the length of the curve and the radius. A. Gray and L. Vanhecke extended Hotelling's theorem to rank one symmetric spaces computing the volumes of the tubes explicitly in these spaces. In the present paper, we generalize these results by showing that every harmonic manifold has the above tube property. We compute the volume of tubes in the DamekRicci spaces. We show that if a Riemannian manifold has the tube property, then it is a 2-stein D'Atri space. We also prove that a symmetric space has the tube property if and only if it is harmonic. Our results answer some questions posed by L. Vanhecke, T. J. Willmore, and G. Thorbergsson.
References
[1] B. Csikós, M. Horváth, Harmonic manifolds and the volume of tubes about curves, Journal of the London Mathematical Society 94: pp. 141-160. (2016)

Speaker: Luis Guijarro (Universidad Autónoma de Madrid, Spain)
Title 1: A primer on Alexandrov spaces
Abstract 1: Alexandrov spaces appear naturally when considering Gromov-Hausdorff limits of manifolds with a lower bound on their curvatures. Although they lack differentiability, their metrics have enough structure to share many of the properties of the smooth case. We will give an overview of these ideas, and show some of the applications that make of Alexandrov spaces such a useful tool in global geometry.

Title 2: Submetries versus submersions
Abstract 2: Submetries are a natural generalization of Riemannian submersions to the context of metric spaces. In this talk we will examine how similar and how different submetries can be with respect to Riemannian submersions. In the first case, we show that submetries between Riemannian manifolds are C^{1,1}-Riemannian submersions; in the second we look at submetries between Alexandrov spaces and give several examples of behavior that differs drastically from the smooth case.

Speaker: Gerhard Knieper (Ruhr-Universität Bochum, Germany)
Title: A survey on non-compact harmonic manifolds
Abstract: A complete Riemannian manifolds is called harmonic iff harmonic functions have the mean value property, i.e. the average of harmonic functions over a geodesic sphere coincide with it's value at the center. In 1944 Lichnerowicz conjectured that all harmonic manifolds are either flat or locally symmetric spaces of rank 1. In 1990 the conjecture has been proved by Z. Szabo for harmonic manifolds with compact universal cover. Furthermore, the conjecture was obtained by Besson, Courtois and Gallot for compact manifolds of strictly negative curvature as an application of their entropy rigidity theorem in combination with the rigidity theorems by Benoist, Foulon and Labourie on stable and unstable foliations.
On the other hand, E. Damek and F. Ricci provided examples showing that in the non-compact case the conjecture is wrong. However, such manifolds do not admit a compact quotient.
In this talk we will present recent results on simply connected, non-compact and non-flat harmonic spaces. In particular for such spaces X the following properties are equivalent: X has rank 1, X has purely exponential volume growth, X is Gromov hyperbolic, the geodesic flow on X is Anosov with respect to the Sasaki metric.
Furthermore we obtain, that no focal points imply the above properties. Combining those results with the above mentioned rigidity theorems shows that the Lichnerowicz conjecture is true for all compact harmonic manifolds without focal points or with Gromov hyperbolic fundamental groups.
Some of the results have been generalized in collaboration with Norbert Peyerimhoff to asymptotically harmonic manifolds which we briefly mention if time permits.

Speaker: Yuri Nikolayevsky (La Trobe University, Australia)
Title 1: Solvable Lie groups of negative Ricci curvature
Abstract 1: The question of which homogeneous manifolds admit a left-invariant metric with the given sign of the curvature is well understood for the sectional curvature and also, for the positive and zero Ricci curvature. The case of negative Ricci curvature is wide open (semisimple examples constructed in the 80's). Our main question is the characterisation of (nonunimodular) solvable Lie groups admitting a left-invariant metric with Ric < 0. We answer this question for solvable Lie algebras whose nilradical is either abelian, or Heisenberg, or filiform. All of them have the same flavour: “there exists a vector Y such that real parts of the restriction of ad(Y) to the nilradical satisfy certain linear inequalities (which depend on the particular nilradical)”. Whether or not this can be generalised to all nilradicals is an open question. 
This is a joint ongoing project with Yurii Nikonorov.

Title 2: Totally geodesic hypersurfaces of homogeneous spaces
Abstract 2: We show that a simply connected Riemannian homogeneous space M which admits a totally geodesic hypersurface F is isometric to either (a) the Riemannian product of a space of constant curvature and a homogeneous space, or (b) the warped product of the Euclidean space and a homogeneous space, or (c)  the twisted product of the line and a homogeneous space (with the warping/twisting function in the last two cases given explicitly). In the first case, the hypersurface F by itself is also the Riemannian product; in the last two cases, it is a leaf of a totally geodesic homogeneous fibration. Case (c) can alternatively be characterised by the fact that M admits a Riemannian submersion onto the universal cover of the group SL(2) equipped with a particular left-invariant metric, and F is the preimage of a two-dimensional solvable totally geodesic subgroup of SL(2).

Speaker: Kouei Sekigawa (Niigata university, Japan)
Title: Notes on the Goldberg conjecture
Abstract: In 1969, S.I.Goldberg raised the conjecture, "Compact almost K"{a}hler Einstein manifolds are it{integrable}, and hence, K"{a}hler manifolds." In the present talk, we shall give a brief survey on the results for the Goldberg conjecture and also introduce some related topics to the same conjecture, for example, "Goldberg conjecture in pseudo-Riemannian setting" and "Odd-dimensional analogies of Goldberg conjecture".

Speaker: Martin Svensson (University of Southern Denmark, Denmark) 
Title 1: Twistor methods for harmonic maps
Abstract 1: In this lecture, I will introduce the concept of a harmonic map between two Riemannian manifolds, which includes familiar concepts such as geodesics, harmonic functions and minimal immersions. I will give a number of examples and discuss different characterisations of harmonic maps. I will then give an introduction to the twistor theory of harmonic maps, in which harmonic maps are constructed through holomorphic maps into auxiliary complex or almost complex manifolds. No prior knowledge of harmonic maps is expected, though I will assume some knowledge of Riemannian and complex geometry.

Title 2: Harmonic maps into $G_2/SO(4)$ and their twistor lifts
Abstract 2: Burstall and Rawnsley have shown how the canonically fibered flag manifolds sit inside the twistor space of a compact, simply connected inner Riemannian symmetric space. It is known that a harmonic map from a surface into an inner Riemannian symmetric space of classical type has a twistor lift into such a flag manifold if and only if it is nilconformal in the sense that its derivative is nilpotent. In my talk, I will show that this result can be generalised to harmonic maps into the exceptional inner symmetric space $G_2/SO(4)$. I will describe the structure of the canonically fibered flag manifolds over this space and the construction of the twistor lifts of nilconformal harmonic maps. I will also show how almost complex maps into $S^6$ can be used to construct harmonic maps into $G_2/SO(4)$. The talk will be based on joint work with John C. Wood.

Speaker: Wilderich Tuschmann (Karlsruher Institut für Technologie, Germany)
Title: Spaces and moduli spaces of metrics with curvature bounds I, II
Abstract: In my talks, I will present and survey recent results about such spaces and moduli spaces of complete
Riemannian metrics with curvature bounds on open and closed manifolds, and also discuss several open problems 
and questions in the field.