D.V. Alekseevsky

Spinor structures on compact homogeneous manifolds   SLIDES

We study spin structures on compact simply-connected homogeneous pseudo-Riemannian manifolds (M = G/H,g) of a compact semisimple Lie group G. The existence of spin structure on a such manifold does not depend on metric and is equivalent to vanishing of the second Stiefel-Whitney class w_2(M). Moreover, if the manifold M admits a complex structure J, then this condition is equivalent to the condition that the first Chern class c_1(M,J) in H^2(M,Z) is even. For a flag manifold F = G/H, the parity of c_1(F,J) does not depend on invariant complex structure. We calculate c_1(F,J) for some "standard" invariant complex structure for all flag manifolds F=G/H where G is a simple compact Lie group ( classical or exceptional). This gives a classification of all spin flag manifolds. We apply this result to construction C-spaces ( a homogeneous torus bundles over flag manifolds) which admit spin structure. The talk is based on a joint work with Ioannis Chrysikos.

C. Böhm

Homogeneous Ricci flows

We provide optimal curvature estimates for homogeneous Ricci flows. Moreover, we show that homogeneous Ricci flow solutions subconverge (after suitable rescalings) in case of finite extinction time to a nonflat, homogeneous gradient shrinking soliton and in the immortal case to a homogeneous nongradient expanding soliton.

R. Bryant

On curvature-homogeneous metrics in dimension 3

A Riemannian manifold (M,g) is said to be {\it curvature-homogeneous} if it is homogeneous to second order, i.e., if, for any two points in M, the Riemannian curvature tensors are equivalent under some isometry of the two tangent spaces. Of course, a locally homogeneous metric is curvature-homogeneous, but the converse is not true in dimensions greater than 2. Already in dimension 3, there are many unanswered questions about the existence and generality of curvature homogeneous metrics, even locally. In this case, curvature-homogeneity is equivalent to having the eigenvalues of the Ricci curvature be constant, which is a system of partial differential equations on the metric. In this talk, I will review what is known about such metrics in dimension 3, particularly the work of O. Kowalski and his collaborators during the 1990s and, more recently, Schmidt and Wolfson. I will show that, for certain values of the eigenvalues of the Ricci tensor, these partial differential equations are integrable by Darboux Method, which yields some surprising relations with classical subjects, such as the theory of holomorphic curves in the complex projective plane and contact curves in the boundary of the complex 2-ball.

T. Colding

Level set flow   SLIDES

The level set method has been used with great success the last thirty years in both pure and applied mathematics to describe evolutions of various physical situations. In mean curvature flow, the evolving hyper surface (front) is thought of as the level set of a function that satisfies a nonlinear degenerate parabolic equation. Solutions have always been defined in the viscosity sense. Viscosity solutions are functions that in general may not even be differentiable (let alone twice differentiable) but satisfy a second order differential equation in a weak sense. For a monotonically advancing front, I will describe why viscosity solutions are in fact twice differentiable and satisfy the equation in the classical sense. The proof weaves together analysis and geometry. Moreover, I explain how the situation becomes very rigid when the second derivative (that is always bounded) is also continuous.

U. Hamenstädt

Spectral properties of hyperbolic 3-manifolds

We relate the first eigenvalue of the Laplacian of a closed hyperbolic 3-manifold to its volume and its Heegaard genus and discuss some spectral properties of the Laplacian on one-forms as well. We describe briefly how these analytic properties relate to some recent open conjectures on covers of such manifolds.

M. Haskins

Exotic Einstein metrics on S^6 and S^3xS^3, nearly Kähler 6-manifolds and G2 cones   SLIDES

H. Hein

Tangent cones of Kähler-Einstein metrcs   SLIDES

A cone in the sense of Riemannian geometry is a warped product manifold C = (0,infty) x L with metric g_C = dr^2 + r^2 g_L, where (L, g_L) is a closed Riemannian manifold called the link of C. If the Riemannian metric g_C is Kähler, it turns out that the 1-point completion of C always has a natural structure of a normal affine algebraic variety with one isolated singularity. This basic fact leads to a rich interplay between the algebraic geometry of singularities of complex varieties and the metric geometry of Kähler manifolds - in particular, of Kähler-Einstein manifolds. I will give an overview of various recent results in this direction.

B. Kleiner

Ricci flow through singularities   SLIDES

It has been a long-standing problem in geometric analysis to find a good definition of generalized solutions to the Ricci flow equation that would formalize the heuristic idea of flowing through singularities. I will discuss a notion in the 3-d case that has good analytical properties, enabling one to prove existence and compactness of solutions, as well as a number of structural results. It may also be used to partly address a question of Perelman concerning the convergence of Ricci flow with surgery to a canonical flow through singularities. This is joint work with John Lott.

C. LeBrun

Mass in Kahler Geometry   SLIDES

Given a complete Riemannian manifold that looks enough like Euclidean space at infinity, physicists have defined a quantity called the "mass" which measures the asymptotic deviation of the geometry from the Euclidean model. In this lecture, I will explain a simple formula, discovered in joint work with Hajo Hein, for the mass of any asymptotically locally Euclidean (ALE) Kahler manifold. For ALE scalar-flat Kahler manifolds, the mass turns out to be a topological invariant, depending only on the underlying smooth manifold, the first Chern class of the complex structure, and the Kahler class of the metric. When the metric is actually AE (asymptotically Euclidean), our formula not only implies a positive mass theorem for Kähler metrics, but also yields a Penrose-type inequality for the mass.

A. Lytchak

Minimal discs in metric space

In the talk I will discuss the solution of the classical Plateau problem in metric spaces, the structure of the arising minimal discs and applications to metric geometry, geometric group theory and analysis on metric spaces. The talk will be based on a series of joint papers with Stefan Wenger.

A. Neves

Towards Yau's conjecture   SLIDES

Yau's conjecture states that any closed three manifold has an infinite number of minimal surfaces. I will explain how a stronger version of that conjecture follows from the multiplicity one conjecture in min-max theory. This is joint work with Fernando Marques.

S. Salamon

Quaternionic geometry in 8 dimensions   SLIDES

I shall present an overview of the theory of manifolds modelled on the quaternions, with emphasis on 8 real dimensions and Sp(2)Sp(1) structures with closed and parallel forms. This will include work (with D. Conti and T. Madsen) concerning metrics with a cohomogeneous-one SU(3) action, and a relationship with metrics of holonomy G_2.

V. Schroeder

Aperiodic geodesics in negative curvature

We define a quantitative measure of the aperiodicity of an orbit of a dynamical system and study this concept for geodesics in hyperbolic manifolds. We can show the existence of "maximally aperiodic" geodesics and we relate this to other geometric and topological invariants of the manifold. This is joint work with Steffen Weil.

V. Tosatti

The Ricci flow on compact Kahler manifolds

The behavior of the Ricci flow on compact Kahler manifolds is intimately related to the complex structure of the manifold. In particular on projective manifolds it has direct connections with the minimal model program in algebraic geometry. It is known that the maximal existence time of the flow can be computed from simple cohomological data. In the case when this is finite, I will give a geometric description of the set where the singularities occur. When the maximal existence time is infinite, I will discuss what is known about metric behavior as time goes to infinity.

B. Wilking

Computing the Eulercharacteristic of positively curved manifolds under logarithmic symmetry assumptions

We show that the Eulercharacteristic of a positively curved n manifold M coincides with the Eulercharacteristic of an n-dimensional compact rank 1 symmetric space provided that the rank of the isometry group of M is larger than 3log_2n.

J. Wolf

Homogeneity for Riemannian Quotient Manifolds   SLIDES

Let M be a simply connected Riemannian homogeneous space and N = Γ/M a Riemannian quotient manifold. The Homogeneity Conjecture says that N is homogeneous if and only if all the elements of Γ are isometries of constant displacement. This conjecture is proved when M is Riemannian (or even Finsler) symmetric, in a number of settings where M has enough negative curvature to force bounded isometries to be trivial, and in several settings where M is a Riemannian normal homogeneous space. I'll survey this and describe some new results proving the conjecture for a class of twistor spaces.