## Workshop on Geometry & Topology 2010

September 16-18, 2010

University of Fribourg, Switzerland
Department of Mathematics
Lecture Room 0.110, Biology Building

## Preliminary program

 Thursday September 16 10:30-11:00 Registration 11:00-12:00 Michael Farber (Durham & ETH Zürich): Stochastic algebraic topology I Lunch break, tables reserved at mensa 13:30-14:30 Iskander Taimanov (Novosibirsk): Surfaces in three-dimensional Lie groups Coffee break 15:00-16:00 Urs Lang (ETH Zürich): Injective hulls of word hyperbolic groups 16:00-16:45 Sebastian Grensing (Kiel & Karlsruhe): Virtual boundaries of higher rank graph-spaces Friday September 17 9:30-10:30 Peter Teichner (Berkeley & MPI Bonn): Equivariant cohomology via super symmetric field theories I 11:00-12:00 Michael Farber (Durham & ETH Zürich): Stochastic algebraic topology II Lunch break, tables reserved at mensa 13:30-14:30 Christoph Böhm (Münster): The second best Einstein metric in higher dimensions Coffee break 15:00-16:00 Boris Botvinnik (Oregon): Isotopy and concordance of metrics of positive scalar curvature 16:00-16:45 Michael Wiemeler (Fribourg): Quasitoric manifolds homeomorphic to homogeneous spaces 18:30 Conference Dinner at Gemelli, Grand-Places 10 (near NH and train station) Saturday September 18 9:30-10:30 Mikiya Masuda (Osaka): Cohomological rigidity problems in toric topology and topological toric manifolds 10:45-11:45 Frank Kutzschebauch (Bern): A knotted minimal surface in \R^4 which is homeomorphic to \R^2 Lunch break, tables reserved at San Marco (Boulevard de Perolles 18) 13:30-14:30 Peter Teichner (Berkeley & MPI Bonn): Equivariant cohomology via super symmetric field theories II 14:30-15:15 Suyoung Choi (Osaka): Classification of real Bott manifolds and acyclic digraphs 15:30 Excursion to Bern

## Abstracts

 Böhm, Christoph (Münster): The second best Einstein metric in higher dimensions Abstract: The round metric on spheres is the "best" Einstein metric, since its Weyl curvature vanishes. In dimensions greater or equal to twelve we describe the second best Einstein metric with positive scalar curvature. Botvinnik, Boris (Oregon): Isotopy and concordance of metrics of positive scalar curvature Abstract: Two positive scalar curvature metrics on a compact manifold are isotopic if they are homotopic through metrics of positive scalar curvature. It is well known that if two metrics $g_0$ and $g_1$ of positive scalar curvature on a closed compact manifold are isotopic, then they are concordant, i.e. there exists a metric $\bar g$ of positive scalar curvature on the cylinder $M\times I$ which is a product metric near the boundary and extending the metrics $g_0$ on $M\times \{0\}$ and $g_1$ on $M\times \{1\}$. We will discuss recent results on the question whether concordance imply isotopy. Choi, Suyoung (Osaka): Classification of real Bott manifolds and acyclic digraphs Abstract: A real Bott manifold is a closed smooth manifold obtained as the total space of an iterated RP1-bundles starting with a point, where each fibration is the projectivization of the Whitney sum of two real line bundles. We show that the diffeomorphism types of real Bott manifolds can be completely characterized in terms of graph operations on directed acyclic graphs. Using this combinatorial interpretation, we find some topological properties of real Bott manifolds. Farber, Michael (Durham & ETH Zürich): Stochastic algebraic topology I & II Abstract: Topological spaces and manifolds are commonly used to model configuration spaces or phase spaces of physical and economical systems. Many current technological challenges, such as those dealing with the modelling, control and design of large systems, lead to topological problems which very often have mixed topological-probabilistic character and therefore require new mathematical tools. In my two talks I will describe several models producing random simplicial complexes and closed smooth manifolds depending on a large number of random parameters and also mechanisms producing random groups. I will focus on recent results and techniques allowing prediction of topological properties of random spaces with high probability. Grensing, Sebastian (Kiel & Karlsruhe): Virtual boundaries of higher rank graph-spaces Abstract: We consider discrete group actions on locally compact Hadamard spaces. Any such action extends to a topological action on the virtual boundary. C. B. Croke and B. Kleiner introduced a class of so-called admissible actions and associated geometric data which determine the topological conjugacy class of the boundary action. They posed the question whether their results hold for a wider class of actions. We show that, for the natural generalization, their question has to be answered in the negative: There is an admissible action of higher rank on a pair of Hadamard spaces with equivalent geometric data and an equivariant quasiisometry which does not extend continuously to the virtual boundary. Kutzschebauch, Frank (Bern): A knotted minimal surface in \R^4 which is homeomorphic to \R^2 Abstract: In this joint work with Sebastian Baader and Erlend Fornaess Wold we solve problem 1.102 (B/C) of Kirby's list on "Problems in low dimensional Topology": We construct proper embeddings of \R^2 into \R^4 whose image is a topologically knotted minimal surface. The proof is a combination of facts from knot theory (due to Rudolph Lee and Stephan Orevkov) and complex analysis (due to Josip Globevnik). Lang, Urs (ETH Zürich): Injective hulls of word hyperbolic groups Abstract: Injective metric spaces, or absolute 1-Lipschitz retracts, share a number of properties with CAT(0) spaces. In the 1960es, J.R.Isbell showed by means of an explicit construction that every metric space possesses a uniquely determined injective hull. We apply this construction to Gromov hyperbolic spaces and groups. The injective hull of a geodesic hyperbolic space $X$ is within bounded distance of $X$. The injective hull of a word hyperbolic group $\Gam$ is a locally finite polyhedral complex with finitely many isometry types of (injective) cells, on which $\Gam$ acts properly and cocompactly by cellular isometries. Some further properties of this complex will be discussed. Masuda, Mikiya (Osaka): Cohomological rigidity problems in toric topology and topological toric manifolds Abstract: Classification of compact smooth toric varieties (which we call toric manifolds) as varieties reduces to classification of their fans as is well-known. However, not much is known for classification of toric manifolds as smooth manifolds. One interesting and naive question is Cohomological rigidity problem for toric manifolds: Are two toric manifolds diffeomorphic (or homeomorphic) if their cohomology rings are isomorphic as graded rings? There are some affirmative solutions and no counterexamples are known to the problem so far. Similar questions can be asked for real toric manifolds and symplectic toric manifolds. In this talk I will discuss the cohomological rigidity problem and related problems. Then I will introduce a new family of manifolds called topological toric manifolds which seems a right topological analogue to toric manifolds. Taimanov, Iskander (Novosibirsk): Surfaces in three-dimensional Lie groups Abstract: We expose the spinor (Weierstrass representation) for surfaces in three-dimensional Lie groups and its relation to the spectral theory of two-dimensional Dirac operators and generalizations of the Willmore functional for surfaces in such ambient spaces. Teichner, Peter (Berkeley & MPI Bonn): Equivariant cohomology via super symmetric field theories I & II Abstract: We will introduce a precise language to describe classical and quantum field theories in terms of certain smooth functors. Then we will compute the space of such (Euclidean) field theories for the smallest interesting dimension of space-time, namely 0|1. It turns out that for pure Sigma-models, this space classifies de Rham cohomology, and for gauged Sigma-models, it leads to equivariant cohomology. Wiemeler, Michael (Fribourg): Quasitoric manifolds homeomorphic to homogeneous spaces Abstract: Quasitoric manifolds are certain 2n-dimensional manifolds on which a n-dimensional torus acts. In this talk we discuss the following question: Which quasitoric manifolds are homeomorphic to homogeneous spaces? If the first Pontrjagin-class of a quasitoric manifold vanishes, then it is a manifold of q-type. Therefore we may apply results of Hauschild on the degree of symmetry of a manifold of q-type. We show that each quasitoric manifold M with p_1(M)=0 which is homeomorphic to a homogeneous space is a product of two-dimensional spheres.

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