### The Geometry and Topology of Three-Manifolds

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They have been studies from these diverse points of view for more than one hundred years. Certain three-dimensional spaces arise in a natural way from the Bianchi groups; these spaces come equipped with a "hyperbolic geometry". This, in a nutshell, means that pairs of straight lines generically diverge from one another unlike Euclidean geometry. Three-dimensional spaces with a hyperbolic geometry are central to the study of general three-dimensional spaces as they represent, in a certain sense, the "generic" examples.

Hyperbolic spaces are among the most difficult to understand, and therefore the proposers aim to elucidate the geometry of these important examples mentioned above.

### Geometry and topology of three-manifolds

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Recent Awards. Presidential and Honorary Awards. The principal areas of research in geometry involve symplectic, Riemannian, and complex manifolds, with applications to and from combinatorics, classical and quantum physics, ordinary and partial differential equations, and representation theory. Research in topology per se is currently concentrated to a large extent on the study of manifolds in low dimensions. Topics of interest include knot theory, 3- and 4-dimensional manifolds, and manifolds with other structures such as symplectic 4-manifolds, contact 3-manifolds, hyperbolic 3-manifolds.

Research problems are often motivated by parts of theoretical physics, and are related to geometric group theory, topological quantum field theories, gauge theory and Seiberg-Witten theory, and higher dimensional topology. The undergraduate courses, Math , , are devoted to different topics in geometry and topology: Math Metric Differential Geometry Math Elementary Differential Topology Math Elementary Algebraic Topology.

There is a 3 semester sequence of graduate courses in geometry, Math , , , two of which are taught each year. Math Riemannian Geometry Math Complex Geometry Math Symplectic Geometry There is a 2 semester sequence in algebraic topology, A,B , taught every year, and a one semester course Math in the foundations of differential topology. Differentiable Manifolds Math A. Algebraic Topology Math B. Algebraic Topology Two or more topics courses are given yearly: Math Hot Topics Math Topics in Topology Math Factorization Algebras Teichner Spring , Math Gauge Theory Wehrheim Spring , Math Ricci Flow Lott Spring , Math Pseudoholomorphic Curves Wehrheim Spring , Math Contact Homology Hutchings Spring , Math Kaehler Geometry Lott Spring , Math Optimal Transport Lott Spring , Math Perturbative Quantization Teichner Spring , Math Mirror Symmetry Auroux Fall , Math Rational Homotopy Theory Teleman.

The Topology seminar is held weekly throughout the year, normally Wednesdays at 4pm. The speakers are normally visitors, but sometimes are resident faculty or graduate students. Three times a year the Bay Area Topology Seminar meets at Stanford fall , Berkeley winter and Davis spring , with two lectures in the afternoon and dinner afterward. There are various seminars related to symplectic geometry. In the first October meeting of each academic year, one of the talks is the Andreas Floer Memorial Lecture, given by a distinguished invited speaker.

The Differential Geometry seminar is held weekly throughout the year, normally Mondays at 5. Jump to Navigation. Courses Undergraduate upper division courses The undergraduate courses, Math , , are devoted to different topics in geometry and topology: Math Elementary Algebraic Topology Graduate courses There is a 3 semester sequence of graduate courses in geometry, Math , , , two of which are taught each year.

Bamler Associate Professor Geometric analysis, differential geometry, topology Robert Bryant Professor Emeritus Nonlinear partial differential equations and differential geometry, exterior differential systems, algebraic geometry, and Finsler geometry Alexander Givental Professor Symplectic and contact geometry, Singularity theory, Mathematical physics Robin C.

- Plautus, Vol. IV: The Little Carthaginian.Pseudolus. The Rope. (Loeb Classical Library No. 260).
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Recent Ph. Melvin Constantin Teleman. University of California, Berkeley. CalNet Login. Search this site:. Mina Aganagic. Ian Agol. Denis Auroux. Symplectic topology and mirror symmetry. Richard H. Geometric analysis, differential geometry, topology. Robert Bryant. Nonlinear partial differential equations and differential geometry, exterior differential systems, algebraic geometry, and Finsler geometry. Alexander Givental.

## Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds

Symplectic and contact geometry, Singularity theory, Mathematical physics. Robin C. Algebraic geometry, History of geometry.

Morris W. Dynamical systems, Neural networks. Wu-Yi Hsiang. Transformation groups, Differential geometry.

Michael Hutchings. Low Dimensional and Symplectic Topology and Geometry. Robion Kirby. The Journal of Geometric Analysis. Let M be a compact 3-manifold whose interior admits a complete hyperbolic structure. Similarly, we let D M be the infimum of the Hausdorff dimensions of limit sets of Kleinian groups whose quotients are homeomorphic to the interior of M. Unable to display preview. Download preview PDF. Skip to main content. Advertisement Hide. Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds.

This process is experimental and the keywords may be updated as the learning algorithm improves. This is a preview of subscription content, log in to check access. Abikoff, W. Google Scholar. Beardon, A. The Hausdorff dimension of singular sets of properly discontinuous groups, Am. Benedetti, R. Bishop, C. Hausdorff dimension and Kleinian groups, Acta Math.

## The Geometric Topology of 3-Manifolds

Bonahon, F. Cobordism of automorphisms of surfaces, Ann. Ecole Norm. Bouts des varietes hyperboliques de dimension 3, Ann. Braam, P. A Kaluza-Klein approach to hyperbolic three-manifolds, L'Enseign. Burger, M.