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Summer Graduate School

Topological and Geometric Structures in Low Dimensions (SLMath) July 21, 2025 - August 01, 2025
Parent Program:
Location: SLMath: Eisenbud Auditorium, Atrium
Organizers LEAD Kenneth Bromberg (University of Utah), Kathryn Mann (Cornell University)

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<p>Laminations arise naturally in hyperbolic geometry and (pseudo-) Anosov flows [Image by Jeffrey Brock]</p>

This school will serve as an introduction to the SLMath semester “Topological and Geometric Structures in Low-Dimensions”.  The school consists of two mini-courses: one on Teichmüller Theory and Hyperbolic 3-Manifolds and the other on Anosov Flows on Geometric 3-Manifolds.  Both topics lie at the interface of low-dimensional geometric topology (specifically, surfaces, foliations, and 3-manifolds) and low-dimensional dynamics.  The first course will be targeted towards students who have completed the standard first year graduate courses in geometry, topology, and analysis while the second course will geared towards more advanced students who are closer to beginning research. However, we expect that all students will benefit from both courses.

School Structure

There will be two lectures each day, as well as two problem sessions. Towards the end of the second week, students will be given the opportunity to present short research or expository talks.


Both courses will assume that students have taken standard first year graduate courses in topology (differentiable and algebraic), analysis (real and complex), and algebra with an emphasis on the first two topics. Some basic differential/Riemannian geometry is helpful but not strictly required. A good source of background material for both courses is the book An Introduction to Geometric Topology by Bruno Martelli. Besides being an excellent book, it also has the advantage that it is freely available on the arXiv. Chapter 1 of this book gives a good review of the background in topology and geometry that is necessary for both courses. While most of the material here is covered in first year courses, this chapter gives a good review for students who are interested in studying low dimensional manifolds. For the course on Teichmüller theory the suggested reading will be Chapters 2 and 3 of Martelli. Chapter 2 gives an an introduction to hyperbolic space, while Chapter 3 discusses hyperbolic manifolds. Students will not be expected to master this material and some of it will be covered in the course. Having some familiarity with hyperbolic space in advance will still be very useful.

For the Anosov flows on geometric manifolds course, students should know about the eight three-dimensional model geometries. Suggested prerequisite reading is either Chapter 12 of Martelli's book, or Sections 1, 4 and 5 of the expository paper The Geometries of 3-manifolds by Peter Scott. Students do not need to understand all proofs of all the results presented there, but should be able to list the 8 three dimensional model geometries, and be able to give an example of a compact 3-manifold modeled on each one.

Application Procedure

For eligibility and how to apply, see the Summer Graduate Schools homepage.

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