Geometry-Topology Group
News
2009-2010 Annual Report, and more
In this document we discuss our philosophy, research projects, and recent (and not so recent) accomplishments.
The blog includes detailed seminar announcements, comments by speakers, and other updates about our activities.
Faculty
Christopher Atkinson, Research Assistant Professor
Chris Atkinson studies low-dimensional geometry and topology. His research provides an algorithmic method for estimating the volume of any non-obtuse hyperbolic polyhedron in terms of the combinatorics of its 1-skeleton. He is currently studying realization spaces and degenerations of hyperbolic polyhedra as well as questions related to the interaction between the geometry and topology of hyperbolic 3-orbifolds.
Akinson received his Ph.D. from the University of Illinois-Chicago, as a student of Ian Agol.
David Futer, Assistant Professor
Dave Futer studies low-dimensional topology and geometry. The central goal of his research is to build a dictionary between combinatorial descriptions of a 3-manifold and that manifold's (typically hyperbolic) geometry. The combinatorial input can be a triangulation, a diagram of a knot, or a surface homeomorphism that specifies a mapping torus. The geometric output involves quantities such as volume, lengths of geodesics, and the area of surfaces. Futer's recent research has also touched on geometric group theory.
Futer received his Ph.D. at Stanford University, as a student of Steve Kerckhoff.
Justin Malestein, Research Assistant Professor
Justin Malestein's research interests lie in low-dimensional (2 or 3) geometry/topology, rigidity theory, and applied math. He does research relating algebraic properties of mapping classes of surfaces and curves on surfaces to their topological/combinatorial properties. He is also currently researching combinatorial aspects of rigidity theory, inorganic crystals known as zeolites and the relation between the two.
Malestein received his Ph.D. at the University of Chicago, as a student of Benson Farb.
Kei Nakamura, Research Assistant Professor
Kei Nakamura's research interests are in low-dimensional topology and geometric group theory. These two closely related areas have always enriched each other. Nakamura primarily focuses on questions regarding 3-manifolds and their Heegaard splittings, knot theory, hyperbolic geometry, hyperbolic and relatively hyperbolic groups, mapping class groups, and the interplay between them.
Nakamura received his Ph.D. at the University of California, Davis as a student of Joel Hass.
Igor Rivin, Professor
Igor Rivin was an undergraduate at the University of Toronto, where he was fortunate enough to study with H. S. M. Coxeter and Ed Bierstone. He went on to study with Bill Thurston at Princeton University, and his checkered career included working with John McCarthy at Stanford (as Applications Director of the QLISP project on parallel symbolic computing), and with Stephen Wofram at Wolfram Research (as Director of Advanced Development for Mathematica, before returning to pure Mathematics. His research interests include hyperbolic geometry, geometry and topology of surfaces, convexity, combinatorial geometry, algebraic groups, probability theory, graph theory, dynamics, finance, and computational crystallography, where he has a long-running research collaboration with Mike Treacy on computational analysis of zeolites.
Louis Theran, Research Assistant Professor
Louis Theran studied with Ileana Streinu at the University of Massachusetts, Amherst, where he was a member of her Linkage Lab. His research interests relate to combinatorial rigidity, which relates the geometric properties objects defined by geometric constraints (e.g., scaffolds) to the combinatorial properties of their incidence structures. Along the way, algorithms, tree decompositions of graphs, and random graphs all come up. At the moment, Louis has been working questions arising in the study of zeolites with Igor Rivin. In the distant past, Louis worked at the OSF research center's web group and later the Nokia Research Center; then he decided to go to college.
Graduate Courses in Geometry and Topology
Math 8061: Smooth Manifolds, David Futer (Fall 2010)
This course will be an introduction to the geometry and topology of smooth manifolds. We will begin the fall semester with the definitions: what does it mean for a space to (smoothly) look just like Rn? We will go on to study vector fields, differential forms (a way to take derivatives and integrals on a manifold), and Riemannian metrics. In the spring semester, we'll study the interplay between the geometry of a manifold and certain ideas from algebraic topology. We will review the idea of the fundamental group and introduce homology - and then relate these algebraic notions to the underlying geometry. If time permits, we will talk a bit about hyperbolic manifolds - a family of manifolds where the interplay between topology and geometry is particularly strong and beautiful.
We will draw from several sources, including Introduction to Smooth Manifolds by John Lee, A Comprehensive Introduction to Differential Geometry (volume 1) by Michael Spivak, and Algebraic Topology by Allen Hatcher.
Math 8700/CIS 9615: Analysis of Algorithms, Louis Theran (Fall 2010)
This course will cover the mathematical prerequisites (combinatorics, graph theory, asymptotic analysis, difference equations) needed for the design and analysis of algorithms, followed by the basic topics such as:
- Sorting
- Searching
- String algorithms
- Dynamic programming
- Basic graph algorithms (shortest paths, minimum cost spanning trees, etc.)
- Network flow algorithms
- Linear Programming
Math 9005: Graph Theory, Louis Theran (Spring 2011)
This course will be an introduction to graph theory. Some of the topics covered will be:
- Trees, connectivity, and matroids
- Matchings and 2-factorizations
- Planarity, duality, and graph drawing
- Random graphs
We'll be using Graph Theory by Diestel as the main course text, with supplements from more specialized monographs (e.g., Matching Theory by Lovasz & Plummer, Matroid Theory by Oxley) as necessary.