Geometry and Topology

Faculty

Ser-Wei Fu, Research Assistant Professor

Ser-Wei studies geometry and topology of low-dimensional manifolds. He is interested in motivations and approaches in the fields of hyperbolic geometry, geometric group theory, geometric structures, ergodic theory, billiards, and Teichmuller theory. His research is focused on the geometric aspect (length) and the topological aspect (intersection number) of closed curves on surfaces. Curves and surfaces, despite basic, still motivates many interdisciplinary and open problems. These problems continue to serve as motivation to develop new tools and techniques.

Fu received his Ph.D. at the University of Illinois, as a student of Chris Leininger.


David Futer, Associate Professor

Dave Futer studies low-dimensional topology and geometry. A central goal of his research is to relate several distict themes in low-dimensional topology: combinatorial descriptions of 3-manifolds, their (typically hyperbolic) geometry, the coarse geometry of fundamental groups, and quantum invariants of knots and links. These distinct points of view turn out to be inter-related in surprising ways.

Futer received his Ph.D. at Stanford University, as a student of Steve Kerckhoff.


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.

Rivin is currently on leave from Temple and is Regius Professor of Mathematics at the University of St. Andrews.


Matthew Stover, Assistant Professor

Broadly, Matthew Stover is interested in the interplay between geometry, topology, number theory, and group theory. Most of his work is on the geometry and topology of locally symmetric manifolds, which are Riemannian manifolds closely related to discrete subgroups of Lie groups, and their generalizations, especially those spaces arising from arithmetic subgroups of algebraic groups. He is especially interested in using techniques from number theory, group theory, and algebraic geometry to understand negatively curved locally symmetric manifolds, like compact hyperbolic n-manifolds. More recently, he has started working in the other direction, applying geometric techniques inspired by ideas in low-dimensional topology to prove theorems in number theory and algebraic geometry. He also has interests in character varieties of finitely generated groups, which are spaces parametrizing representations into a fixed Lie groups, particularly for those groups appearing in low-dimensional topology.

Stover received his Ph.D. at the University of Texas, where he was a student of Alan W. Reid.


Graduate Students

Dianbin Bao

Dianbin Bao is studying identities between Hecke eigenforms, and their arithmetic consequences, for congruence subgroups of the modular group. His thesis advisor is Matthew Stover.


Tim Morris

Tim Morris is pursuing dissertation research on arithetic manifolds, hyperbolic geometry, character varieties of 3-manifold groups, and some geometric group theory. His thesis advisor is Matthew Stover.


Thomas Ng

Thomas Ng is pursuing dissertation research on the interplay between geometric group theory and low-dimensional topology. He is interested in algorithmic properties of groups that arise in the study of surfaces and 3-manifolds from the coarse geometry of their actions on various combinatorial objects and complexes. His thesis advisor is Dave Futer.


William Worden

William Worden is pursuing dissertation research on the hyperbolic geometry of knot complements and other 3-manifolds. He is interested in canonical triangulations of knot complements, and in particular the link between combinatorial/ geometric properties of the triangulation, and combinatorial properties of the knot. He is also interested in surface homeomorphisms, veering triangulations of punctured surface bundles (as described by Ian Agol), and commensurability of 3-manifolds. His thesis advisor is Dave Futer.


Postdoctoral Program Alumni

Christopher Atkinson, Research Assistant Professor (2009-12)

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. at the University of Illinois-Chicago, as a student of Ian Agol. He is currently an Assistant Professor at the University of Minnesota, Morris.


Justin Malestein, Research Assistant Professor (2009-12)

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. He is currently an Assistant Professor at the University of Oklahoma.


Kei Nakamura, Research Assistant Professor (2009-13)

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. He is currently a Research Associate at Rutgers University.


Brian Rushton, Research Assistant Professor (2012-15)

Brian Rushton studies geometric group theory and low-dimensional topology, focusing on finite subdivision rules for 3-manifolds. If we lived in a small 3-manifold, we would see ourselves reflected across the sky in a fractal-like pattern. Finite subdivision rules give a formula for such a pattern, and they are used to translate combinatorial data into analytic data.

Rushton received his Ph.D. at Brigham Young University, as a student of James Cannon. He is currently an Assistant Professor at Brigham Young University, Hawaii.


Louis Theran, Research Assistant Professor (2009-11)

Louis Theran's 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.

Theran received his Ph.D. at the Univeristy of Massachusetts, Amherst, as a student of Ileana Streinu. He is currently a Lecturer at the University of St Andrews.


Graduate Student Alumni

Michael Dobbins

Michael Dobbins' primary research interests are discrete geometry, convexity, combinatorics, topology, and foundations. His preferred programming language is Haskell.

Dobbins received his Ph.D. in 2011, as a student of Igor Rivin. He is currently an Assistant Professor at Binghamton University.


Christian Millichap

Christian Millichap is pursuing research in the area of low-dimensional topology and hyperbolic geometry. He is interested in the properties of hyperbolic 3-manifolds that have a number of geometric invariants in common, but are non-isometric. In particular, he studies how many hyperbolic 3-manifolds can have the same volume or the same geodesic lengths, the types of constructions used to build such geometrically similar hyperbolic 3-manifolds, and relations between geometrically similar hyperbolic 3-manifolds and commensurability.

Millichap received his Ph.D. in 2015, as a student of Dave Futer. He is currently an Assistant Professor at Linfield College.


Graduate Courses in Geometry and Topology

Math 8061-62: Smooth Manifolds

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.

The textbooks for this course are Introduction to Smooth Manifolds by John Lee and Algebraic Topology by Allen Hatcher.


Math 8161: Point-Set Topology

Topics for this course include topological spaces, metric spaces, separation axioms, Urysohn Lemma and Metrization Theorem, continuity, the Tychonoff Theorem, compactification, the fundamental group, covering spaces.

Textbook: Topology, by James R. Munkres.


Math 9005: Graph Theory

This course will be an introduction to graph theory. Some of the topics covered will be:

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.


Math 9023-24: Knot Theory and Low Dimensional Topology

This course will survey the modern theory of knots, coming at it from several very distinct points of view. We will start at the beginning with projection diagrams and the tabulation problem. We will proceed to several classical polynomial invariants, which can be constructed via the combinatorics of diagrams, via representation theory, or via the topology of the knot complement. We will touch on braid groups and mapping class groups, and use these groups to show that every (closed, orientable) 3-manifold can be constructed via knots. Finally, we will use these constructions to gain a glimpse of several skein-theoretic and quantum invariants of 3-manifolds.

Textbooks for this course include An Introduction to Knot Theory, by W.B. Raymond Lickorish, and Knots, Links, Brads, and 3-Manifolds, by V.V. Prasolov and A.B. Sossinsky.


Math 9061-62: Lie Groups

This will be a course on the basic theory of Lie groups and Lie algebras, with a geometric point of view. We will cover the structure theory and classification, along with the rich geometry associated with these groups via the geometry of symmetric spaces. In particular, we will study the classification of semisimple Lie groups and apply that to classify low-dimensional homogeneous spaces, which are spaces that `look the same' at every point. The course will use some Riemannian geometry, but will also introduce the necessary concepts as we go along.


Math 9210: Mapping Class Groups and Braid Groups

This course will focus on the symmetry groups of manifolds and cell complexes. We will look at both the geometric and algebraic properties of the mapping class group, which can roughly be thought of as the group of symmetries of a surface. One of the ways in which we will study the group is by designing a certain cell complex on which it acts by isometries. This principle has wide generalizations: one can derive a great deal of information about a group by constructing a geometric object for the group to act on.

In the spring semester, we will focus our attention on the braid groups, which are a special class of mapping class groups. In particular, we will study the representation theory of these groups, with connections to invariants of knots and links.