Geometry and Topology
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
Dave Futer studies low-dimensional topology and geometry. A central
goal of his research is to relate several distict themes in
their (typically hyperbolic)
geometry, the coarse geometry of fundamental groups, and
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 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
Thurston at Princeton University, and his checkered career
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
Rivin is currently on leave from Temple and is
Regius Professor of Mathematics at the University of St. Andrews.
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
which are Riemannian manifolds
closely related to discrete subgroups of Lie groups,
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.
Dianbin Bao is studying identities between Hecke eigenforms, and their arithmetic
consequences, for congruence subgroups of the modular group.
His thesis advisor is
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
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 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
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
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.
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.
Rushton, Research Assistant Professor (2012-15)
Brian Rushton studies geometric group theory and low-dimensional topology,
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
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
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
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
at the University
of St Andrews.
Graduate Student Alumni
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 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
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.
by James R. Munkres.
Math 9005: Graph Theory
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.
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,
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.