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### Fall 2011 Tutorial

### Category Theory

**Description:**Category theory has found ubiquitous uses in modern mathematics. Most of the subjects which have been profoundly influenced are however rather technical and typically lie outside the standard undergraduate curriculum. On the other hand, insights derived from a categorical point of view could be very useful when learning mathematics at any level. The first part of this course will introduce the subject by building on many familiar examples from basic algebra and topology. We will explain the meaning of categories, functors, natural transformations, limits, colimits, Yoneda's lemma, representability, and adjunctions. In order to make our exposition more meaningful and enjoyable, each definition will be followed by an explicit example in a category the students are acquainted with. Once the basis of the subject has been thoroughly covered, we will venture into a more advanced topic. The aim in this second part will be to bring together all ideas developed earlier in one or more lengthier and more realistic examples. Potential topics include, but are not limited to: the correspondence between (1+1)-dimensional topological quantum field theories and Frobenius algebras, the concept of classifying/moduli spaces in topology and algebraic geometry, abelian categories and a little homological algebra, higher categories. Students background and interests will be the main parameters taken into consideration when choosing topics.

**Prerequisites:**Math 122 will be essential.

**Contact:**: Atanas Atanasov (nasko@math.harvard.edu)

### Spring 2012 Tutorials

### Groupoids in Topology

**Description:**For most mathematicians the study of symmetry means basically the study of groups, but there are plenty of objects whose full symmetry is not quite captured by groups. Consider, for example, a finite rectangular portion of a grid: it's group of symmetries is the same as that for the outer rectangle, ignoring the inner grid lines. With groupoids one can capture the local sameness of the grid around different points, even in a finite portion, showing the relation to the symmetry group of the infinite grid. Groupoids have many applications in mathematics, far, far more than we can discuss in one tutorial; so we will initially focus on their use in algebraic topology. Each space has a fundamental groupoid, which contains all the fundamental groups for all the different base points and their relations. We will discuss the fundamental groupoid versions of at least covering space theory and Van Kampen's theorem. In both cases the groupoid variant is not really harder to develop than the group one but gives much nicer and more widely applicable theorems. Compared to the groupoid van Kampen, the classical theorem for groups is a weakling that can't even be used to compute the fundamental group of the circle! After that depending on time and people's interest we can discuss some of the many other uses of groupoids in topology, combinatorics, algebraic geometry, and differential geometry.

**Prerequisites:**Algebra at the level of Math 122. Some familiarity with the fundamental group (e.g. Math 131) is desirable but not required.

**Contact:**Omar Antolin Camarena (oantolin@math.harvard.edu)

### The Figure Eight Knot : Introduction to Hyperbolic Geometry

In 1872 Felix Klein published his celebrated "Erlanger Program". His vision was to study geometry in a unified way by examining groups of geometric transformations. For example, the concept of congruence of two triangles on the plane could be defined by asserting the existence of an isometry of the plane, a transformation that preserves distances, that carries one triangle into the other. It is no surprise that the group of such rigid motions of the plane is naturally associated with Euclidean 2-space. Felix Klein also introduced the term Hyperbolic Geometry to describe one of the non- Euclidean geometries first introduced by Gauss, Bolyai and Lobachevskii. About 30 years ago, William Thurston revolutionized the field of 3-dimensional topology with his Geometrization Theorem and the unexpected appearance of hyperbolic geometry in purely topological problems. A space is called hyperbolic if it admits a metric of constant negative curvature. Today hyperbolic geometry forms an important chapter of 3-dimensional topology and geometry that also has deep and influential connections with such diverse fields of mathematics as complex analysis, dynamical systems, algebraic geometry and group theory. Complements of knots in Euclidean 3-space provide us with lots of examples of 3-manifolds, most of which can be studied using tools from hyperbolic geometry. In this tutorial we will study in detail some of these examples, such as the figure eight knot complement. Discrete groups of hyperbolic isometries play the same role as the isometries of the plane in Euclidean geometry and are called Kleinian groups. The intrinsic beauty, both mathematical and visual, of these objects will be our leading theme for this tutorial.The tutorial will be an introduction to hyperbolic geometry in dimensions 2 and 3 with a concrete goal to understand the geometry of the figure eight knot complement. We will start by looking at the geometry of the hyperbolic plane, and we will introduce several models to study it - the disk and upper-half plane models and the hyperboloid and projective models, each one has its own advantages. We will classify hyperbolic isometries and study discrete groups formed by them along with their limit sets. We will see how these objects relate to hyperbolic surfaces (also Riemann surfaces or complex algebraic curves) and how to use them to understand their geometry. One of the gems here is Poincare's polygon theorem. Having developed enough intuition on hyperbolic geometry in dimension two we will go one dimension up and follow the same strategy. We will emphasize examples along the way, in particular we will give a careful description of the figure eight knot complement and related examples. We will finish with the statement of Thurston's Geometrization theorem for knot complements. Extra topics we could discuss include : (Ford) Fundamental domains, arithmetic Kleinian groups, Margulis' lemma and the geometry of cusps, Algebraic vs Geometric limits of Kleinian groups, Fenchel- Nielsen coordinates, Quasi-Fuchsian groups and Geodesic Laminations.

**Prerequisites:**You should know basic point set topology, as well as about groups and group actions as discussed in Math 122. It would be extremely useful if you know about fundamental groups and covering spaces, but I would be willing to help you review or go through this material if you are not familiar with it. You do NOT need to know manifold topology, (co)homology or differential geometry.

**Texts:**"Low-Dimensional Geometry" by Francis Bonahon : Large part of the tutorial will follow this book. It's very accessible and I highly recommend it. "Three-Dimensional Geometry and Topology, volume 1" by William P. Thurston : Anything by Thurston is great, I suggest you browse through this book even if you don't follow the details. More books to be added later.

**Contact:**Stergios Antonakoudis (stergios@math.harvard.edu)

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