# Learning seminar in geometry and topology

__Speaker:__ Jennifer Vaughan

__Date:__ Dec 1, 2017

__Title:__ Introduction to Momentum Maps

__Abstract:__ I will review symplectic manifolds, Lie group actions, and Hamiltonian vector fields, culminating in the definition of a momentum map and an explanation of its properties. Time permitting, I will discuss the Marsden-Weinstein symplectic reduction theorem.

__Speaker:__ Marc Ethier

__Date:__ Nov 24th, 2017

__Title:__ An introduction to persistent homology with applications to shape analysis

__Abstract:__ Persistent homology was developed starting in the 1990s to study the duration of topological properties along a filtration of spaces. It proved very useful in applications to shape recognition and analysis, by allowing one to recognize noise from geometrically relevant data. In this talk, I will describe the theory of persistent homology and some of its extensions such as multidimensional persistence, and show some interesting applications of the theory to applied problems.

**Notes**
**Monodromy Visualization (java download)**

__Speaker:__ Adam Clay

__Date:__ Nov 17, 2017

__Title:__ Free subgroups of groups of homeomorphisms

__Abstract:__ The goal of this talk is to contrast two fundamental results concerning the dynamics of group actions on the unit interval, namely the abundance of free subgroups in Homeo_+([0,1]) and the absence of nonabelian free subgroups in PL_+([0,1]). The proofs of both abundance and absence of free subgroups are quite short, so if time permits I will provide an outline of both.

**Notes**

__Speaker:__ Adriana Ciupeanu

__Date:__ Nov 3, 2017

__Title:__ Introduction to Hyperbolic Metric Spaces

__Abstract:__ In this talk, firstly, we will introduce quasi-isometry and give examples of quasi-isometric metric spaces. Secondly, we look at geodesic metric spaces, Gromov product, slim and thin triangles, with the goal of defining hyperbolic metric spaces.

**Notes**

__Speaker:__ Derek Krepski

__Date:__ Oct 20, 2017

__Title:__ The holonomy correspondence continued

__Abstract:__ After reviewing/recalling some elementary definitions, we introduce the moduli space of surface group representations and the moduli space of flat connections. Then we will see that these are actually the same, thanks to the holonomy correspondence.

**Notes**

__Speaker:__ Derek Krepski

__Date:__ Oct 13, 2017

__Title:__ The holonomy correspondence

__Abstract:__ After reviewing/recalling some elementary definitions, we introduce the moduli space of surface group representations and the moduli space of flat connections. Then we will see that these are actually the same, thanks to the holonomy correspondence.

**Notes**

__Speaker:__ Adam Clay

__Date:__ Sept 29, 2017

__Title:__ The ping-pong lemma

__Abstract:__ The ping-pong lemma is a fundamental tool in the study of group actions. It allows one to identify certain groups as free products and to embed free products into other more interesting groups. I will discuss the lemma in both its basic and more general forms, and will provide plenty of examples and applications.

**Notes**

__Speaker:__ Eric Schippers

__Date:__ Sept 22, 2017

__Title:__ Teichmuller space of compact surfaces

__Abstract:__ Teichmuller space is a moduli space of equivalence classes of Riemann surfaces. In the first half of the talk I will give an overview of the problem of complex analytic equivalence of Riemann surfaces, and its relation to the Teichmuller space in the special case of compact surfaces. The second half of the talk will illustrate the first half with the specific example of genus one tori.

**Notes**

__Speaker:__ Azer Akhmedov

__Date:__ February 19, 2016

__Title:__ On bi-orderability of knots.

__Abstract:__ It is well known that every knot is left-orderable (i.e. the fundamental group of the complement of the knot in S^3 is left-orderable). On the other hand, not every knot is bi-orderable, e.g. non-trivial torus knots are never bi-orderable. A beautiful result by B.Perron and D.Rolfsen (2003) claims that if a knot is fibered and all roots of the Alexander polynomial are real and positive then the knot is bi-orderable. This result applies, for example, to the Figure-8 knot showing its bi-orderability; the Alexander polynomial of it equals t^2-3t+1. In recent years, A.Clay and D.Rolfsen have obtained new results which allows to decide the bi-orderability for most of the knots with a small crossing number.
We develop tools which is effective in treating a broad class of knots. We are able to show that the knots 6_2 and 7_6 are not bi-orderable. These are the only two knots up to crossing number 7 whose bi-orderability was not known. This is a joint work with Cody Martin.

**Notes 1**

**Notes 2**

__Speaker:__ Colin Desmarais

__Date:__ November 26, 2015

__Title:__ An algorithm for determining non bi-orderability of knots.

__Abstract:__ Determining when two knots are equivalent (one can be transformed to the other via an ambient isotopy) is central to knot theory. One method for determining equivalence is by studying knot invariants: properties of a knot invariant under ambient isotopy. The Alexander polynomial is an example of a knot invariant. Another knot invariant is bi-orderability: A knot K is called bi-orderable if its knot group is bi-orderable, meaning it can be given an ordering that is invariant under multiplication by the left and the right. The knot is called non bi-orderable otherwise.

The bi-orderability of many knots remain unknown. I will discuss results that determine bi-orderability for certain knots. These results rely heavily on studying the roots of the Alexander polynomial.

Next, I will demonstrate an algorithm for determining if a group is non bi-orderable. I will discuss an implementation of this algorithm using properties specific to knot groups for improvements on running-time. Using this computer program, it was shown that if K is an alternating knot with fewer than ten crossings whose Alexander polynomial has no real positive roots, then K is non bi-orderable.

**Notes**

__Speaker:__ Adam Clay

__Date:__ November 12, 2015

__Title:__ Introduction to the space of left-orderings II

__Abstract:__ From any left-orderable group G one can create a compact topological space called “the space of left-orderings,” denoted by LO(G). Algebraic properties of the underlying group are reflected in the topology of this space, however even groups which are well-understood can produce spaces of left-orderings that are poorly understood. In this talk, I will explain the construction LO(G), investigate its structure in some simple cases and explain how it has served as a great tool in proving some open conjectures.

**Notes**

__Speaker:__ Adam Clay

__Date:__ November 5, 2015

__Title:__ Introduction to the space of left-orderings I

__Abstract:__ From any left-orderable group G one can create a compact topological space called “the space of left-orderings,” denoted by LO(G). Algebraic properties of the underlying group are reflected in the topology of this space, however even groups which are well-understood can produce spaces of left-orderings that are poorly understood. In this talk, I will explain the construction LO(G), investigate its structure in some simple cases and explain how it has served as a great tool in proving some open conjectures.

**Notes**

__Speaker:__ Juliana Theodoro de Lima

__Date:__ October 29, 2015

__Title:__An introduction to braid groups on surfaces II

__Abstract:__ This talk will provide an introduction to the braid groups on surfaces from a geometrical perspective, and will deal with the case of braids over non-orientable surfaces.

**Notes**

__Speaker:__ Derek Krepski

__Date:__ October 22, 2015

__Title:__ Universal vector bundles II

__Abstract:__ This talk is a brief introduction to vector bundles. After reviewing some basic definitions/examples, we’ll introduce some natural vector bundles over Grassmannian manifolds and discuss how these vector bundles are "universal", meaning that (almost) all vector bundles can be obtain from members of this family.

**Notes**

__Speaker:__ Eric Schippers

__Date:__ October 15, 2015

__Title:__ Hyperbolic Geometry in Complex Analysis

__Abstract:__ Hyperbolic geometry developed together with complex analysis, through work of Fuchs, Poincare, Schwarz and others. In spite of this, the simple and elegant connections between the two fields are not very well known except to specialists. We give an overview of hyperbolic geometry in the unit disk. We then give examples of theorems in complex analysis in which hyperbolic geometry (or more generally, negatively curved metrics) plays a central role.

**Notes**

__Speaker:__ Derek Krepski

__Date:__ October 8, 2015

__Title:__ Universal vector bundles I

__Abstract:__ This talk is a brief introduction to vector bundles. After reviewing some basic definitions/examples, we’ll introduce some natural vector bundles over Grassmannian manifolds and discuss how these vector bundles are "universal", meaning that (almost) all vector bundles can be obtained from members of this family.

**Notes**

__Speaker:__ Serhii Dovhyi

__Date:__ October 1, 2015

__Title:__ Dehn surgery

__Abstract:__ A knot is an embedding of the circle into S^3. The complement of a knot is then a compact 3-manifold. There is a method of constructing a new manifold from the knot complement called Dehn surgery. In this talk will be describe such method.

**Notes**

__Speaker:__ Juliana Theodoro de Lima

__Date:__ Sept 24, 2015

__Title:__ An introduction to braid groups on surfaces

__Abstract:__ This talk will provide an introduction to the braid groups on surfaces from a geometrical perspective. We will see a presentation for such group and by this way, we will see the differences between the presentation of the surface braid groups on surfaces and the presentation of braid group given by Artin, in 1925.

**Notes**

__Speaker:__ Adam Clay

__Date:__ Sept 17, 2015

__Title:__ Introduction to group growth

__Abstract:__ Given a set of generators for a group, it’s natural to ask how many elements of the group can be written as a product of k or fewer generators, for some positive integer k. As k varies, so will the answer to this question, and this observation leads naturally to the idea of a "growth function" of a group: it is a function that describes the size of the set of group elements that can be written as a product of length k or less.

In this talk, I will give a rigorous definition of the growth function of a group, and compute the growth functions of some of the groups that we encounter in daily life. I will also discuss equivalence of growth functions, so that our results don’t depend on a choice of generating set.

**Notes**

__Speaker:__ Eric Schippers

__Date:__ Nov 27, 2014

__Title:__ Fuchsian groups and the uniformization theorem

__Abstract:__ The uniformization theorem is a deep generalization of the
Riemann mapping theorem. It says that the universal cover of any
Riemann surface is conformally equivalent to the disk, plane, or
sphere. Aside from a handful of exceptions, the covering is
the disk.

Fuchsian groups are the groups of complex analytic deck
transformations of this covering. They also happen to be
groups preserving hyperbolic distance, and thus can be seen
to give tilings of the disk with respect to this non-Euclidean
geometry.

In this talk I will give a friendly non-technical introduction
to these concepts.

**Notes**

__Speaker:__ Adam Clay

__Date:__ Nov 21, 2014

__Title:__ Dimension and the fundamental group of a manifold

__Abstract:__ In this talk I will discuss some properties of the fundamental group that are influenced by the dimension of the underlying manifold. For example, we know exactly what groups will arise as the fundamental group of a 1 or 2 dimensional manifold, because the manifolds in these dimensions are completely classified. On the other hand, if we’re given a k-manifold where k>2, what special properties of its fundamental group can we expect for particular values of k? This talk is based on notes of Stefan Friedl.

**Notes**

__Speaker:__ Eric Harper (McMaster)

__Date:__ Oct 30, 2014

__Title:__ Alexander invariants of knot groups

__Abstract:__ In this talk we will introduce Alexander invariants of knots by way of their knot group. By using a Wirtinger presentation of a knot group we can
draw many conclusions about their Alexander invariants, including the vanishing of the zeroth elementary ideal, principality of the first elementary ideal, and normalization of the Alexander polynomial. Time-permitting we will discuss the twisted versions of the above constructions.

**Notes**

__Speaker:__ R. Padmanabhan

__Date:__ Oct 16, 2014

__Title:__ Geometry using Group Theory

__Abstract:__ While visualization is the source of all geometry, group theory is a part of abstract algebra. However, these two topics interact beautifully as expounded by Felix Klein in his famous Erlanger Program recalled by Eric in his talk last week. We plan to use the familiar calculus of reflections as a backdrop to demonstrate the expressive power of group theory to systematically build geometric concepts by just using the algebra of groups. This will show that group theory and plane geometry are but two sides of the same concept. The axioms we use were introduced by Frederich Backhmann to describe the Hjelmslev planes.

This talk was originally given way back in 2006 in the Rings & Modules Seminar.

References

[1] Bachmann, F. Aufbau der Geometrie aus dem Spiegelungswbegriff, Springer Verlag, Berlin, 1983.

[2] Bachmann, F et al. Absolute Geometry, Foundations of Mathematics, Volume II, MIT Press, Cambridge, 1983.

[2] Szczerba, L. W., "Interpretations of elementary theories" pages 129-145 in Logic and Foundations of Mathematics, Reidel, Boston 1977

[3] Tarski, A., "What is elementary geometry?", Studies in Logic and the Foundations of Mathematics, p.16-29, North- Holland, 1959

**Notes**

**Handout**

__Speaker:__ Derek Krepski

__Date:__ Oct 9, 2014

__Title:__ The coadjoint representation of a Lie group

__Abstract:__ Following Dr. Schippers’ talk last week on Lie groups and Lie algebras, this talk will introduce the coadjoint representation of a Lie group through some concrete examples. If time permits, we'll also introduce Poisson brackets on manifolds to provide further perspective on our examples.

**Notes**

__Speaker:__ Eric Schippers

__Date:__ Oct 2, 2014

__Title:__ An introduction to Lie Groups

__Abstract:__ I will give a overview of some of the ideas of the
theory of Lie groups. In order to avoid technicalities I
will illustrate these ideas with matrix groups. The focus will
be on elementary properties: definitions, Lie algebras, Lie
brackets and the exponential map. The talk can be understood
by an advanced undergraduate.

**Notes**

__Speaker:__ Adam Clay

__Date:__ Sept 25, 2014

__Title:__ Introduction to Dehn Surgery

__Abstract:__ Using the Poincaré conjecture as motivation, I’ll introduce a construction of Max Dehn, now known as Dehn surgery, as a way of constructing counterexamples to Poincaré ’s first conjecture (but not his second conjecture, which is the famous one). Dehn’s construction was later (~1960's) shown to be an extremely versatile way of constructing 3-manifolds, and so has become a fundamental tool in the study of low-dimensional topology.

**Notes**

__Speaker:__ Derek Krepski

__Date:__ March 17, 2014

__Title:__ The Hopf Invariant

__Abstract:__ This talk will introduce the Hopf invariant, a classical topic in algebraic topology/homotopy theory, and discuss its relation to the non-existence of bilinear multiplications on $R^n$ (without zero divisors) for $n$ other than 1, 2, 4 and 8 (where of course such multiplications exist, in the real numbers, complex numbers, quaternions and octonians, respectively).

**Notes**

__Speaker:__ Adam Clay

__Date:__ March 10, 2014

__Title:__ An introduction to classical knot theory II

__Abstract:__ This talk will give a quick summary of the classification of surfaces, and introduce Seifert’s algorithm for constructing spanning surfaces from knot diagrams (aka Seifert surfaces). I’ll also define knot genus, and show how to compute the genus of a large class of knots using Seifert’s algorithm.

**Notes**

__Speaker:__ Adam Clay

__Date:__ March 3, 2014

__Title:__ An introduction to classical knot theory

__Abstract:__ This talk will begin by introducing some of the basic constructions of knot theory, such as knot equivalence, knot diagrams, and Reidemeister moves. Following that, I’ll define what is meant by a knot invariant, and explain one of the earliest attempts at creating knot invariants (namely, colourability of a knot diagram) and its later generalization to an abstract algebraic object called a ‘quandle’.

**Notes**

__Speaker:__ Eric Schippers

__Date:__ February 24, 2014

__Title:__ An introduction to Riemann surfaces

__Abstract:__ This talk will be a gentle introduction to Riemann surfaces, aimed
at the level of graduate students. I will give definitions, motivation,
and examples. If time allows I will discuss the idea of a Teichmuller
space using the example of the moduli space of tori.

**Notes**

__Speaker:__ Derek Krepski

__Date:__ February 10, 2014

__Title:__ The Schur-Horn theorem and symplectic geometry

__Abstract:__ The goal of this talk is to introduce symplectic geometry (a branch of differential geometry) by casting some results about Hermitian matrices with prescribed eigenvalues (originally due to Schur in the 1920's and Horn in the 1950's) as a corollary of a big theorem in symplectic geometry.

**Notes**

__Speaker:__ Adam Clay

__Date:__ February 3, 2014

__Title:__ An introduction to the braid groups

__Abstract:__ This talk will provide an introduction to the braid groups from several different perspectives. I will show how they can be defined by generators and relations, as isotopy classes of arcs or surface homeomorphisms, or as fundamental groups. Each perspective allows for different geometric or algebraic intuition, which often allows for straightforward insight into complex proofs. For each approach introduced I will highlight some of the results that intuitively (though perhaps not easily) follow from the definitions.

**Notes**