Learning seminar in geometry and topology
Speaker:From particles to strings: an introduction to 2d CFT, vertex algebras and their noncommutative generalization.
Date: December 2, 2024
Title: 3-manifold group ABCs
Abstract: The traditional method of studying particle physics is to write down the Lagrangian of a system, use the path integral method to deduce the Feynman rules, then use Feynman diagrams to calculate the scattering amplitudes, finally, compare with the calculated scattering amplitudes with the results of generated by the collider. However, the method has severe troubles both in mathematics and in physics. In 1980's, Graeme Segal studied necessary conditions for the path integral to make sense and axiomatized the two-dimensional conformal field theory (2d CFT hereafter) in pure mathematical terms. It did not take long for people to realize that vertex algebras are the starting point for a mathematically rigorous construction of 2d CFTs. I will briefly go over these ideas, then discuss a noncommutative generalization of vertex algebra that is expected to play essential roles in more general quantum field theories.
Speaker:Junyu Lu
Date: November 18, 2024
Title: 3-manifold group ABCs
Abstract: The fundamental groups of surfaces are relatively well understood. Any finitely presented group can be the fundamental group of a 4-manifold, and hence we have a serious obstacle to understand high-dimensional manifolds. This talk is about basic properties of the fundamental groups of 3-manifolds, aka 3-manifolds groups, which sits in-between. With a focus on the deficiency, we are interested in those properties of such 3-manifold groups that be formulated purely group-theoretically, e.g. torsion-freeness, residually finiteness, linearity, etc.
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Speaker:Eric Schippers
Date: November 4, 2024
Title: Spaces of one-forms on Riemann surfaces
Abstract: A Riemann surface is a two-dimensional manifold with enough structure to make sense of complex analysis. The shape of a two-dimensional manifold up to diffeomorphism can be detected by one-forms on the surface through what is called cohomology. On the other hand, to detect when two Riemann surfaces are the same up to complex analytic diffeormophism, we need more information. Namely, we need to see how the cohomology interacts with the space of complex analytic one-forms. I will explain the main ideas and illustrate them with examples.
Speaker:Tommy Cai
Date: October 21, 2024
Title: Generalized torsion in amalgams and 3-manifold groups
Abstract: A group is left-orderable (bi-orderable) if there is a total order invariant under multiplication from the LHS (resp. both sides). A group is generalized torsion free if no product of conjugates of any nontrivial element is trivial. We will recall a sufficient condition ensuring an amalgam of two groups to be generalized torsion free. As applications, we prove two results: (1) there is a 3-manifold whose fundamental group is generalized torsion free and not orderable and (2) there is a group which is generalized torsion free and not left orderable. These results resolve a conjecture of Motegi and Teragaito and Problem 16.48 of the Kourovka Notebook (which is also Question 2.1 in Unsolved problems in ordered and orderable groups, arXiv:0906.2621), respectively. In this talk, we will focus on the proof of the first result. This is a joint work with Adam Clay.
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Speaker:Michael Astwood
Date: October 7, 2024
Title: Geometric Mechanics, Integrability, and the Kepler Problem
Abstract: In this talk we will discuss the Kepler problem, which seeks to determine qualitative descriptions for Hamiltonian systems. The natural setting for a Hamiltonian system is as a dynamical system on a symplectic manifold. The notions of configuration space, phase space and Hamiltonian flow will be introduced, as well as the concept of a Poisson algebra. In order to understand the Kepler dynamical system qualitatively, we will introduce symmetries and infinitesimal symmetries of Hamiltonian systems, and use them along with Noether's theorem to reduce the dimensionality of the Kepler problem. If time permits, the results of the presenters MSc thesis will be discussed, which generalize the Kepler problem to pseudo-Riemannian surfaces of revolution.
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Speaker: Tommy Cai
Date: April 9, 2024
Title: Generalised torsion in amalgams
Abstract: A generalized torsion element in a group is a nontrivial element such that a product of its conjugate is trivial. An amalgam G:=A\ast_CB$ of two groups A and B, amalgamated by their common subgroup $C$ is a group defined as following. Let A, B be the fundamental groups of two path-connected topological spaces X and Y respectively. Assume two connected open subspaces of X and Y respectively are homeomorphic with fundamental group C. Then the amalgam G=A\ast_CB is defined to be the fundamental group of the disjoint union of X and Y with Z_1 and Z_2 identified. For an amalgam G=A\ast_CB of two subgroups amalgamated by a common subgroup C, we give a necessary and sufficient condition for the factor groups A and B to be free of generalized torsion elements of G. Combining with results of Heuer-Chen and Ito-Montegi-Teragaito, we have a sufficient condition for G to be free of generalized torsion elements. We then consider several applications of this result, including giving a Bludov-Glass type result about amalgams with matching bi-orderings and providing many groups which are free of generalized torsion element but not bi-orderable. This is a joint work with Adam Clay.
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Speaker: Derek Krepski
Date: March 26, 2024
Title: Lie groups, Lie algebras, and Lie’s third theorem
Abstract: Lie groups are quite literally the objects that embody symmetry in differential geometry: they are groups that are also smooth manifolds. To every Lie group one naturally gets a Lie algebra by considering the tangent space at the identity element of the group. Conversely, Lie’s third theorem says every finite dimensional Lie algebra is obtained in this way. In this talk, I will sketch an elementary proof of Lie’s third theorem (assuming a hard theorem due to Ado) as an application of Frobenius’ Theorem on integrable distributions.
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Speaker: Dana Kapoostinsky
Date: March 12, 2024
Title: Hyperbolic groups have solvable word problem
Abstract: The aim of this talk will be to show that hyperbolic groups have solvable word problem (an algebraic phenomenon) using geometric techniques. Any group can be associated to a metric space via its Cayley graph. Using a notion of $\delta-$slim triangles as the defining characteristic for hyperbolicity, we will show that certain non-geodesic paths can always be tamed or approximated by neighboring geodesic paths in the space. This geometric property ultimately yields a Dehn group presentation, for which there exists a shortening algorithm that solves the word problem.
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Speaker: Junyu Lu
Date: February 27, 2024
Title: A step towards the character variety and the A-polynomial of a knot
Abstract: For the most part, 3-manifolds are determined by their fundamental groups. The set of representations of the fundamental group potentially encodes deep properties of a 3-manifold. In this talk, we will introduce the character variety—the set of all representations under certain equivalence relation. With a focus on the peripheral subgroup of a knot group, we will also introduce the A-polynomial, which unveils certain aspects of the character variety. Time permitting, we will explore some properties of the character variety and A-polynomials. The talk is accessible to senior undergraduates with a foundation in topology and algebra.
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Speaker: Adam Clay
Date: February 13, 2024
Title: Introduction to Thompson's group F
Abstract: This talk will introduce Thompson's group F in two different ways, as a subgroup of the homeomorphisms of the unit interval and as a group made from certain operations on pairs of binary trees. Using these perspectives, I will indicate how you can start to uncover many of the interesting algebraic properties of F, including (time permitting) an fun trick due to Koberda, Kim and Lodha that shows many copies of F are hidden in every group of homeomorphisms of every manifold.
Speaker: Fei Qi
Date: December 4, 2023
Title: From particles to strings: an introduction to quantum field theory
Abstract: The traditional method of studying particle physics is to write down the Lagrangian of a system, use the path integral method to deduce the Feynman rules, then use Feynman diagrams to calculate the scattering amplitudes, finally, compare with the calculated scattering amplitudes with the results of generated by the collider. However, experiments found that aside from the well-understood subatomic particles in the standard model, there might exist many more beyond. Even if we restrict to the standard model, the Lagrangian is enormously complicated. String theory arises as a possible solution. Particles are complexified from a zero-dimensional point to a one-dimensional string. Different types of particles are unified as different vibration states of a string. The quantum field theory is built upon this interpretation for computing the scattering amplitudes. In this talk, I will discuss Atiyah and Segal's mathematical axiomatization of quantum field theory and the role of vertex algebras and representation theory in the study. I will also report on my recent work on the deformation quantization.
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Speaker: Junyu Lu
Date: November 20, 2023
Title: A Layman’s Introduction to Knots and Jones Polynomials
Abstract: A classical knot is an embedding of a simple closed curve in 3-sphere. Two knots are equivalent if they can be deformed into each other without breaking or self-intersecting themselves. The study of knots was motivated by an attempt to classify 3-manifolds, but has become its own area of mathematical interest. In this talk, we will give a basic introduction to knot theory and meet the Jones polynomials, one of the most important knot invariants. Time permitting, we will discuss some open questions around Jones polynomials. All the materials in this talk are quite accessible, even to senior undergraduates.
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Speaker: Dinamo Djounvouna
Date: November 6, 2023
Title: A Short Introduction to Multisymplectic Geometry
Abstract: Multisymplectic geometry, or n-plectic geometry, extends classical symplectic geometry by using closed, nondegenerate n+1-forms (where n≥1). Historically, it emerged to underpin first-order classical field theories and offers a broader view than symplectic geometry alone.
In this framework, we explore 2-plectic manifolds characterized by closed nondegenerate 3-forms. This geometry connects higher symplectic concepts with string theory, impacting modern physics.
Notably, multisymplectic geometry goes beyond mechanical systems. Semi-simple Lie groups inherently possess 2-plectic structures, while oriented n+1-dimensional manifolds exhibit n-plectic properties. Like Lie algebras in symplectic geometry, n-plectic manifolds yield Lie n-algebras.
Join us in this talk, where we introduce Lie L-infinity algebras as a generalization of conventional Lie algebras.
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Speaker: Derek Krepski
Date: October 23, 2023
Title: A short introduction to symplectic geometry
Abstract: In this talk, we give a short introduction to symplectic geometry. We will introduce and try to motivate the basic definitions in the subject: symplectic manifolds, Hamiltonian vector fields, and the Poisson bracket.
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Speaker: Eric Schippers
Date: October 6, 2023
Title: Complex structures in two real dimensions
Abstract: A complex structure on a two-dimensional surface provides a well-defined notion of angle at every point. It is the minimal structure necessary to make sense of complex analytic maps. There are several ways to define a complex structure on a Riemann surface -- for example, as an atlas of charts into the complex plane, as an almost complex structure (rotation by pi/2), or as an equivalence class of Riemannian metrics. We outline these approaches to complex structure, and explain their equivalence. The equivalence is tied to the existence of isothermal coordinates, a geometric concept originating with Gauss. If time allows we will touch on the Beltrami equation, which underpins the modern theory of moduli spaces of Riemann surfaces
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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.
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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.
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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).
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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.
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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’.
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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.
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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.
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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.
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