18.900 | Spring 2023 | Undergraduate

Geometry and Topology in the Plane

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Two bees winding their ways among 2 flowers surrounded by red polygons.

Chapter I: Bees winding their ways among flowers (winding numbers). (Image courtesy of Talia Blum. Used with permission.)

My experience is that students initially feel more comfortable with straight lines than curved ones. Therefore, in this chapter and the following two, we talk about polygons and polygonal loops. The starting point is the area of a polygon, a subject which is already interesting from a delightfully elementary kindergarten cut-and-paste perspective. From that, we reach (in a slightly sneaky way) the main notion of the chapter, winding numbers. The last lecture takes a peek at the topology of loops going beyond the winding number (more experienced readers will realize that it discusses conjugacy classes in the free group, without using the words “group” or “conjugacy class”).

Among the many available expositions for the material in Lecture 1, I have mostly used I. Zakharevich, Scissors congruence (lecture notes from the 2013 Chicago REU, transcribed by E. Elmanto and H. Chan). A source of intriguing examples is G. Frederickson, Dissections: plain and fancy, Cambridge Univ. Press, 1997. The proof of Pick’s theorem in Lecture 2 follows C. Blatter, Another proof of Pick’s area theorem, Math. Magazine 70(3), p. 200, 1997.

Lecture 1: Cutting and Pasting Polygons

Lecture 2: Integer Polygons

Lecture 3: The Shoelace Formula and the Winding Number

Lecture 4: The Winding Number (continued)

Lecture 5: Loops Avoiding Two Points

A polygon is in front of a mirror with lots of polygons inside.

Chapter II: A polygon admiring itself in the mirror (tiling by mirrored copies). (Image courtesy of Talia Blum. Used with permission.)

The simple idea of a ball (of size zero) bouncing off the walls of a polygon has given rise to an astonishingly rich mathematical theory, combining geometry and ergodic theory. Here, we stay near the surface of the theory, but that still allows us to explore several important notions. One of these is tiling the plane by reflected copies of a polygon (polygonal tilings are a staple of geometry teaching, but in this course we won’t consider them as a subject in its own right). The second notion is phase space and its area. Switching to that viewpoint can be a bit of a struggle, and our exposition is quite compressed; but the theoretical insight is well worth it (as a symplectic geometer, it’s obvious I would say that). The last lecture considers curved walls, which is the first time we leave polygons behind. I’m not sure I was successful in writing that lecture, but it introduces another important idea (finding trajectories as extrema), and also gives you a reason for remembering what you might have learned in school about plane conics.

I have been guided by S. Tabachnikov, Geometry and billiards, Amer. Math. Soc. 2005, and the much more advanced H. Masur and S. Tabachnikov, Rational billiards and flat structures, in Handbook of dynamical systems, vol. I, Elsevier, 2002, p. 1015–1089. The discussion of the Fagnano trajectory follows H. Coxeter and S. Greitzer, Geometry revisited, MAA Publications, 1967.

Lecture 6: Introduction to Billiards

Lecture 7: Phase Space

Lecture 8: Billiards in Curved Domains

Is this chapter about analysis? Well, yes and no. The topic is resonance modes, which are the eigenfrequencies (eigenvalues, up to a square root) of the Laplace operator for polygonal or other regions. Usually, one goes into a certain amount of analysis to show that this notion makes good theoretical sense, but here we think of the subject as spectral geometry, which means focusing on the interaction between the resonance modes and the geometry of the region. In fact, we really only look at the lowest resonance mode. There is a substantial math (and physics) relation between this and the previous chapter, but I can’t bring that out here. Instead, we only observe one parallel, which is that tiling-by-reflection comes back as an idea. Next, the extremal characterization of the lowest eigenvalue is something that every student of mathematics should know, and also particularly useful in this situation. Finally, we look at Steiner symmetrization, which is exactly the kind of topic this course is aiming for: entirely elementary, requiring a bit of creativity to use, and surprising in its appearance here. I am ashamed to say I had never heard of it prior to preparing this chapter!

A general reference is Chapters 5 and 7 of G. Polya and G. Szegö, Isoperimetric inequalities in mathematical physics, Princeton Univ. Press, 1951. I’ve also benefited from the more recent survey R. Laugesen and B. Siudeja, Triangles and other special domains, in: Shape optimization and spectral theory, De Gruyter Open, 2017, p. 149–200. I read about the Ritz (or Rayleigh-Ritz) method in J. Levandoski, Lecture notes for Math 220B (Applied PDE), Stanford, Summer 2003.

Lecture 9: First Computations

Lecture 10: An extremal characterization

Lecture 11: Symmetrization

A man walking a pig around a lamppost.

Chapter IV: A man walking a pig around a lamppost (dog-man-lamppost theorem). (Image courtesy of Talia Blum. Used with permission.)

Winding numbers for differentiable loops are standard content in both topology and complex analysis classes, but their application to real (Lecture 13) and complex (Lecture 14) equations seems to be the part of this course that students find most difficult. Partly, this is because of the lingering impression that such equations are there in order to be solved explicitly, which is what the topological method presented here is intended to avoid. The other stumbling block seems to be that the logic of “this term is small and we can neglect it” sits uneasily with some math students, in spite of many efforts to promote it at various levels of teaching. One could try to ease the transition by spreading out the material over more lectures, but I find that this leads to preliminary discussions which are not particularly meaningful or satisfying by themselves. The last lecture is an excursion into three dimensions. At that point, one could insert an entire chapter on the topology of knots and links, but I am simply not the right person to write such a chapter.

If you want to learn more about winding numbers, a good option is J. Roe, Winding around, Amer. Math. Soc., 2015.

Lecture 12: Smooth Loops

Lecture 13: Equations in Two Variables

Lecture 14: Complex Polynomials

Lecture 15: The Linking Number

Six ants walking across a tree stump.

Chapter IX: Ants shrinking as they walk across a tree stump (geometries with varying scales). (Image courtesy of Talia Blum. Used with permission.)

Building on the example of the hyperbolic metric, we now consider metrics that are stretched by an arbitrary function (in technical terms, conformally flat metrics). The discussion focuses on the geodesic equation. Originally this is presented without any motivation, and you may feel that you landed in a nightmare version of a differential equations class. It takes quite a bit of computation to even see that, for hyperbolic geometry, the equation reproduces our previous notion of geodesic (experienced readers may notice that this computation uses Noether’s theorem, meaning the conserved quantities associated to the symmetries of the hyperbolic metric). Still, I hope that intuition eventually accretes around the formulae. The (standard) proof of the general Gauss-Bonnet theorem involves going back to the rotation numbers from much earlier in the class. Concerning the very last lecture, which tries to tie up things by looking back at combinatorial surfaces and polygonal billiards, I have to confess that so far I’ve never reached this material in class, so it’s more a dream of a lecture.

Lecture 37: The Geodesic Equation

Lecture 38: Behavior(s) of Geodesics

Lecture 39: Curvature

Lecture 40: Geometry of Combinatorial Surfaces

A fish jumping out of the fish tank and two fish are still in the tank.

Chapter V: Fishes with round and spiky tails (immersed and non-immersed loops). (Image courtesy of Talia Blum. Used with permission.)

I sometimes get puzzled questions about the content of this chapter. The rotation number seems fine, and Whitney’s formula allows us to relate it to the shape of the immersed curve, fulfilling an obvious need. But Arnold invariants are hard to learn about from either Google or GPT, so they must be pretty pointless? Setting aside the question of one’s faith in the internet, the landscape of immersed curves is quite complicated, and Arnold invariants give one a kind of rough coordinate system. They are also a first introduction to a method which is widespread in knot theory, namely the construction of invariants that are primarily characterized by how they change under various geometric transitions. And finally, they are fun and a little tricky to work with, which is what we care about most!

Besides V. Arnold, Topological invariants of plane curves and caustics, Amer. Math. Soc., 1991, I have used O. Viro, Generic immersions of circle to surfaces and complex topology of real algebraic curves. In Topology of real algebraic varieties and related topics, Amer. Math. Soc., 1996, p. 231-252, and A. Shumakovich, Explicit formulas for strangeness of plane curves, St. Petersburg Math. J. 7 (1996), p. 445-472.

Lecture 16: Immersed Loops and the Rotation Number

Lecture 17: Arnold Invariants

Lecture 18: Arnold Invariants (continued)

An eagle flying between the stalagtites.

Chapter VI: Stalagmites joining with stalagtites (perturbing nodes). (Image courtesy of Talia Blum. Used with permission.)

This long chapter is a visual introduction to algebraic geometry, done entirely over the real numbers rather than the complex numbers, and postponing projective space to the end. Since we assume neither knowledge of the implicit function theorem nor any algebra background, the discussion is necessarily informal. Some of the terminology is nonstandard, such as my use of “oval,” and many statements are far from their optimal versions. Lectures 20, 24-25, and 27 are still in a bit of a raw state; please be aware of that.

The relation between mechanical linkages and algebraic curves is a classical topic, still represented in some modern textbooks (such as C. Gibson, Elementary geometry of algebraic curves, Cambridge Univ. Press, 1998; which I have also used as a source of examples of algebraic curves). Lectures 21–25 are influenced by Viro’s survey articles, in particular O. Viro, Introduction to topology of real algebraic varieties (a chapter of an unpublished book project) and O. Viro, Dequantization of real algebraic geometry on logarithmic paper. European Congress of Mathematics, Vol. I, 135-146, Birkhauser, 2001. My source for configurations of lines was the wonderful classical book of Hilbert-Cohn-Vossen.

Lecture 19: Introduction to Algebraic Curves

Lecture 20: Mechanical Linkages and Polynomial Equations

Lecture 21: Intersections of Algebraic Curves

Lecture 22: Nonsingular Curves

Lecture 23: Singular Points

Lecture 24: Patchworking

Lecture 25: Tropical Geometry

Lecture 26: Projective Geometry

Lecture 27: Algebraic Curves in the Projective Plane

Five pigs in five triangular pastures.

Chapter VII: Triangular pastures (triangulations). (Image courtesy of Talia Blum. Used with permission.)

This chapter is the most algebro-topological part of the course. However, we don’t talk about topological spaces at all, instead keeping to the combinatorial tradition of simplicial complexes (in two dimensions). Arguably, this misses the point entirely, but one can hope that students’ intuition will in some way make up for that. The notion of Betti numbers is easily accessible in the combinatorial context, requiring only some linear algebra, the main drawback being that examples would be more fun with a computer than when done by hand.

Midway through the chapter, we start reaching back to some of the earlier material, not so much in order to use previous theorems, as to take advantage of the visual intuition we’ve built. In that vein, we encounter combinatorial models of the projective plane; and we also introduce loops on abstract complexes, using the previous idea of winding numbers to launch a more general notion. (Experienced readers will notice that this is an ultra-compressed discussion of first homology/cohomology and their pairing.) That final part (Lectures 32-33) is more of a draft, and still needs to mature.

In preparing Lecture 28, I have used S.-W. Cheng, T. Dey, J. Shewchuk, Delaunay mesh generation, CRC Press, 2013, as a reference for Delaunay triangulations. I apologize to all computational geometers for the amateurish notion of “shape complex,” which was vaguely inspired by Edelsbrunner’s alpha complex; see H. Edelsbrunner, A short course in computational geometry and topology, Springer, 2014. Similarly, Lecture 30 is a caricature of G. Carlsson, T. Ishkanov, V. de Silva, A. Zomorodian, On the local behavior of spaces of natural images, Intern. J. of Computer Vision 76 (2008), 1-12. The proof of the evenness of Euler characteristic of an orientable surface was contributed by the sages of the internet: https://mathoverflow.net/questions/310335/closed-orientable-surfaces-have-even-Euler-characteristic.

It’s the kind of “how does one come up with that” argument that I usually avoid, but this one was too pretty to skip.

Lecture 28: Delaunay Triangulations

Lecture 29: Betti Numbers

Lecture 30: Betti Numbers (continued)

Lecture 31: Surfaces

Lecture 32: Combinatorial Loops

Lecture 33: Combinatorial Winding Numbers and the Boundary Operators

Two frogs on two plants.

Chapter VIII: Frogs and their tongues (hyperbolic geodesics). (Image courtesy of Talia Blum. Used with permission.)

My vague impression is that most undergraduate geometry teaching is centered either on Euclidean/spherical/hyperbolic geometries, or on curved surfaces. As you’ve realized by now, this course is not of either kind, but we have time for a bit of each subject.

There are various ways of drawing the hyperbolic plane in the ordinary Euclidean one; obviously none of them works perfectly. Here we stick with the half-plane model, which is what you are most likely to see where hyperbolic geometry intersects with other parts of mathematics such as number theory. The downside is that we have to omit regular hyperbolic polygons, since those don’t look particularly symmetric in the half-plane picture. I’m not a regular user of hyperbolic geometry, so I breathe a sigh of relief when we reach its symmetries, which fit into a more general scheme of group actions. Students seem to have the opposite reaction: they often find it hard to use such transformations creatively to simplify computations. If their school teachers have spent long hours discussing Euclidean transformations, to get them into the required Felix-Klein-ian mindset, those seeds may have got lost in the thicket. (As I write this, I am ruefully looking out onto my front yard, hence the metaphor.) But really, any blame rests with the class design, as this chapter seems to be lacking “je ne sais quoi” to give it a clearer intellectual direction.

I have found C. Series (with S. Maloni and K. Farooq), Hyperbolic geometry (Lecture notes for the University of Warwick course MA 448), 2010, a helpful reference, especially for the various formulae involved. Of course, there are many other textbooks on hyperbolic geometry, usually aiming much higher than we do here.

Lecture 34: The Hyperbolic Plane

Lecture 35: Arclengths and Areas

Lecture 36: Hyperbolic Isometries

Course Meeting Times

Lectures: 3 sessions / week, 50 minutes / session

Prerequisites

In MIT terms, the prerequisites are 18.03 Differential Equations or 18.06 Linear Algebra. Linear algebra figures briefly in chapter III, and then plays a more substantial role in chapter VII. Differential equations (well, one of them) play an important role in chapter IX. Complex numbers are important in lecture 14 and in chapter VIII. On a more basic level, students are expected to be familiar with 18.02 Multivariable Calculus.

Course Description

This course introduces students to selected aspects of geometry and topology, using concepts that can be visualized easily. We mix geometric topics (such as hyperbolic geometry or billiards) and more topological ones (such as loops in the plane). The course is suitable for students with no prior exposure to differential geometry or topology. Think of it as a moderate hike, overlooking various parts of the geometry and topology landscape. Bits are flat, bits are uphill, there are occasional rocky parts (may be different for everyone), but none that are designed to be cliff faces. The class moves from one topic to the next quickly, so it’s important to learn continuously.

Lectures

Each lecture is designed for 50 minutes. With three sessions per week, there is a little more material here than fits into a semester. The spring 2023 version omitted lectures 20, 24–25, 38, and 40 (earlier versions of the class have omitted Chapter III; generally, the course setup is pretty modular, for instance one could skip lectures 8 or 15 or 17–18 without any adverse effects elsewhere).

Topics

Here’s a brief summary of each chapter:

  • Polygons and their areas, Pick’s theorem, winding numbers
  • Billiards in polygons, phase space, Liouville’s theorem
  • Eigenfrequencies of the Laplace operators on domains, especially the lowest one
  • Differentiable loops, their winding number in the plane, application to solving systems of equations
  • Immersed loops, the rotation number, and Arnold invariants
  • Algebraic curves, the theorems of Bezout and Harnack, nodal singularities, the construction of algebraic curves via patchworking and tropical geometry, projective geometry
  • Triangulations and two-dimensional complexes, Betti numbers, combinatorial surfaces, combinatorial loops
  • Hyperbolic geometry, lengths and areas in the hyperbolic plane, the Gauss-Bonnet theorem
  • Curved geometries, the definition of geodesics and their behavior, curvature and the general Gauss-Bonnet theorem, curvature for combinatorial surfaces

Homework/Exams/Grading

As taught in spring 2023, the class was assessed by a mixture of comprehension questions (30%), more challenging problem sets (30%), and exams (40%). The comprehension questions are part of this OCW site. [Problem sets and exams are not available to OCW users.] They are intended to ensure that there’s been no misunderstanding of the basic concepts and computational methods taught in the class.

Course Info

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Departments
As Taught In
Spring 2023
Learning Resource Types
Lecture Notes
Problem Sets