USC Fall 2022

- Primary: Kyler Siegel (kyler.siegel@usc.edu)
- Teaching assistant: Gin Park (ginpark@usc.edu)

We will typically meet on Mondays, Wednesdays, and Fridays, 2:00-2:50pm, in KAP 166 (see the schedule below for academic holidays and other modifications). In cases of sickness, it will also be possible to view the lectures online - see the $“$USC Zoom Pro Meetings$”$ tab on Blackboard for the link. We may occasionally meet only on Zoom.

This course is an introduction to low dimensional geometry, through the lens of Felix Klein's $“$Erlangen Program$”$. We will investigate geometric properties of surfaces in detail, with a special focus on the fundamental trichotomy between Euclidean, elliptic, and hyperbolic geometries. Along the way we will introduce key notions such as geodesics, curvature, and isometries, and we will also naturally encounter various important notions in topology and group theory. Towards the end we will peak into the vast world of three-dimensional geometry, and we will try to approach the question $“$What is the shape of our universe?$”$.

If for some reason you aren't currently registered for this course on Blackboard please email Kyler to make sure you receive class emails.

We will assume proficiency in multivariable calculus. Knowledge of linear algebra and complex numbers is also recommended. Some familiarity with topology and group theory will also be very helpful, although we will not assume you have seen these.

Some of the topics we may cover over the course of the semester (time permitting) include:

- Euclidean geometry and the parallel postulate (and the failure thereof)
- elliptic geometry
- hyperbolic geometry
- basics of complex analysis
- isometry groups and Möbius transformations
- geodesics and curvature
- global geometry of surfaces and the Gauss-Bonnet theorem
- gluing constructions and fundamental domains
- tessellations
- three-manifolds and cosmic topology

- [H18]
*Geometry, with an introduction to cosmic topology*- Hitchman

Note that the digital version is freely available here. If you're like me and you prefer have a paper copy which you can hold in your hands, the print version is also quite inexpensive (at least as far as textbooks go).

- [B09]
*Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots*- Bonahon - [RS05]
*Geometry and topology*- Reid and Szendroi - [S98]
*Geometry: plane and fancy*- Singer

In addition to the lectures, there will be weekly problem sets. These will be listed on this website (see the schedule below) and handed in via Gradescope. Late homeworks will not be accepted. However, we will drop the lowest problem set score.

Much of the course material will be developed in the problem sets. It is very important to do all of the problem sets to the best of your ability and to challenge yourself to solve the problems on your own, as this is the most effective way to absorb the material. We expect students to devote a significant amount of time to the problem sets.

While working on the problem sets, you are allowed to consult or collaborate with your peers, as well as textbooks and the internet (apart from cheating websites such as Chegg or Cramster). However, you must write down attributions for any peer, textbook, website etc from which you took any significant ideas. Moreover, you must attempt all problems on your own and your submitted solutions must be written out originally and individually. Submissions which are copied or suspiciously similar are subject to being rejected and potential disciplinary action.

There will be one midterm (most likely during regular class time) and one final exam (scheduled by the registrar). All exams are graded on a curve, so please don't stress too much about the raw score.

Problem sets: 35%, midterm exam: 25%, final exam: 40%. We will drop the lowest problem set grade.

- Kyler: Wednesday 10-11am and 3:15-4:15pm in KAP 400C
- Gin: Tuesday 11am-1pm and Friday 10am-11am in the math center (KAP 265)

$\#$ | Date | Material | References | Problem set |
---|---|---|---|---|

1 | Monday 8/22/22 | Course introduction. Euclid's postulates and example theorems. History of the parallel postulate. Glimpse at hyperbolic and elliptic geometries. | [H18] §1.1,§1.2, [RS05] §9.1 | |

2 | Wednesday 8/24/22 | Recapitaluation of last time. More foreshadowing of the trichomoty: Euclidean, elliptic, hyperbolic. Gauss-Bonnet theorem and curvature of surfaces. Basics of complex numbers. | [H18] §1.3,§2.1 | |

3 | Friday 8/26/22 | More basics of complex numbers: modulus and argument, Euler's formula, polar form, multiplication and division. | [H18] §2.1,§2.2,§2.3,§2.4 | |

4 | Monday 8/29/22 | Equations of lines and circles via complex numbers. Basic naive set theory: sets, maps, injections, surjections, bijections. Rotations about arbitrary points in the point. | [H18] §2.4,§3.1 | |

5 | Wednesday 8/31/22 | Reflections about lines in the plane. Metric spaces and isometries. The Euclidean metric and Euclidean isometries. | [H18] §3.1,§3.2 | Pset 2 (due Weds 9/7/22 by 11:59pm) |

6 | Friday 9/2/22 | More on the Euclidean metric. The Euclidean inner product and the Euclidean norm. Euclidean geodesics and the Euclidean metric as the length of the shortest path between two points. The group of isometries of a metric space. Examples of Euclidean isometries. | ||

Monday 9/5/22: Labor day - no class | ||||

7 | Wednesday 9/7/22 | More on the triangle inequality for the Euclidean metric. Proof sketch that Euclidean geodesics are straight lines. Linear and affine transformations. Colinearity is preserved by Euclidean isometries. | [RS05] §1.6,§1.7 | Pset 3 (due Weds 9/14/22 by 11:59pm) |

8 | Friday 9/9/22 | More discussion of linear and affine linear transformations. Proof that Euclidean isometries are affine transformations. | [RS05] §1.7,§1.8,§1.9 | |

9 | Monday 9/12/22 | Orthogonal transformations and the orthogonal group. Normal form for orthogonal matrices. Classification of Euclidean isometries. Specialization to low dimensions. | [RS05] §1.10,§1.11,§1.12,§1.13,§1.14,§1.15 | |

10 | Wednesday 9/14/22 | Further discussion of the normal formal theorem. The normal forms in the case $n=2$. Linear (in)dependence and bases. Eigenvectors and eigenvalues. | [RS05] §1.11 | Pset 4 (due Fri 9/23/22 by 11:59pm) |

11 | Friday 9/16/22 | Orientation-preserving and orientation-reversing isometries. The group $SO(3)$ of three-dimensional rotations. Euler's theorem on compositions of rotations. Inversions about circles. The extended complex plane. | [RS05] §1.15, [H18] §3.2, §3.3 | |

12 | Monday 9/19/22 | More on inversions about circles. Inversions send clines to clines. General complex equation of a cline. Inversions preserve angle magnitudes. | [H18] §3.2, §3.3 | |

13 | Wednesday 9/21/22 | Review of inversions about clines. Inversions preserve angle magnitudes. Möbius tranformations. Möbius transformations are invertible. Möbius transformations as compositions of an even number of inversions. | [H18] §3.4 | |

14 | Friday 9/23/22 | Proof that Möbius transformations are compositions of an even number of inversions. Stereographic projection. | [H18] §3.4 | Pset 5 (due Fri 9/30/22 by 11:59pm) |

15 | Monday 9/26/22 | More on Möbius transformations. Checking that they form a group under composition. Identifying the group of all Möbius transformations with $PSL(2,\mathbb{R})$. Fixed points and the fundamental theorem of Möbius transformations. | [H18] §3.4 | |

16 | Wednesday 9/28/22 | Groups of transformations. Symmetric points about a cline. Criterion for two perpendicular clines. The cross ratio and its invariance under Möbius transformations. | [H18] §3.2, §3.4 | |

17 | Friday 9/30/22 | More on perpendicular clines. Inversion preserve symmetry points. Definition of a "geometry". The Poincaré disk model for hyperbolic space. The group of hyperbolic transformations. Inverting about clines perpendicular to the circle at infinity. | [H18] §3.2, §3.4, §4.1, §4.2, §5.1 | Pset 6 (due Fri 10/7/22 by 11:59pm) |

18 | Monday 10/3/22 | Review of the Poincaré disk model. Proof that inversions preserve angle magnitudes via secants limiting to tangent lines. | [H18] §3.2,§5.1 | |

19 | Wednesday 10/5/22 | Proof that every hyperbolic transformation is a composition of two hyperbolic reflections. | [H18] §5.1,§5.2 | |

20 | Friday 10/7/22 | Transformations taking a point in $\mathbb{D}$ to $0$ and a point in $S^1_{\infty}$ to $1$. Hyperbolic lines and triangles. Any two hyperbolic lines are congruent. Parallel hyperbolic lines and failture of the parallel postulate. Desirata for the hyperbolic distance function. | [H18] §5.2,§5.3 | Pset 7 (due Mon 10/17/22 by 11:59pm) |

21 | Monday 10/10/22 | Desirate for the hyperbolic distance function continued. Deriving the hyperbolic distance function. | [H18] §5.3 | |

22 | Wednesday 10/12/22 | Deriving the hyperbolic distance function continued. Interpretation in terms of the cross ratio and explicit formula. Deriving the hyperbolic length of a curve in $\mathbb{D}$. | [H18] §5.3 | |

Friday 10/14/22: Fall recess - no class | ||||

23 | Monday 10/17/22 | midterm review session | ||

24 | Wednesday 10/19/22 | midterm exam during class | midterm 1 sample solutions | |

25 | Friday 10/21/22 | Computing hyperbolic area. Hyperbolic triangles up to congruence. Areas of hyperbolic triangles (assuming the formula in the $\tfrac{2}{3}$-ideal case. | [H18] §5.4 | |

26 | Monday 10/24/22 | Computing the area of a hyperbolic circle. There are no "hyperbolic rectangles". The hyperbolic Pythagorean theorem. Hyperbolic right-angled hexagons. Trichotomy for hyperbolic transformations: elliptic, parabolic, hyperbolic. | [H18] §5.4 | Pset 8 (due Mon 10/31/22 by 11:59pm) |

27 | Wednesday 10/26/22 | More on the elliptic / parabolic / hyperbolic trichotomy. Characterization in terms of two hyperbolic reflections. | ||

28 | Friday 10/28/22 | The upper half plane model $\mathbb{U}$ and its transformation group. The Möbius transformations relating the Poincaré disk model with the upper half plane model. Lines, distances, and areas in the upper half plane model. Proof of the area formula for a $\tfrac{2}{3}$-ideal hyperbolic triangle. | [H18] §5.5 | |

29 | Monday 10/31/22 | Recap on the upper half plane model and its equivalence with the Poincaré disk model. Understanding transformations of hyperbolic transformations of the upper half plane model in terms of Möbius transformations with real coefficients. The trichotomy in terms of the trace of the corresponding matrix. Group theoretic discussion of $PSL(2,\mathbb{R})$. | [H18] §5.5 | Pset 9 (due Weds 11/9/22 by 11:59pm) |

30 | Wednesday 11/2/22 | Spherical and elliptic geometry. Antipodal points and elliptic transformations. Great circles in $S^2$ and $\mathbb{C}_+$. | [H18] §6.1, §6.2 | |

31 | Friday 11/4/22 | Spherical and elliptic geometry. Models for projective space. | [H18] §6.1, §6.2 | |

32 | Monday 11/7/22 | More on spherical and elliptic geometry. Recalling the group $SO(3)$ of rotations of the two-sphere. Unique elliptic line joining two points. Any two elliptic lines intersect. | [H18] §6.1, §6.2 | |

33 | Wednesday 11/9/22 | Recap of elliptic geometry. Normal form for an elliptic transformation. Measurements in elliptic geometry: lengths, areas, and distances. Area of a lune. | [H18] §6.3 | Pset 10 (due Weds 11/16/22 by 11:59pm) |

Friday 11/11/22: Veterans day - no class | ||||

34 | Monday 11/14/22 | Area of elliptic triangles. Equivalence relations. Real projective space $\mathbb{RP}^n$. Complex projective space $\mathbb{CP}^n$. | ||

35 | Wednesday 11/16/22 | More on real and complex projective spaces. Action of $GL(2,\mathbb{C})$ on $\mathbb{C}^2$ descends to Möbius transformations on $\mathbb{CP}^1$. Introduction to topology and definition of homeomorphism. Surfaces and examples. | ||

36 | Friday 11/18/22 | More on topology and surfaces. Surfaces with boundary. Examples and non-examples. The Möbius strip and orientable versus nonorientable surfaces. The Klein bottle. Connect sums. The orientable surface $\Sigma_g$ of genus $g$ and the nonorientable (a.k.a. crosshap) surface $C_g$ of genus $g$. The classification of surfaces theorem. | [H18] §7.5 | |

37 | Monday 11/21/22 | Recap on the classification of surfaces. Cell complexes and the Euler characteristic. Identifying surfaces by their Euler characteristic and orientability. | [H18] §7.5 | §7.5 |

Wednesday 11/23/22: Thanksgiving break - no class | Pset 11 (due Fri 12/2/22 by 11:59pm) | |||

Friday 11/25/22: Thanksgiving break - no class | ||||

38 | Monday 11/28/22 | Constructing surfaces by gluing edges of polygons. Pictures for positive, zero, and negative curvature. | §7.1,§7.5 | |

39 | Wednesday 11/30/22 | Definition of curvature. Computing curvature in Euclidean, spherical, and hyperbolic geometry. The Poincaré disk model with radius $R$. The Gauss-Bonnet formula | §7.5,§7.6 | |

40 | Friday 12/2/22 |
Existence of geometries with constant curvature for all surfaces. Verification of Gauss-Bonnet. | §7.6 | |

Friday 12/9/22: 2-4pm | final exam (check schedule) |

Image sources: here.