Math 341: Topics in Geometry, Spring 2020
MWF 1:10-2pm, Chem 301 (and via zoom starting March 15)
Office Hours: Scheduled here or email for additional times
Text: Hyperbolic geometry by Birger Iversen (link requires Reed login)
Final presentation assignment
- The assignment.
- Friday, 3 April: Email team preferences.
- Monday, 13 April: Email topic proposal.
- Reading period, 4-8 May: Practice presentations.
- 13 May, 8am-12pm: Presentations.
Week 1: January 27 - January 31
- M: Welcome and warmup.
- W: Syllabus. What is hyperbolic geometry?
- F: Quadratic forms and their isometries.
Week 2: February 3 - February 7
- M: Witt's theorem.
- W: Real, positive definite, and parabolic forms.
- F: The pseudosphere. HW1 due. (Solution hints.)
Week 3: February 10 - February 14
- M: The Lorentz group.
- W: Euclidean geometry.
- F: Spherical geometry. HW2 due. (Solution hints.)
Week 4: February 17 - February 21
- M: More spheres.
- W: Hyperbolic space.
- F: The Klein disk. HW3 due. (Solution hints.)
Week 5: February 24 - February 28
- M: Möbius transformations.
- W: More Möbius transformations.
- F: The Poincaré disk and half-space. HW4 due. (Solution hints.)
Week 6: March 2 - March 6
- M: More Poincarism. Exam 1 distributed.
- W: The Riemann sphere.
- F: More Riemann sphere. Exam 1 due. (Solution hints.)
Week 7: March 9 - March 13
- M: The action of $\mathrm{SL}_2(\mathbb{R})$. Animations.
- W: Vector calculus on $\mathfrak{sl}_2(\mathbb{R})$.
- F: Class cancelled. Watch this for a hyperbolic geometry fix.
Week 8: March 16 - March 20
Class now meeting via zoom.
Instructions for setting up zoom on your computer, tablet, or phone are available via the "Getting Started" links here. (Do this before our first meeting! 🤓)
Homework submission is now via gradescope. You should have received an email saying you were added to our gradescope page. Please contact me if this is not the case or you have trouble with submissions.
- M: Pencils of geodesics. HW5 due (DEADLINE EXTENDED TO MARCH 30). Geodesics sage worksheet. (Click the green "Open in CoCalc" button to render the images.) Video lecture.
- W:
Classification of isometries.Class cancelled. - F: Class Cancelled. See this page (especially the "Communications to the Community" sidebar) for the latest info on Reed's response to the novel coronavirus pandemic. I will provide information on Math 341's remote meetings and homework submission soon.
Spring Break
- Fill out this survey by Wednesday, March 25.
Week 9: March 30 - April 3
See March 26 email for updated course procedures!
- M: Classification of isometries. HW5 due (via gradescope).
- W: The special linear group.
- F: Trigonometry. Email presentation team preferences by end-of-day.
Week 10: April 6 - April 10
- M: Angle of parallelism, right-angled pentagons, and right-angled hexagons.
- W: Hyperbolic area.
- F: Discrete subgroups. HW6 due.
Week 11: April 13 - April 17
- M: Cusps and horocyclic compactification.
- W: The modular group and its fundamental domain.
- F: Locally finite and convex fundamental domains. HW7 due.
Week 12: April 20 - April 24
- M: Dirichlet domains.
- W: Compact polygons.
- F: Poincaré's theorem. HW8 due.
Week 13: April 27 - May 1
- M: Hyperbolic geometry in special relativity with Joel Franklin: notes and chalkboards. Final exam distributed.
- W: Triangle groups. (No video; read the notes.)
- F: The Klein quartic. (No video; read the linked website and we will have a wide-ranging discussion in class.) Final exam due via gradescope.
Final Presentations: May 13, 9:30am-12pm via Zoom
Additional resources
- Hyperbolic geometry by Cannon, Floyd, Kenyon, and Parry.
- DIY hyperbolic geometry by Mann.
- Too many triangles by Numberphile / Segerman.
- SL(View) by Dumas.
- Directions for making hyperbolic models by Henderson.
- Make hyperbolic tilings of images by Christersson.
- Euclid: the game.
- NonEuclid: the nongame by Castellanos.
- Hyperbolic hypernom by Hart, Hawksley, Segerman, Stay, and Matsumoto.
- Hyperbolic VR with Ray Tracing by Nelson, Segerman, and Woodard.
- Isometry classes of hyperbolic 3-space by Nelson.
- Decomposing $\mathrm{SL}_2(\mathbb{R})$ by K. Conrad.
- Right-angled hexagons in GeoGebra by Chas.
- Fundamental domain drawer by Verrill. (Java applet and C program - Can you make them work?)
- From an octagon to a genus 2 surface by Leys.
- The modular group in action by Gargava.
- Interactive Dedekind tesselation by Kocik.
- The Klein quartic by Baez.
The $\LaTeX$ document preparation system
Poor handwriting? Love escape characters? Too much free time? Try $\LaTeX$!
- $\LaTeX$ at Reed.
- A short guide [pdf] to writing mathematics with $\LaTeX$.
- Change .pdf to .tex in the URL of (nearly) any file accessed from this page to get the $\LaTeX$ source code.
Kyle M. Ormsby