Let \(d\) be a path metric on \(X\). A geodesic in the metric space \((X,d)\) is a curve \(\gamma\) in \(X\) such that for every \(P\in\gamma\) there exists \(\epsilon>0\) with the property that if \(Q_0,Q_1\in\gamma\) satisfy \(d(P,Q_0)<\epsilon\) and \(d(P,Q_1)<\epsilon\), then the section of \(\gamma\cap B_d(P,\epsilon)\) joining \(Q_0\) to \(Q_1\) has length \(d(Q_0,Q_1)\).
The following is a tentative schedule, to be updated as the term progresses. We're starting with a fairly ambitious plan, and may need to modulate according to student needs.
While topics are tentative, due dates are not.
Last update:
| Date | Topic(s) | Reading | Notes | Comments |
|---|---|---|---|---|
| Jan 8 (M) | Euclidean geometry | Ch 1 | Jan 8 | |
| Jan 10 (W) | Some isometries of the hyperbolic plane | Sec 2.1-2.2 | Jan 10 | |
| Jan 15 (M) | Martin Luther King, Jr. Day | |||
| Jan 17 (W) | Geodesics | Sec 2.3 | Jan 17 | |
| Jan 22 (M) | All isometries of \((\mathbb{H}^2,d_{\mathrm{hyp}})\) and (anti)linear fractional maps | Sec 2.4-2.5.1 | Jan 22 | HW 1 due (thru 1/17) |
| Jan 24 (W) | More on (A)LFMs and the hyperbolic norm | Sec 2.5.2-2.6 | Jan 24 | |
| Jan 29 (M) | Spherical geometry | Ch 3 | Jan 29 | |
| Jan 31 (W) | Quotient semi-metrics | Sec 4.1-4.2 | Jan 31 | |
| Feb 5 (M) | Homework digest | HW 2 due (thru 1/31) | ||
| Feb 7 (W) | Polygon gluing | Sec 4.3 & 4.5 | Feb 7 | |
| Feb 12 (M) | Some Euclidean surfaces | Sec 5.1 | Feb 12 | |
| Feb 14 (W) | The surface of genus 2 and the projective plane | Sec 5.2-5.3 | Feb 14 | |
| Feb 19 (M) | The cylinder and the Möbius strip | Sec 5.4 | Feb 19 | HW 3 due (thru 2/14) |
| Feb 21 (W) | The once-punctured torus | Sec 5.5 | Feb 21 | |
| Feb 26 (M) | Triangular pillowcases | Sec 5.6 | Feb 26 | |
| Feb 28 (W) | Tessellations | Sec 6.1-6.2 | Feb 28 | |
| Mar 4 (M) | Midterm | HW 4 (thru 2/28) | ||
| Mar 6 (W) | The tessellation theorem | Sec 6.3 | Mar 6 | |
| Mar 11 (M) | Completeness and compactness | Sec 6.4.1 | Mar 11 | |
| Mar 13 (W) | Compactness, bounded polygons | Sec 6.4.2-6.5.1 | Mar 13 | |
| Mar 18 (M) | Spring break | |||
| Mar 20 (W) | Spring break | |||
| Mar 25 (M) | Tessellations by polygons | Sec 6.5.2-6.6 | Mar 25 | HW 5 due (thru 3/13) |
| Mar 27 (W) | Incomplete hyperbolic surfaces | Sec 6.7 | Mar 27 | |
| Apr 1 (M) | Group actions and quotient spaces | Sec 7.1-7.2 | Apr 1 | |
| Apr 3 (W) | Fundamental domains and Dirichlet domains | Sec 7.3-7.4 | Apr 3 | |
| Apr 8 (M) | Three-dimensional hyperbolic space | Ch 9 | Apr 8 | HW 6 due (thru 4/3) |
| Apr 10 (W) | Knots | Sec 12.1 | Apr 10 | |
| Apr 15 (M) | Geometrization for knot complements | Sec 12.2 | Apr 15 | |
| Apr 17 (W) | Mostow's rigidity theorem | Sec 12.3 | Apr 17 | |
| Apr 22 (M) | Geometrization for 3-manifolds | Sec 12.5 | Apr 22 | |
| Apr 24 (W) | Art exhibition (12pm-1pm) | Skiles 202 | ||
| Apr 26 (F) | Term paper due | HW 7 due (thru 4/22) |
