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Austin Christian

Georgia Tech Mathematics

Math 4441: Differential Geometry (Fall 2022)

Instructor: Austin Christian
Office hours:
   Sun 5pm-6pm Zoom
   Mon 10:30am-11:45am Clough 248
   Thur 8pm-9pm Zoom
Meeting time: MWF 9:30am-10:20am
Meeting location: Skiles 368
Syllabus    Discord
a gif showing surfaces of negative, 0, and positive Gaussian curvature

Announcements

Schedule

The following is a tentative, nonbinding schedule. It will be updated as the term progresses.

Topics covered Notes Materials
Week 1
(8/22-8/26)
M: Course goals and first definitions
W: Tangent vectors and arc length as geometric invariants
F: Activity 1 (linear algebra)
Aug 22
Aug 24
Homework 1 (due 9/9)
homework-1.tex
Activity 1 (due 8/31)
activity-1.tex
Week 2
(8/29-9/2)
M: Arc length parametrizations and Frenet-Serret apparatus
W: Frenet-Serret apparatus day 2
F: Activity 2 (Frenet-Serret)
Aug 29
Aug 31
Activity 2 (due 9/7)
activity-2.tex
frenet-serret.nb
Week 3
(9/5-9/9)
M: Labor Day
W: Line integrals and 2D Frenet-Serret
F: Activity 3 (evolutes)
Sept 7 Homework 2 (due 9/23)
homework-2.tex
Activity 3 (due 9/14)
activity-3.tex
evolutes.nb
Week 4
(9/12-9/16)
M: Rotation index day 1
W: Rotation index day 2
F: Activity 4 (convexity)
Sept 12
Sept 14
Activity 4 (due 9/21)
activity-4.tex
Week 5
(9/19-9/23)
M: Isoperimetric inequality
W: Isoperimetric wrap-up/multivariable calculus review
F: Surfaces definitions/Activity 5 (calculus on surfaces)
Sept 19
Sept 21
Activity 5 (due 10/3)
activity-5.tex
surfaces.nb
Week 6
(9/26-9/30)
M: More surfaces/activity 5
W: Tangent spaces/midterm 1 review
F: Midterm 1 (through isoperimetric inequality)
Sept 26
Sept 28
Practice midterm 1
Practice midterm solutions
Week 7
(10/3-10/7)
M: First fundamental form day 1
W: First fundamental form day 2
F: Activity 6 (map projections)
Oct 3
Oct 5
Homework 3 (due 10/14)
homework-3.tex
Activity 6 (due 10/12)
activity-6.tex
map-projections.nb
Week 8
(10/10-10/14)
M: First fundamental form day 3
W: Normal and geodesic curvatures day 1
F: Activity 7 (Darboux frame)
Oct 10
Oct 12
Activity 7 (due 10/19)
activity-7.tex
Week 9
(10/17-10/21)
M: Fall break
W: Normal and geodesic curvatures day 2
F: Activity 8 (Foucault pendulum)
Oct 19 Homework 4 (due 10/31)
homework-4.tex
Activity 8 (due 10/26)
activity-8.tex
foucault.nb
Week 10
(10/24-10/28)
M: Geodesic curvature is intrinsic
W: Existence and uniqueness of geodesics
F: Activity 9 (Clairaut's relation)
Oct 24
Oct 26
Activity 9 (due 11/2)
activity-9.tex
Week 11
(10/31-11/4)
M: Length minimizing curves day 1
W: Length minimizing curves day 2
F: Activity 10 (directional derivatives)
Oct 31
Nov 2
Homework 5 (due 11/14)
homework-5.tex
Activity 10 (due 11/9)
activity-10.tex
Week 12
(11/7-11/11)
M: How does a surface curve?
W: Properties of the Weingarten map, or writing things down
F: Activity 11 (Weingarten map)
Nov 7
Nov 9
Activity 11 (due 11/16)
activity-11.tex
weingarten.nb
Week 13
(11/14-11/18)
M: The second fundamental form
W: Gaussian curvature
F: Midterm 2 review (hopefully)
Nov 14
Nov 16
Practice midterm 2
Practice midterm solutions
Week 14
(11/21-11/25)
M: Midterm 2
W: Thanksgiving break
F: Thanksgiving break
Week 15
(11/28-12/2)
M: Gaussian curvature as a stretch factor
W: Theorema Egregium
F: Activity 12 (A rotation index for surfaces)
Nov 28
Nov 30
Homework 6 (due 12/12)
homework-6.tex
Activity 12 (due 12/7)
activity-12.tex
Week 16
(12/5-12/9)
M: The Gauss-Bonnet theorem (final class meeting)
F: Term paper due
Dec 5

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