MATH 426, Introduction to modern geometry

Syllabus

• HWA-14, due 14:30, Mon. Apr. 25 (scan and upload it to CANVAS).
  • Skim thru chapters 7--14.
  • Exercises: 9.16, 11.8, 12.2, 13.3, 14.2.

• Midterm in class, Fri. Apr 22, five problems:
  • two problems from HWA 08--13;
  • two theorems or its parts from the following list, possibly with related definitions:
    9.3, 9.13,
    10.1, 10.10, 10.25,
    11.3, 11.4, 11.5,
    12.1, 12.9.
    
  • one new problem.
• You may prepare and use a cheat sheet --- one sided standard paper size, include diagrams and formulas, but no words. Do not make it cryptic.

• HWA-13, due 14:30, Mon. Apr. 18 (scan and upload it to CANVAS).
  • Read: Chapter 13 and 14;
  • Exercises: 10.32, 12.15, 13.1, 14.5, 14.7.
• HWA-12, due 14:30, Mon. Apr. 11 (scan and upload it to CANVAS).
  • Read: Chapter 11 and 12;
  • Exercises: 10.11, 10.26, 11.1, 11.6, 12.4.
• HWA-11, due 14:30, Mon. Apr. 4 (scan and upload it to CANVAS).
  • Read: Chapter 10 and 11;
  • Exercises: 9.8, 9.17, 10.9, 10.19, 10.27.
• HWA-10, due 14:30, Mon. Mar. 28 (scan and upload it to CANVAS).
  • Read: Chapters 9 and 10;
  • Exercises: 9.7, 9.9, 9.15, 9.19, 10.5.
• HWA-09, due 14:30, Mon. Mar. 21 (scan and upload it to CANVAS).
  • Read: Chapters 8 and 9;
  • Exercises: 8.4, 8.8, 9.1, 9.2, 9.4.
• HWA-08, due 14:30, Mon. Mar. 14 (scan and upload it to CANVAS).
  • Read: Chapters 7 and 8;
  • Exercises: 7.3, 7.6, 7.7, 7.8(a), 8.5
• Q-07, due 14:30, Wed, Mar. 4 (scan and upload it to CANVAS).
  • Write a parametrization for an evolvent of unit circle.
  • Let \(\gamma\) be a unit speed plane curve with curvature less than 1 at all points. Imagine that a unit disc rolls along \(\gamma\) without sliding, write an equation for a curve traced by a point on its boundary in terms of \(\gamma\) and its Frenet frame \(T,N\).
• HWA-07, due 14:30, Mon. Feb. 28 Wed, Mar 2 (scan and upload it to CANVAS).
  • Exercises: 2.23, 3.27(a,c), 5.4, 6.12(a), and the following:
    Let \(\gamma\) be a plane curve with without vertices and \(\omega\) be its evolute. Find the Frenet frame and signed curvature of \(\omega\) in terms of Frenet frame of and signed curvature of \(\gamma\).

• Midterm in class, Fri. Feb 25 Mon. Feb 28, five problems:
  • two problems from HWA 01--06;
  • two theorems or its parts from the following list, possibly with related definitions:
    2.10,2.20, 3.11, 3.17, 3.20, 3.23, 3.25, 5.2, 5.5(smooth case), 5.11, 6.10, 6.14(an intermediate statement in its proof), 6.17.
  • one new problem.
• You may prepare and use a cheat sheet --- one sided standard paper size, include diagrams and formulas, but no words. Do not make it cryptic.

• HWA-06, due 14:30, Mon, Feb 21 (scan and upload it to CANVAS).
  • Read and understand: Chapter 6;
  • Skim thru chapters 1--6; prepare questions.
  • Exercises: 6.8, 6.11, 6.15, 6.18, and the following:
    Assume that a smooth regular simple closed plane curve \(\gamma\) is contained in a square with side 2 and surrounds a square with diagonal 2. Show that \(\gamma\) contains a point with curvature 1.
• HWA-05, due 14:30, Mon, Feb 14 (scan and upload it to CANVAS).
  • Read and understand: Chapter 5;
  • Read: Chapter 6;
  • Prepare one question;
  • Exercises: 4.7, 5.6, 5.7, 5.15, 6.2.
• HWA-04, due 14:30, Mon, Feb 7 (scan and upload it to CANVAS).
  • Read and understand: Chapters 4--5;
  • Prepare one question;
  • Exercises: 3.29, 4.1, 4.3, 4.6, 5.1.
• HWA-03, due 14:30, Mon, Jan 31 (scan and upload it to CANVAS).
  • Read and understand: Chapters 3--4;
  • Prepare one question;
  • Exercises: 3.8, 3.15, 3.18, 3.24, 3.28.
• HWA-02, due 14:30, Mon, Jan 24 (scan and upload it to CANVAS).
  • Read: Chapters 2--3;
  • Prepare one question;
  • Exercises: 2.9, 2.13, 2.22, 3.2, 3.4.
• HWA-01, due 14:30, Fri, Jan 14 (scan and upload it to CANVAS).
  • Read: Chapters 1--2;
  • Prepare one question;
  • Exercises: 1.5, 1.6, 1.7, 2.4, 2.5.

• Extra credit
  • Some extra credit is given for finding misprint/mistake in the book (the score depend on the type of mistake).
  • Advanced exercises: 1.3, 2.6, 3.7, 3.13, 4.12, 5.10, 6.4, 7.8(d), 8.11, 9.21, 9.22, 10.23, 10.24, 10.34, and the following:
    (A) Construct a smooth regular closed space curve without parallel tangent lines.
    (B) Assume a smooth regular closed space curve has positive torsion. Show that its tangent indicatrix has a self-intersection.
    (C) Show that a smooth regular spherical curve with nonvanishing torsion has no self-intersections.