 Siu-Cheong Lau

MS

I have moved to BU Blackboard and this webpage will no longer be updated.

Spring 2021

MA 722: Differential Topology 2 (TH 9:30-10:45)

Intersection theory, Lefschetz fixed point theory, integration on manifolds, vector fields and flows, and Frobenius' theorem. This year, we will introduce symplectic topology.

Fall 2020

MA 225: Multivariable Calculus (TH 12:30-1:45)

To see how calculus applies in practical situations described by more than one variable, we study: Vectors, lines, planes, parameterization of curves and surfaces, partial derivatives, directional derivatives and the gradient, optimization and critical point analysis, including constrained optimization and the Method of Lagrange Multipliers, integration over curves, surfaces and solid regions using Cartesian, polar, cylindrical, and spherical coordinates, divergence and curl of vector fields, and the Greens, Stokess, and Divergence Theorems.

MA 563: Differential Geometry (TH 2:00-3:15 @ CAS 324)

The exterior differential calculus and its applications to curves and surfaces in 3-space and to various notions of curvature.

Spring 2020

MA 822: Topics in Geometry (TH 2-3:15)

This time we focus on quiver representations. This is a very interesting subject which is intersection of many branches of mathematics, including algebraic and symplectic geometry, algebra and representation theory, graph theory, mirror symmetry and Donaldson-Thomas invariants, gauge theory and so on.

Fall 2019

MA 225: Multivariable Calculus (TH 12:30-1:45)

To see how calculus applies in practical situations described by more than one variable, we study: Vectors, lines, planes, parameterization of curves and surfaces, partial derivatives, directional derivatives and the gradient, optimization and critical point analysis, including constrained optimization and the Method of Lagrange Multipliers, integration over curves, surfaces and solid regions using Cartesian, polar, cylindrical, and spherical coordinates, divergence and curl of vector fields, and the Greens, Stokess, and Divergence Theorems.

Assignment

Week 13: (no need to turn in. These topics ARE INCLUDED in final exam)

1. 21.Stokes theorem: Section 16.7 Q1, 5, 7, 13, 17.

2. 22.Divergence theorem: Section 16.8 Q5, 7, 13.

Week 12: (due Dec 4 or 5)

1. 20.Surface integrals: Section 16.6 Q19, 21, 25, 39.
Curl and divergence: Section 16.8 Q1, 2.

Week 11: (due Dec 4 or 5)

Note: In Section 16.4, given a vector field and a curve in the plane, there are two main quantities:

(1) Flux across a curve;

(2) Path integral along a curve (or called the "circulation" when the curve is a loop).

We have focused on (2).

To avoid confusion, (1) "flux across a curve" has been skipped and will not appear in exam.

Indeed, (1) is simply the path integral of the vector field rotated by 90 degrees.

(1) is a 1-dimensional analog of "flux across a surface", which will be introduced next week.

1. 19.Curls and Green's Theorem: Section 16.3 Q1, 5, 7, 27.
Section 16.4 The circulation part of Q5, 9, 15; 21, 27.

Week 10: (due Nov 20 or 21)

Note: Section 16.1 has been skipped and will not appear in exam.
It is about integration of FUNCTION (rather than vector field) along a curve.
This is analogous to integration of FUNCTION over a surface in Section 15.8.
It may cause confusion with integration of VECTOR FIELD in Section 16.2. Thus we skip Section 16.1.

1. 17.Line integral of vector fields: Section 16.1 Q. 1-8 (matching);
Section 16.2 Q1, 4, 7, 11, 19, 38, 39, 47.

2. 18.Fundamental theorem for line integral: Section 16.3 Q13, 25, 29, 34.

Week 9: (due Nov 13 or 14)

1. 15.Substitution and surface area: Section 15.4 Q1, 9, 27, 29;
Section 15.8 Q2, 9, 18;    Question 3,4 in this file (please click).

2. 16.Triple integrals: Section 15.5 Q7, 10, 23, 25, 41;
Section 15.7 Q1, 15, 17, 21, 33.

Week 8: (due Nov 6 or 7)

1. 14.Double integrals: Section 15.1 Q9, 17, 19, 25;
Section 15.2 Q1, 6, 14, 19, 26, 33, 47, 57;
Section 15.3 Q3, 5, 13.

Week 7: (due Oct 30 or 31)

1. 12.Lagrange multipliers: Section 14.8 Q1, 6, 9, 12, 22, 26, 29, 47.

2. 13.Global extrema: Section 14.7 Q31, 40, 42.

Week 6: (due Oct 23 or 24)

1. 11.Critical points: Section 14.7 Q1, 9, 13, 17, 23, 24, 29, 45, 50, 56.

Week 5: (due Oct 16 or 17)

1. 9.Directional derivative: Section 14.5 Q1, 4, 7, 11, 15, 17, 19, 22, 25, 27, 29.

10.Tangent plane and linearlization: Section 14.6 Q1a, 5a, 9, 14, 19, 21, 23, 25, 27, 29.

Week 4: (due Oct 9 or 10)

1. 7.Partial derivatives: Section 14.3 Q1, 7, 12, 19, 21, 29, 42, 45, 52, 65, 75, 83, 90.

8.Chain rule: Section 14.4 Q1, 4, 7, 11, 26, 29, 33, 41, 43, 44.

Optional:An application of gradient to Artificial Intelligence. (This does not count into your score.)

Week 3: Mid-Term Test 1. No Assignments.

Week 2: (due Sep 25 or 26)

1. 5.Parametric surfaces: Q1 to 6 in this file (please click).
Section 14.1 Q14, ALSO parametrize the graph of f;
Q31-36 (counted as one question);
Parametrize the level surfaces {f(x,y,z)=1} in Q53, 59;
Write down the parametrized surfaces in Q77, 80 as level surfaces {f(x,y,z)=0}.

2. 6.Multivariable functions: Section 14.1 Q5, 6.

Week 1: (due Sep 18 or 19) (PLEASE IGNORE the star symbols in the scanned pages.)

1. 3.Lines, planes and quadrics: Section 12.5 Q2, 6, 8, 19, 21, 22, 23, 31, 35, 41, 45, 53, 57;
Section 12.6 Q1 to 12, 13, 17, 21, 25, 27.

2. 4.Curves and arc length: Section 13.1 Q3, 4, 6, 11, 15, 22, 23a,b;
Section 13.3 Q1, 9, 13, 18a,b.

Week 0: (due Sep 11 or 12) (Please buy your own textbook. For the first two weeks, we provide the scanned pages of the questions. PLEASE IGNORE the star symbols in the scanned pages.)

1. 1.Coordinates, distance and vectors: Section 12.1 Q7, 14, 20, 21, 27, 29, 43, 51, 58;
Section 12.2 Q6, 15, 22, 23, 33, 37, 40, 41.

2. 2.Dot product and cross product: Section 12.3 Q1, 5, 18, 25, 26;
Section 12.4 Q3, 15, 20, 23, 27, 28, 30, 31, 39.

3. Optional:An application of dot product to Artificial Intelligence. (This does not count into your score.)

MA 731: Lie groups and Lie algebras (Tue Thu 3:30-5)

Lie groups and introduction to representation theory.
Lie theory appears everywhere: in geometry and topology, in particle physiscs, in engineering and robotics...

Syllabus

HW questions are located in the end of the notes.

Due on Sep 12:    Examples of Lie groups.

Due on Sep 19:    Closed subgroup theorem.

Due on Sep 26:    Adjoint action.

Due on Oct 3:    Matrix exponential.      BCH formula.

Due on Oct 10:    Lie correspondence.      Levi decomposition.

Due on Oct 24:    Representations.

Due on Oct 31:    Schur's Lemma.

Due on Nov 7:    sl(3,C).

Due on Nov 14:    Weyl group for sl(3,C).

Due on Nov 21:    Semi-simple Lie algebra.

Due on Dec 5:    Root system      Rrepresentations of semi-simple Lie algebra.

Spring 2019

MA 722: Differential Topology 2

Morse theory.

Fall 2018

MA 563: Differential Geometry (MWF 11:15-12:05 @ CAS 201)

The exterior differential calculus and its applications to curves and surfaces in 3-space and to various notions of curvature.

Assignment

Due on Sep 14:    Arc length.

Due on Sep 21:    Curvature.

Due on Sep 28:    Spherical curves.

Due on Oct 5:    Curves in general dimensions.

Due on Oct 12:    Regular surfaces.

Due on Oct 19:    Gauss and mean curvatures and Einstein notations.

Due on Nov 2:    Levi-Civita connection.

Due on Nov 9:    Intrinsic geometry.

Due on Nov 16:  Fundamental theorem for surfaces.

Due on Nov 30:  Differential forms: Q. 1,2,6. You are advised to finish all this Friday (no class).

Due on Dec 7:  More on differential forms.

Due on Dec 12 in class:  Gauss-Bonnet Theorem.

MA 225: Multivariable Calculus (MWF 1:25-2:15 @ SCI 113)

To see how calculus applies in practical situations described by more than one variable, we study: Vectors, lines, planes, parameterization of curves and surfaces, partial derivatives, directional derivatives and the gradient, optimization and critical point analysis, including constrained optimization and the Method of Lagrange Multipliers, integration over curves, surfaces and solid regions using Cartesian, polar, cylindrical, and spherical coordinates, divergence and curl of vector fields, and the Greens, Stokess, and Divergence Theorems.

Assignment

Week 0: (due Sep 13 or 14) (PLEASE IGNORE the star symbols in the scanned pages)

1. 1.Coordinates and distance: Section 12.1 Q7, 14, 16, 20, 21, 27, 29, 43, 51, 58.

2. 2.Vectors and dot product: Section 12.2 Q6, 15, 22, 23, 33, 37, 40, 41;
Section 12.3 Q1, 5, 18, 25, 26.

Week 1: (due Sep 20 or 21) (PLEASE IGNORE the star symbols in the scanned pages)

1. 3.Cross product: Section 12.4 Q3, 6, 13, 15, 20, 23, 27, 28, 30, 31, 39, 41.

2. 4.Lines and planes: Section 12.5 Q2, 6, 8, 19, 21, 22, 23, 32, 35, 41, 45, 53, 57, 61.

3. 5.Level surfaces and quadrics: Section 12.6 Q1 to 12, 13, 17, 21, 25, 27.

Week 2: (due Sep 27 or 28)

1. 6.Curves: Section 13.1 Q3, 4, 6, 11, 14, 15, 22, 23.

2. 7.Arc length: Section 13.3 Q1, 9, 11, 13, 18.

3. 8.Curvature and torsion: Section 13.4 Q3, 4, 7, 10; Section 13.5 Q8, 10.

Week 3: (due Oct 4 or 5)

1. 9.Spherical coordinates: Q1 to 6 in this file (please click).

2. 10.Parametric surfaces: Section 14.1 Q14, ALSO parametrize the graph of f;
Q31-36 (counted as one question);
Parametrize the level surfaces {f(x,y,z)=1} in Q53, 59;
Write down the parametrized surfaces in Q77, 80 as level surfaces {f(x,y,z)=0}.

3. 11.Functions and continuity: Section 14.2 Q9, 13, 21, 32, 41, 61, 68.

Week 4: No assignment after Mid-Term 1.

Week 5: (due Oct 18 or 19)

1. 12.Partial derivatives: Section 14.3 Q1, 7, 12, 19, 21, 29, 42, 45, 52, 65, 75, 83, 90.

2. 13.Chain rule: Section 14.4 Q1, 4, 7, 11, 26, 29, 33, 43, 44.

3. 14.Directional derivative: Section 14.5 Q1, 4, 7, 11, 15, 17, 19, 22, 25, 27, 29.

Week 6: (due Oct 25 or 26)

1. 15. Gradient and tangent plane: Section 14.6 Q1a, 5a, 9, 14, 19, 21, 23, 25, 27, 29.

16. Critical points: Section 14.7 Q1, 9, 13, 17, 23, 24, 29, 45, 50, 52, 56.

Week 7: (due Nov 1 or 2)

18. Lagrange multipliers: Section 14.8 Q1, 6, 9, 12, 22, 26, 29, 47.

19. Global extrema: Section 14.7 Q31, 40, 42.

1. 20.Taylor’s formula: Section 14.9 Q3.  This exercise is optional.  Taylor’s formula will not appear in exams.

Week 8: (due Nov 8 or 9)

21. Double integrals: Section 15.1 Q9, 17, 19, 25, 36;
Section 15.2 Q1, 6, 14, 19, 26, 33, 47, 57;
Section 15.3 Q3, 5.

Week 9: (due Nov 15 or 16)

1. 22.Polar integration: Section 15.4 Q1, 9, 27, 29.

2. 23.Surface area: Question 1,2,3 in this file (please click).

3. 24.Triple integrals: Section 15.5 Q7, 10, 23, 25, 41;
Section 15.7 Q1, 15, 17, 21, 33.

Week 10: (due Nov 29 or 30)

1. 25.Substitution: Section 15.8 Q2, 9, 18.

2. 26.Vector fields: Section 16.2 Q1, 4, 39.

3. 27.Line integrals: Section 16.2 Q7, 11, 19, 38, 47.

4. 28.Fundamental theorem for line integral: Section 16.3 Q13, 25, 29, 34.

Week 11: (due Dec 6 or 7)

1. 29.Conservative fields and curl: Section 16.3 Q1, 5, 7, 25, 27.

2. 30.Greens theorem: Section 16.4 Q5, 9, 15 (just need the circulation for 5,9,15), 21, 27.

Week 12: (due Dec 12 in class)

1. 31.Curl and divergence: Section 16.8 Q1, 2.

2. 32.Surface integrals: Section 16.6 Q19, 21, 25, 39.

3. 33.Stokes theorem: Section 16.7 Q1, 5, 7, 13, 17.

4. 34.Divergence theorem: Section 16.8 Q5, 7, 13.

Lecture 1: Manifolds

Lecture 2: Embedding theorem

Lecture 3: Vector bundles

Lecture 4: Tangent bundle

Lecture 5: De Rham cohomology

Lecture 6: Cartan formula for flows

Lecture 7: Integration

Lecture 8: Mayer-Vietoris sequence

Lecture 9: Poincare duality

Lecture 10: Symplectic structure

Lecture 12: Poincare recurrence

Lecture 13: Moser Argument

Lecture 14: Weinstein neighborhood theorem

Lecture 15: Action-angle coordinates

Lecture 16: Metrics and connections

Lecture 17: Geodesics

Lecture 18: Curvature

Assignment

Assignment 1 (due on Sep 12): Manifolds and submanifolds

Assignment 2 (due on Sep 19): Vector bundle and tangent bundle

Assignment 3 (due on Sep 26): De Rham cohomology

Assignment 4 (due on Oct 3): Poincare lemma and Mayer-Vietoris sequences

Assignment 5 (due on Oct 10): Poincare duality, Symplectic structure

Assignment 6 (due on Oct 17): Lagrangian submanifolds

Assignment 7 (due on Oct 24): Poincare recurrence

Assignment 8 (due on Oct 31): Moser Argument

Assignment 9 (due on Nov 7): Action-angle coordinates

Assignment 10 (due on Nov 14): Metrics and connections

Assignment 11 (due on Nov 21): Metrics and connections (2)

Assignment 12 (no need to hand in): Geodesics

Assignment 13 (no need to hand in): Curvature

Math 230a Differential Geometry

Fall 2012

Math 115 Method of Analysis

Math 276y SYZ Mirror Symmetry

Spring 2013

Math 136: Differential Geometry (Tu, Th 10-11:30)

The exterior differential calculus and its applications to curves and surfaces in 3-space and to various notions of curvature. Introduction to Riemannian geometry in higher dimensions.

Fall 2013

Math 115: Methods of Analysis

Complex functions; Fourier analysis; Hilbert spaces and operators; Laplaces equations; Bessel and Legendre functions; Sturm-Liouville theory.

Math 91r: Supervised reading and research

Project researcher: Rolando La Placa

Main reference: Atiyah’s book “The geometry and physics of knots

Aim: To study topological quantum field theory (TQFT) from a mathematical point of view, and learn important related geometric constructions.  They include:

1. -Algebraic topology

2. -Differential geometry

3. -Riemann surface and divisors

4. -Symplectic geometry and geometric quantization

5. -Knot theory

6. -Representation theory

7. -Lie group and Lie algebras

Spring 2014

Math 280y: Topics in Symplectic Geometry (M, W 2:30-4)

Symplectic geometry has grown into an important branch of mathematics due to its intimate relationship with physics. A focus on symplectic enumerative invariants and Lagrangian Floer theory, which have great developments in recent years brought by string theory and mirror symmetry.

Fall 2014

Math 115: Methods of Analysis

Complex functions; Fourier analysis; Hilbert spaces and operators; Laplaces equations; Bessel and Legendre functions; Sturm-Liouville theory.

Math 21a: Multivariable Calculus

To see how calculus applies in practical situations described by more than one variable, we study: Vectors, lines, planes, parameterization of curves and surfaces, partial derivatives, directional derivatives and the gradient, optimization and critical point analysis, including constrained optimization and the Method of Lagrange Multipliers, integration over curves, surfaces and solid regions using Cartesian, polar, cylindrical, and spherical coordinates, divergence and curl of vector fields, and the Greens, Stokess, and Divergence Theorems.

Spring 2015

Math 261: Topics in Symplectic Geometry (Tu, Th 2:30-4)

An investigation of geometric aspects of mirror symmetry in the SYZ approach using Lagrangian intersection theory.

Spring 2016

Math 822: Topics in Geometry (Tu, Th 9:30-11)

The main theme this year is toric geometry and mirror symmetry.

Math 722: Differential Topology 2 (Tu, Th 11-12:30)

This course is a continuation of MATH 721: Differential topology 1.  We will introduce the Mayer-Vietoris sequence, which is the central tool to compute de Rham cohomology.  As an important application of de Rham theory, we will define the Poncare dual of a submanifold and the Thom class of a vector bundle.  If time is allowed, we will take our first visit to the spectral sequence, a foundational modern technique in homology and homotopy theory.

Fall 2016

Math 725: Differential Geometry 1 (Tu, Th 11-12:30)

Geometry of surfaces in Euclidean space; geodesics and curvature of Riemannian manifolds; topological restrictions on curvature; manifolds; vector bundles; Higgs fields.

Spring 2017

MA 225: Multivariate Calculus (MWF 11:15-12:05)

Vectors, lines, planes. Multiple integration, cylindrical and spherical coordinates. Partial derivatives, directional derivatives, scalar and vector fields, the gradient, potentials, approximation, multivariate minimization, Stokes's and related theorems.

Syllabus

Mock Final Exam (The final exam will be mainly about integration, from Week 8 to 12.)  Answer (Please finish it yourself before checking the answers.)

Final exam: May 12 12:30-2:30 at COM 101 (College of Communication, 640 Commonwealth Avenue).

Assignment

Week 0: (due Jan 26 or 27)

1. 1.Coordinates and distance: Section 12.1 Q7, 13, 16, 20, 21, 27, 29, 43, 51, 57.

Week 1: (due Feb 2 or 3)

1. 2.Vectors and dot product: Section 12.2 Q5, 15, 22, 23, 33, 35, 39, 41;
Section 12.3 Q1, 5, 8, 18, 25, 26.

2. 3.Cross product: Section 12.4 Q3, 6, 11, 16, 21, 23, 27, 29, 30, 31, 37, 42.

3. 4.Lines and planes: Section 12.5 Q4, 6, 8, 17, 21, 22, 23, 31, 33, 39, 45, 53, 57, 61.

Week 2: (due Feb 9 or 10)

1. 5.Level surfaces and quadrics: Section 12.6 Q1 to 12, 13, 17, 21, 25, 27, 31.

2. 6.Curves: Section 13.1 Q1, 4, 6, 11, 14, 15, 22, 23a, 23b, 23c, 23d, 23e.

3. 7.Arc length: Section 13.3 Q2, 6, 9, 11, 13, 18.

Week 3: (due Feb 16 or 17)

1. 8.Curvature and torsion: Section 13.4 Q1, 3, 5, 7, 10, 13; Section 13.5 Q8, 10.

2. 9.Spherical coordinates: Q1 to 6 in this file (please click).

3. 10.Parametric surfaces: Section 14.1 Q14, also parametrize the graph of f;
Q31-36 (counted as one question);
parametrize the level surfaces {f(x,y,z)=1} in Q53, 55, 59, 60;
write the parametrized surfaces in Q77, 80 as level surfaces {f(x,y,z)=0}.

Week 4: (due Feb 23 or 24)

1. 11.Functions and continuity: Section 14.1 Q14, 15, 31 to 36, 53, 57, 59.
Section 14.2 Q9, 13, 21, 32, 41, 61, 68.

Week 5: (due Mar 2 or 3)

12. Partial derivatives: Section 14.3 Q1, 7, 12, 19, 21, 29, 42, 45, 52, 65.

1. 13.PDE: Section 14.3 Q75, 83, 90.

2. 14.Chain rule: Section 14.4 Q1, 4, 7, 11, 26, 29, 33, 43, 44.

Week 6: (due Mar 16 or 17)

15. Directional derivative: Section 14.5 Q1, 4, 7, 11, 15, 17, 19, 22, 25, 27, 29.

16. Gradient and tangent plane: Section 14.6 Q1a, 5a, 9, 19, 21, 25, 27, 29.

1. 17.Critical points: Section 14.7 Q1, 9, 13, 23, 24, 29, 45, 50, 52, 56.

Week 7: (due Mar 23 or 24)

18. Lagrange multipliers: Section 14.8 Q1, 6, 9, 12, 22, 29, 47.

19. Global extrema: Section 14.7 Q31, 40, 42.

1. 20.Taylor’s formula: Section 14.9 Q3.  This exercise is optional.  Taylor’s formula will not appear in exams.

Week 8: (due Apr 6 or 7)

21. Double integrals: Section 15.1 Q9, 17, 19, 25, 36;
Section 15.2 Q1, 6, 14, 19, 26, 33, 47, 57;
Section 15.3 Q3, 5.

22. Polar integration: Section 15.4 Q1, 9, 27, 29.

1. 23.Surface area: all questions in this file (please click).

2. 24.Triple integrals: Section 15.5 Q7, 10, 23, 25, 41;
Section 15.7 Q1, 15, 17, 21, 33.

Mock Mid-Term Test 2 (The mid-term will be on Section 11-23.)

Week 9: (due Apr 13 or 14)

1. 25.Substitution: Section 15.8 Q2, 9, 13, 14, 18.

2. 26.Integration of functions on curves and surfaces: Section 16.1 Q1-8 (as a whole), 9, 12;
Section 16.5 Q1, 7, 17;
Section 16.6 Q1, 3, 17.

3. 27.Vector fields: Section 16.2 Q1, 4, 39, 40.

Week 10: (due Apr 20 or 21)

28. Line integrals: Section 16.2 Q7, 11, 17, 19, 23, 38, 47, 49, 53.

1. 29.Line integral theorem: Section 16.3 Q13, 25, 29, 31, 34.

2. 30.Conservative fields, potential and curl: Section 16.3 Q1, 3, 5, 7, 27.

Week 11: (due Apr 27 or 28)

31. Greens theorem: Section 16.4 Q1, 5, 9, 15, 21, 25, 27.

1. 32.Curl and divergence: Section 16.8 Q1, 2, 3.

Week 12: (due May 4 or 5)

33. Surface integrals: Section 16.6 Q19, 21, 23, 25, 39.

34. Stokes theorem: Section 16.7 Q1, 5, 7, 13, 15, 17.

1. 35.Divergence theorem: Section 16.8 Q5, 7, 9, 13.

Fall 2017

MA 123: Calculus I (MWF 1:25-2:15)

Limits; derivatives; differentiation of algebraic functions. Applications to maxima, minima, and convexity of functions. The definite integral; the fundamental theorem of integral calculus; applications of integration.

MA 563: Differential Geometry (MWF 11:15-12:05)

The exterior differential calculus and its applications to curves and surfaces in 3-space and to various notions of curvature.

Homework 6 (due on Friday Oct 20)

Homework 5 (due on Friday Oct 20)

Appendix: Differential forms 1 2

Ch 4: Gauss-Bonnet formula

Ch 2: Surfaces

Ch 1: Curves

Syllabus

Spring 2018

MA 731: Lie groups and Lie algebras (Tue Thu 3:30-5)

Lie groups and introduction to representation theory.

Syllabus

HW questions are located in the end of the notes.

Due date Feb 1:    Examples of Lie groups.

Due date Feb 8:    Closed subgroup theorem.

Due date Feb 15:  Adjoint action.

Due date Feb 22:  Matrix exponential.  (Choose one to hand in.)

Lie Algebra.  (Choose one to hand in.)

Due date Mar 1:    Representation

Due date Mar 15:  Schur's Lemma

Due date Mar 28:  SL(3,C). (Choose one to hand in.)

Weyl group for SL(3,C).  (Choose one to hand in.)

Due date Apr 5 :  Semi-simple Lie algebra

Due date Apr 12 :  Abstract Root System 1

Due date Apr 19 :  Abstract Root System 2

Representation of semi-simple Lie algebra  (Assignment is optional)