Siu-Cheong Lau

MS

 

人生幾何｜幾何人生

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.

Syllabus


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.

Syllabus


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.

Syllabus


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.

Syllabus


01. Quiver representation

02. Cohomology

03. Gabriel's Theorem

04. Kac generalization of Gabriel's Theorem

05. GIT moduli

06. Toric quiver varieties

07. Hochschild cohomology of quiver algebra

08. Topology of framed quiver varieties

09. Nakajima quiver varieties

10. Quiver algebra from mirror symmetry


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.

Syllabus


Mock Final Exam     


Mock Mid-Term Test 1      Answer


Mock Mid-Term Test 2      Answer


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.

Syllabus

01. Morse functions.

02. Smale condition.

03. Morse complex.

04. Properties of Morse homology.

05. Family Morse theory.

06. Distribution.

07. Frobenius Theorem.

08. Toric varieties.

09. Toric morphisms.

10. Topology of toric varieties.

11. Toric line bundles.

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.

Syllabus

Mock Mid-Term Test

Mock Exam

Ch 1: Curves

Ch 2: Surfaces

Ch 3: Intrinsic geometry

Ch 4: Gauss-Bonnet formula

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.

Syllabus


Mock Mid-Term Test 1


Mock Mid-Term Test 2


Answer to Mock Mid-Term Test 2


Answer to Mid-Term Test 2


Mock Final Exam Answers

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.


Answers to Mid-Term Test 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.


An expanding list of something to keep in mind in DG

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 11: Lagrangian submanifolds and symplectomorphisms

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

Syllabus

Math 115 Method of Analysis

Mock mid-term 1

Mock mid-term 2

Math 276y SYZ Mirror Symmetry

Spring 2013

Lecture 1: Overview of developments in SYZ mirror symmetry

Lecture 2: Deformation space of special Lagrangians

Lecture 3: Semi-flat mirror symmetry

Lecture 4: SYZ transform of branes

Lecture 5: Examples of special Lagrangian fibrations

Lecture 6: Lagrangian Floer theory on compact toric manifolds

Lecture 7: Open mirror theorem for compact toric Gorenstein orbifolds

Lecture 8: SYZ for toric Calabi-Yau orbifolds

Lecture 9: Generalized SYZ construction

Lecture 10: Gross-Siebert program

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.

Syllabus

Syllabus

Ch 1: Curves

Ch 2: Surfaces

Ch 3: Intrinsic geometry of surfaces

Ch 4: Gauss-Bonnet formula

Ch 5: Manifolds

Ch 6: Tangent bundle

Ch 7: Vector bundles

Ch 8: Metrics and connections

Ch 9: Curvature

Ch 1: Holomorphic functions

Ch 2: Power Series

Ch 3: Laurent Series and residue

Ch 4: ODE

Ch 5: Applications of ODE

Ch 6: Laplace transform

Ch 7: Applications of Laplace transform

Ch 8: Fourier series

Ch 9: Sturm-Liouville problems

Ch 10: Bessel function

Ch 11: Laplace equation

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.

Hand-written notes:
Polygon enumerations via mirror symmetry
Notes on Joyce’s paper on stability 1
Notes on Joyce’s paper on stability 2
Quadratic differentials and Fukaya categories 1
Quadratic differentials and Fukaya categories 2
Seidel elements
Gross-Siebert program for toric Calabi-Yau manifolds

Fall 2014

Math 115: Methods of Analysis

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

Ch 1: Holomorphic functions

Ch 2: Power Series

Ch 3: Laurent Series and residue

Ch 4: ODE

Ch 5: Applications of ODE

Ch 6: Laplace transform

Ch 7: Applications of Laplace transform

Ch 8: Fourier series

Ch 9: Sturm-Liouville problems

Ch 10: Bessel function

Ch 11: Laplace equation

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.

Syllabus

00 Motivation

01 Toric variety from gluing

02 Toric morphism

03 Topology

04 Line bundles

05 Vanishing and Riemann-Roch Theorem

06 GIT quotient

07 Symplectic quotient

08 Introduction to mirror symmetry

09 SYZ

10 Holomorphic discs

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.

Syllabus

00 Motivation

01 De Rham cohomology

02 Homotopy

03 Mayer-Vietoris sequence

04 Compactly supported de Rham cohomology

05 Poincare duality

06 Degree

07 Singular homology

08 de Rham theorem and Poincare dual

09 Distribution

10 Foliation

11 Lie subgroup and exponential map

12 Closed subgroup theorem

13 Integrating infinitesimal actions

14 Lie correspondence

15 Normal subgroup

16 Quotient manifold

17 Discrete quotient

18 Homogeneous space

19 Symplectic structure

20 Lagrangian submanifolds and symplectomorphisms

21 Poincare recurrence

22 Moser argument

23 Weinstein neighborhood theorem



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.

Syllabus

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}.


Mock Mid-Term Test 1    Answer


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 11


Homework 10


Homework 9


Homework 8


Homework 7


Mock Mid-Term Test    Sketch of Answer


Homework 6 (due on Friday Oct 20)


Homework 5 (due on Friday Oct 20)


Homework 4


Homework 3


Homework 2


Homework 1


Appendix: Differential forms 1 2


Ch 4: Gauss-Bonnet formula


Ch 3: Connection and geodesics


Ch 2: Surfaces


Ch 1: Curves


Syllabus


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)