Postgraduate Courses

On successful completion of the course, students will be able to:

• 1.
  Analyze if a given set is countable or uncountable, construct bijections between two sets, and analyze the cardinality of the Cantor set.
• 2.
  Analyze if a function defined on a set is a metric or a norm, analyze if a sequence in a metric or norm space converges, determine whether or not two metrics or two norms are equivalent, analyze if a metric space is of the second category, and apply the Baire Category Theorem in different contexts.
• 3.
  Analyze if a given subset of a metric space is open or closed, compact, connected, totally bounded, and construct the relative metric.
• 4.
  Analyze if a given mapping from a metric space into another metric space is continuous and uniformly continuous by using different continuity characterizations, and determine if a given map is a homeomorphism.
• 5.
  Analyze if a metric space is complete by using nested set theorem and Balzono-Weierstrass Theorem.
• 6.
  Apply fixed point theorem in the context of differential equations.

On successful completion of the course, students will be able to:

• 1.
  Formulate the quantities that describe space curves.
• 2.
  Present standard examples of curves that are interesting in high school education.
• 3.
  Formulate a curve as the motion of a point particle.
• 4.
  Formulate the quantities that describe surfaces.
• 5.
  Present standard examples of surfaces that are interesting in high school education.
• 6.
  Formulate arc-length in the language of Riemannian metrics.
• 7.
  Formulate the use of normal vectors and curvature in the description of surfaces.
• 8.
  Apply the theory of curves, and surfaces to physical problems.

On successful completion of the course, students will be able to:

• 1.
  Use Peano axioms to define natural numbers and their arithmetic rigorously.
• 2.
  Use the method of Dedekind cuts to rigorously define the field of real numbers.
• 3.
  Define complex numbers and apply them to algebraic and geometric problems.
• 4.
  Describe the connection between polynomials and geometric objects defined by them.  
• 5.
  Work with various special polynomials and show a variety of their applications.

On successful completion of the course, students will be able to:

• 1.
  Formulate precisely the mathematical axioms which a group (and ring) must satisfy.
• 2.
  Derive certain more advanced properties of groups and rings from the axioms.
• 3.
  Identify when a group or ring occurs in a physical or mathematical situation, and infer useful information based on properties satisfied by groups and rings.
• 4.
  Apply algebraic concepts to organize and solve problems.
• 5.
  Explain how abstract algebra is used to solve certain classical questions of geometry and algebra.
• 6.
  Explain how abstract algebra is used to prove certain number theory results, and how these results are used for encryption purposes in modern day communications.

On successful completion of the course, students will be able to:

• 1.
  Explain the axiomatic nature of Euclid’s Elements and its shortcomings and show how Hilbert's axioms provide a rigorous foundation for Euclidean geometry.
• 2.
  Formulate the parallel postulate and explain its role in the development of hyperbolic geometry.
• 3.
  Use geometric transformations to characterize various geometries.
• 4.
  Use hyperbolic trigonometry to study various properties of hyperbolic geometry. 

On successful completion of the course, students will be able to:

• 1.
  Explain the reasons behind combinatorial principles and applications.
• 2.
  Design combinatorial problems using the principles taught in class.
• 3.
  Present projects related to certain theorems or problems in combinatorics.
• 4.
  Construct problems to show the usefulness of the principles for teaching purposes.

On successful completion of the course, students will be able to:

• 1.
  Identify the core ideas on analyzing data.
• 2.
  Communicate effectively the statistical results to both lay and expert audiences utilizing appropriate information and suitable technology.
• 3.
  Apply rigorous, analytic and highly numerate methods to analyze and solve problems in daily life and at work.
• 4.
  Carry out objective analysis and prediction of quantitative information with independent judgement.

On successful completion of the course, students will be able to:

• 1.
  Recognize various techniques to solve certain types of math problems.
• 2.
  Present solutions and compare the strength and weakness of the techniques.
• 3.
  Modify problems to create new problems.
• 4.
  Identify conditions in solving problems to investigate further applications.

On successful completion of the course, students will be able to:

• 1.
  Apply theoretical knowledge, principles and techniques to mathematical physics, mathematical biology and traffic flows.
• 2.
  Recognize roles of mathematics in solving some mathematical physics, mathematical biology and traffic flows.
• 3.
  Apply various theorems to study ideal mathematical models, such as ideal 2-dimensional fluid flows, ideal electrostatic force fields, steady state temperature, etc.
• 4.
  Present solutions of some physical problems in a systematic way.

On successful completion of the course, students will be able to:

• 1.
  Apply theoretical knowledge, principles and techniques to physical problems.
• 2.
  Recognize roles of mathematics in scientific computation.
• 3.
  Implement various numerical algorithms on computers and apply them to real life problems.
• 4.
  Present numerical output from a computer code in a systematic way.

On successful completion of the course, students will be able to:

• 1.
  Explain advanced mathematical theories, concepts and principles using precise mathematical language.
• 2.
  Apply independent judgment to investigative mathematical work.
• 3.
  Apply a rigorous logical and analytic approach to execute tasks and solve mathematical problems.
• 4.
  Communicate mathematical concepts and methods effectively to a range of audiences, both orally and in writing.

On successful completion of the course, students will be able to:

• 1.
  Explain and apply mathematical knowledge of the selected topics of current interest which may not be covered by existing courses.
• 2.
  Apply independent judgment to investigative mathematical work.
• 3.
  Present findings and solutions in a systematic way.