Math 321(W), Introduction to Advanced Mathematics

Instructor: Pavel Guerzhoy

The class meets on Mondays, Wednesdays, and Fridays on 10:30 - 11:20am at 413, Keller Hall

Office: 501, Keller Hall (5-th floor)
tel: (808)-956-6533
e-mail: pavel(at)math(dot)hawaii(dot)edu (usually, I respond to e-mail messages within a day)
Office hours: MW 12-1pm & 3:30-4:30pm
General Expectations
The Department of Mathematics has a general expectations statement, which we are assumed to follow in this class.
In this class we use the

lecture notes by Paolo Aluffi.

I may write some additional materials on my own, and post it on this web-site.

Course Objective
The primary goal of this class to develop a way to approach mathematics as professional mathematicians do. Most of the previous mathematical experience of the students is related to Calculus classes. These classes are primarily directed to acquisition of tools. For this reason, although very useful, Calculus classes tend to create a misleading impression about mathematics. In this class we will learn that mathematics is not about specific tools, but about ideas and arguments. These ideas and arguments are frequently expressed as proofs . The class is designated as writing-intensive. As a consequence, mathematical writing, and particularly, the writing of clear and correct proofs is a subject of emphasis. The topics covered in this class are of substantial importance for most upper division courses.
Calculus Math243 or Math253A (or concurrent) or consent.

Grading Policy
The course contains a combination of concepts, ideas and techniques which the students must be able to apply in solving specific problems. Most of these problems require proving or disproving certain statements. Since this is a writing-intensive class, we simultaneously learn how to write mathematical texts.

At the end of the day, your grade will reflect your ability to solve specific problems. This assumes your ability to write down a simple proof so that it is mathematically and grammatically acceptable. More specifically, the following rules are to be taken.

  • Final exam takes place on Monday May 11 at 9:45 --11:45am, and will count for 30% of the final grade. The exam is cumulative (it covers all the material).
  • Five writing homework assignments count together for 40% of the final grade.
  • One midterm quiz. The grade for the quiz counts for 30% of the final grade.
  • Regular homework assignments consist of problem sets marked as ``Homework" in the notes. It is assumed that students are working on these problems as soon as the corresponding section is covered in class. These assignments will never be collected and graded. However, those students who skip working on these problem sets very seriously have pretty high chances to fail the class.

    Contents and writing homework assignments

    This table is continuously under construction, and reflects progress of the class.

    Date Sections Extras Writing Homework Assignement
    Mon, Jan 12 INTRO
    Wed, Jan 14 1.1--1.5
    Fri, Jan 16 1.6 -- 2.2
    Wed, Jan 21 2.3 -- 3.3
    Fri, Jan 23 proof samples
    Mon, Jan 26 section 4
    Wed, Jan 28 section 5 to 5.2
    Fri, Jan 30 section 6 to 6.1-6.2
    Mon, Feb 2 sections 7.2,7.3
    Wed, Feb 4 section 8
    Fri, Feb 6 section 9
    Mon, Feb 9 from naturals to integers HW1 deadline
    Wed, Feb 11 from naturals to integers from natural to integers
    Fri, Feb 13 from integers to rationals, 9.2
    Wed, Feb 18 section 10 to definition 10.9
    Fri, Feb 20 section 10
    Mon, Feb 23 section 11
    Wed, Feb 25 section 12 HW1 redo deadline
    Fri, Feb 27 sections 13
    Mon, Mar 2 sections 13,14
    Wed, Mar 4 section 13
    Fri, Mar 6 Real numbers real numbers 1 HW2 deadline
    Mon, Mar 9 midterm practice questions
    Wed, Mar 11 Cauchy sequences
    Fri, Mar 13 Real numbers
    Mon, Mar 16 Real numbers real numbers 2
    Wed, Mar 18 Real numbers real numbers 3
    Fri, Mar 20 HW2 redo deadline
    Mon, Mar 30 Cardinalities cardinalities HW3 deadline
    Wed, Apr 1 Denumerable sets denumerable sets
    Fri, Apr 3 Denumerable sets
    Mon, Apr 6 Countability of rationals countability of rationals
    Wed, Apr 8 Countability of rationals
    Fri, Apr 10 Cardinality of a power set HW4 deadline
    Mon, Apr 13 Continuum and numerables
    Wed, Apr 15 Continuum and finite
    Fri, Apr 17 HW3 redo deadline
    Mon, Apr 20 Interval and line
    Wed, Apr 22 Unions of continuum sets
    Fri, Apr 24 Product of continuum sets
    Mon, Apr 27 Continuum of an interval of reals) HW4 redo deadline
    Wed, Apr 29 Continuum of an interval of reals (continued)
    Fri, May 1 Cantor - Bernstein - Schroder
    Mon, May 4 review for final practice problems for final
    Wed, May 6 review for final