Math 321(W), Introduction to Advanced Mathematics

Instructor: Pavel Guerzhoy

The class meets Tuesdays and Thursdays 9 - 10:15am in 414, Keller Hall


e-mail: pavel(at)math(dot)hawaii(dot)edu (usually, I respond to e-mail messages within a day)
Office hours: TuTr 2-3pm


A detailed instruction on the homework assignments





General Expectations
The Department of Mathematics has a general expectations statement, which we are assumed to follow in this class.

Reading
In this class, we use the draft of a textbook by Paolo Aluffi MODICUM MATHEMATICUM





As a supplementary material, the textbook by Richard Hammac Book of Proof is recommended.


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 consitute general mathematical literacy and are of substantial importance for most upper division mathematics proof-based courses. Specifically, the students will learn some language and terminology common throughout mathematics, basics of proof-building, and naive set theory. That will allow us to meaningfully address a question of the nature of an object as fundamental as real numbers. The class lays a foundation for leraning core areas of mathematics including but not limited to algebra, analysis, and topology.

Prerequisites
Calculus Math 243 (or concurrent) or 253A (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 according to the University final exams schedule, and will count for 30% of the final grade. The exam is cumulative (it covers all the material). There will be no make-ups for the final.
  • Four writing homework assignments count together for 40% of the final grade.
  • Two midterm tests. The (average) grade for the tests counts for 30% of the final grade.
  • Regular homework assignments consist of problem sets marked as "Homework" in the textbook. 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 solving these problem sets have pretty high chances to fail the class.







  • Contents and Homework assignments

    This table will be updated regularly. Please check it often!

    Remarks on the non-graded homework exercises.






    Date Sections Extras Writing Homework Assignement
    Tue, Jan 11 INTRO, Section 1.1
    Thu, Jan 13 Section 1.1, 1.2
    Tue, Jan 18 Section 1.2,1.3
    Thu, Jan 20 Section 1.3 proof samples
    Tue Jan 21 Section 2.1
    Thu Jan 27 Section 2.1 (Exerc 2.1.10,2.1.11)
    Tue Feb 1 2.2
    Thu Feb 3 2.3
    Tue Feb 8 2.4 HW 1 due Feb 8
    Thu Feb 10 from naturals to integers
    Thu Feb 15 Section 3.1
    Tue Feb 17 First Midterm Test practice problems for midterm
    Tue Feb 22 3.2 HW 1 redo due Feb 22
    Thu Feb 24 3.3
    Tue Mar 1 3.4
    Thu Mar 3 3.4,3.5
    Tue Mar 8 3.5
    Thu Mar 10 Section 4.1 HW 2 due Mar 14
    Tue Mar 22 4.2
    Thu Mar 24 Second Midterm Test practice problems for the second midterm
    Tue Mar 29 4.3 HW 2 redo due Mar 28
    Thu Mar 31 4.3
    Tue Apr 5 4.4
    Thu Apr 7 4.4
    Tue Apr 12 Section 5.1
    Thu Apr 14 5.2
    Tues Apr 19 5.3 HW3 due Apr 19
    Thu Apr 21 5.4
    Tue Apr 26
    Thu Apr 28
    Tue May 3 redo HW3 due May 6
    Thu May 12 (9:45-11:45) FINAL EXAM practice problems for the final exam