Welcome to M192: First-year Seminar in Mathematics

Fall Semester 2006


Instructor:	Dr. Holger Kley	Time:	TR 4:10--5:00
Hours:	Mondays at 9, Wednesdays at 4, and by appointment.	Place:	Shepardson 118


Announcements	Description	Schedule	Homework	Projects	Grading	Handouts	Links

Announcements:


TYPO ALERT! HW 4 Problem 2 has been corrected. Please get the new version of HW set.	9/30
CSU Math Club: first meeting, October 4th. Read the announcement.	9/28
8/30 Office hours cancelled, but please email me your questions.	8/29

Description:

The concept of proof plays a central role in mathematics. Understanding what does and does not constitute a proof is an intellectual hurdle that every successful student of mathematics must overcome. On the other hand, this understanding becomes an intellectual delight that every mathematician revels in. The purpose of this seminar is not to teach you ``how to do proofs''---that comes gradually with the experience gained in junior- and senior-level mathematics courses---but rather, to provide you with some of the key language, concepts and techniques that are commonly used in mathematical proofs. Perhaps the "Prince of Mathematics" said it best:

I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible. Karl Friedrich Gauss (1777--1855)

Several self-contained guest lectures on specific mathematical problems will supplement our work on proofs.

Schedule of Lectures (tentative):

	Date	Lecture	Notes
Week 1	Aug. 22	Introduction to university mathematics. The bridges of Königsberg
Week 1	Aug. 24	First order logic: truth tables.	HW 1 assigned
Week 2	Aug. 29	First order logic: some examples involving prime numbers.
Week 2	Aug. 31	Guest speaker: Dr. Rick Miranda, Dean of the College of Natural Sciences will speak on "The Buffon Needle Problem."	HW 1 due.
Week 3	Sept. 5	Deduction and proof by contradiction. Examples involving rational and irrational numbers. Euclid's proof: there are infinitely many prime numbers	HW 2 assigned.
Week 3	Sept. 7	Sets and logic: Russell's paradox.
Week 4	Sep. 12	More about sets.	HW 2 due.
Week 4	Sep. 14	Guest speaker: Dr. Dan Rudolph, Yates Professor of Mathematics, will speak on "Love, Money, and Combinatorics."	HW 3 assigned
Week 5	Sep. 19	Finite, countably infinite, and uncountable sets.
Week 5	Sep. 21	Introduction to the Principle of Mathematical Induction	HW 3 due
Week 6	Sep. 26	More on induction: complete induction.	HW 4 assigned
Week 6	Sep. 28	Euler's theorem revisited.
Week 7	Oct. 3		HW 4 due.
Week 7	Oct. 5	Set theory as entertainment	Projects due.
Week 8	Oct. 10	Guest speaker: Dr. Bob Liebler, Professor of Mathematics
Week 8	Oct. 12	Poster Session; refreshments served!	Room t.b.a.

Homework:

There will be four problem sets based on material presented during the lectures. Please see the schedule of for the actual assignments and their due dates.
You are encouraged to discuss problems among yourselves, that is, to collaborate. In order for collaboration to be beneficial for all participants, however, it is important that you read, think about, and start to solve the problems on your own first. That way, you will be bringing ideas to the table as too. Each student should submit their own solution. If you feel that your solution is the result of extensive collaboration with a classmate, please point that out and include the name of your coauthor.

Homework assignments will be posted in the Handouts section of this page.

Projects and Posters: The very best of modern mathematics.

Work with a partner.
Briefly answer the following: What are the Abel Prize, the Wolf Prize or the Fields Medal? (Choose one.) Who receives this award and what for?
Then: Select a recipient of one of these awards and prepare a brief (3--4 page) biography. You might consider answering the following: Where was (s)he educated? Where is (s)he now? What did (s)he accomplish in order to receive the award? Please use care in proofreading, citing your sources, etc.
Due date: Thursday, October 5.
Finally: You and your collaborator from the course will prepare a poster version of your project. On Thursay, October 12th, you will present your work to your classmates in a poster session. A panel of judges will choose the top posters for display in the Weber building. Some guildelines for preparing your poster are here.

Grading:

Course grades will be determined based on homework (60%), your paper (20%) and your poster (20%).

Handouts:


Class Policies	8/22	HW 1	8/24
HW 2	9/5	HW 3	9/14
HW 4	9/26	Poster Guidelines	9/26

Links:

If you know of any good resources that should appear here, please let me know.


Upcoming events:
Mathematics in the news:	The cover story "Math will Rock your World" of the 1/23/06 issue of Business Week examines the expanding role of mathematics in the business sector.	The article "Manifold Destiny" by S. Nasar and D. Gruber in the 8/28/06 issue of The New Yorker examines the recent solution of a long-standing mathematical problems, and the controversy that such solutions may bring.
Mathematical organizations:	The American Mathematical Society (AMS) was founded to further mathematical research and scholarship.	The mission of the Mathematical Association of America (MAA) is "to advance the mathematical sciences, especially at the collegiate level."
Mathematical organizations:	The Society for Industrial and Applied Mathematics (SIAM) exists to ensure the strongest interactions between mathematics and other scientific and technological communities
Some useful reference sites:	SAGE: the CSU library's online catalogue.	MathSciNet: electronic database of mathematics publications, maintained by the American Mathematical Society. NOTE: you must access this site from a CSU IP address.
Some useful reference sites:	mathworld: an online mathematical encyclopedia maintained by Wolfram Research.	Wikipedia Mathematics: a free, user-created and -edited encyclopedia.
Some sites about university mathematics:	Misconceptions about learning mathematics: Prof. Steven Zucker (Johns Hopkins University)
Math tutoring at CSU:	Tutorial Hall of the College of Natural Sciences	private mathematics tutors: a list maintained by the department of mathematics.

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