... Of that kingdom this is the key.

Nursery Rhyme

Nursery Rhyme

Mathematicians like the mathematics that most people are not aware about.

The course answers to the question **What is the mathematics that
mathematicians like?**

Students who will also find this mathematics attractive will be helped to
enter it.

Each lecture is aimed to surprise and reshape the students' perception of mathematics. The course is organized as a collection of short, important, and self-contained subjects, which are mostly independent from each other. They are picked up from various parts of mathematics: number theory, geometry, set theory, combinatorics, topology, optimization theory, etc. and complement the content of advanced mathematical courses.

No preliminary knowledge of advanced mathematics is required.

The course is targeted towards students of the following three types:

- freshmen who have not yet decided what major to choose
- beginning math majors
- junior and senior math majors

A beginner math major would learn plenty of nice mathematics and would be helped to decide if she/he is ready to join the honors program in mathematics immediately.

The course will provide students with a rich collection of classical mathematics of prime importance, which otherwise may not reach students, because is not included in the main math courses.

The book is available from Amazon for about $ 13.

Certainly, the course will not follow the book section by section, but the style would be similar.

From the introduction to this book:

*
We select a sequence of subjects, each one complete in itself,
none requiring more than an hour to read and understand.
The subjects are independent, so that one need not remember
what has gone before when reading any chapter.''*

- The Sequence of Prime Numbers
- Traversing Nets of Curves
- Some Maximum Problems
- Incommensurable Segments and Irrational Numbers
- A Minimum Property of the Pedal Triangle
- A Second Proof of the Same Minimum Property
- The Theory of Sets
- Some Combinatorial Problems
- On Waring's Problem
- On Closed Self-Intersecting Curves
- Is the Factorization of a Number into Prime Factors Unique?
- The Four-Color Problem
- The Regular Polyhedrons
- Pythagorean Numbers and Fermat's Theorem
- The Theorem of the Arithmetic and Geometric Means
- The Spanning Circle of a Finite Set of Points
- Approximating Irrational Numbers by Means of Rational Numbers
- Producing Rectilinear Motion by Means of Linkages
- Perfect Numbers
- Euler's Proof of the Infinitude of the Prime Numbers
- Fundamental Principles of Maximum Problems
- The Figure of Greatest Area with a Given Perimeter
- Periodic Decimal Fractions
- A Characteristic Property of the Circle
- Curves of Constant Breadth
- The Indispensability of the Compass for the Constructions of Elementary Geometry
- A Property of the Number 30
- An Improved Inequality

Notes and Remarks

- Score of level 4 or better on the math placement exam,
- or MAT 123 passed with a grade C or better,
- or a permission by the instructor.

Priority given to students in the University's honors programs.

Notice that the prerequisites do not include any Calculus. High-school
algebra suffices.

Though the course is not easy: as any mathematics, it relies on proofs.

This is why **MAT 200: Logic, Language and Proof** is
an advisory co-requisite.

Oleg Viro

Professor, Ph.D. 1974, Doctor Phys-Mat.Sci. 1983, both from Leningrad State University

Arrived at Stony Brook in 2007.

Professor, Ph.D. 1974, Doctor Phys-Mat.Sci. 1983, both from Leningrad State University

Arrived at Stony Brook in 2007.

Office: Math Tower 5-110

Phone: (631) 632-8286

Email: oleg.viro AT math.stonybrook.edu

Web page: www.math.stonybrook.edu/~oleg

Research fields: Topology and Geometry,

especially low-dimensional topology
and real algebraic geometry.

Mondays and Wednesdays 5:30pm - 6:50pm in Library W4530,

Mondays and Wednesdays 7:00pm - 8:00pm in Math Tower 5-110.

The final grade will be based on

- Homeworks 20%,
- Midterm 40%,
- Final exam 40%.

In MAT 150, the main purpose of homeworks is to keep students aware about the material of lectures: definitions, examples, statements of theorems, some proofs. More creative challenging problems (e.g., compose a problem or generalize a theory given in a lecture) will be proposed, but rarely and only as non-compulsory bonus problems.

The exams, both midterm and final will serve the same purpose: to verify whether the material is familiar.

If you have a physical, psychological, medical, or learning disability that
may impact your course work, please contact Disability Support
Services or call (631) 632-6748. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential.
Students who require assistance during emergency evacuation are encouraged
to discuss their needs with their professors and Disability Support
Services. For procedures and information go to the Evacuation
Guide for People with Physical Disabilities.

Each student must pursue his or her academic goals honestly and be
personally accountable for all submitted work. Representing another
person's work as your own is always wrong. Faculty are required to report
any suspected instances of academic dishonesty to the Academic Judiciary.
For more comprehensive information on academic integrity, including
categories of academic dishonesty, please refer to the academic judiciary
website.

Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Judicial Affairs any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn.

The statement and proof of theorem which was called Theorem 3 in Lecture 1 were not formally distinguished in the textbook. The proof was not quite clear. The homework required to find them in the textbook, and to answer to the following two questions: What is not said in the proof? What is number $p$ in the proof of Theorem 3? and list the differences between the proofs in the lecture and textbook.

The number $p$ is not defined in the textbook. One can guess what $p$ is. Moreover, the numbers in the formula that defines $M$ are not specified either. They cannot be the same as in the proof of Theorem 3 in Lecture 1.

The textbook is a beautiful classical book. A good mathematical book may, and even should, be somewhere unclear. This gives to a student an opportunity to think, fix the vague piece and improve self-esteem.

The third problem (bonus problem) in Homework 1:

What sets of primes can you prove to be infinite?

- the sets of primes in arithmetic series $4,8,12,16, \dots,4n,\dots$; $1,5,9,\dots,4n+1,\dots$; $6,10,14,18, \dots,4n+2\dots$; $3,8,11,\dots,4n-1,\dots$?
- the sets of primes with the last digit $3$ or $7$ (i.e., having the last digit $3$ or $7$ in the decimal notation)?

The pdf file was used as a background for xournal. This allowed to write over the screen. Almost all the proofs were inserted during the lecture.