The Millennium Prize Problems –

Making Millions the Hard Way

 

Math 17 Winter Term 2016 - An Introduction to Mathematics Beyond Calculus

 

Virtually every mathematician has, at some point, been asked:

 

“What do mathematicians do?”

 

The famous 1940 essay A Mathematician’s Apology by British mathematician G.H. Hardy offers the following conceptual answer:

 

“A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”

 

Math 17 will exemplify these everlasting ideas and provide an overview of the “queen of sciences”, as German mathematician C.F. Gauss (reportedly) called mathematics.

 

The course will explain the various branches of mathematics with the help of the seven Millennium Prize Problems of the Clay Mathematics Institute – each worth $1,000,000:

 

P vs NP Problem                                            

(unsolved, formulated in 1971)    

Yang–Mills and Mass Gap

(unsolved, going back to the 1950s)           

Riemann Hypothesis

(unsolved, formulated in 1859)

Poincaré Conjecture

(solved,     formulated in 1904)

Birch and Swinnerton-Dyer Conjecture

(unsolved, formulated in 1965)

Hodge Conjecture

(unsolved, formulated in 1952)

Navier–Stokes Equation

(unsolved, going back to the 1840s)

 

Math 17 is primarily aimed at first-year and sophomore students who have completed Math 8 (with ease), Math 11, Math 12, or Math 13. It is intended to prepare and inspire you to major in mathematics.

 

If you have any question, please get in contact with:

 

Instructor:

Peter Herbrich

Office:

Kemeny Hall 334

Email:

peter.herbrich@dartmouth.edu

 

 

Weekly Schedule

 

 

Monday

 

Wednesday

Friday

1

What is Mathematics?

Logic

Set Theory

2

Functions and Relations

Theory of Computation

P vs NP Problem

3

Groups

Rings and Fields

Polynomials

4

Linear Algebra

Representation Theory

Yang–Mills and Mass Gap

5

Algebraic Number Theory

Analytic Number Theory

Riemann Hypothesis

6

Topology

Differential Geometry

Poincaré Conjecture

7

Classification of Surfaces

Algebraic Geometry

Birch and Swinnerton-Dyer Conjecture

8

Complex Geometry

Algebraic Topology

Hodge Conjecture

9

Differential Equations

Chaos Theory

Navier–Stokes Equation