Freshman Seminar-SPRING 2015  THE MILLENNIUM PROBLEMS   The Seven Greatest Unsolved Mathematical Puzzles of Our Time  

Tuesdays 11:00-11:50 AM 129 Social Science Hall (SSL129) (syllabus}

First class meeting is on March 31, 2015 in SSL 129 11:00-11:50 AM

FRESHMAN SEMINAR PROGRAM-DIVISION OF UNDERGRADUATE EDUCATION SPRING (SPRING 2015) Scroll down for a description

MAIN REFERENCES 

"The Millennium Problems," by Keith Devlin, 2002. (not in Ayala Science Library) (Amazon.com---cheap)

"Love and Math," by Edward Frenkel, 2013. Available online for UCI students (here)

"Love and Math," by Edward Frenkel, 2013. Amazon.com<

Mark Ronan, Symmetry and the Monster. One of the greatest quests of Mathematics Ayala Science Library

Mark Ronan, Symmetry and the Monster. One of the greatest quests of Mathematics Amazon.com

Marcus du Sautoy, Symmetry. A Journey into the Patterns of Nature (not in Ayala Science Library) Newport Beach Public Library

Marcus du Sautoy, Symmetry. A Journey into the Patterns of Nature Amazon.com ($0.20 +$3.99 shipping)

SUMMARY for March 31, 2015. Class orientation; the Prime Number Theorem (Marina High School, pp. 1-13)

REFERENCES for March 31, 2015. 

The Prime Number Theorem---Marina High School, June 7, 2011 (click here)

FRESHMAN SEMINAR WINTER 2005---PRIME OBSESSION (Some details, with pictures)

The Prime Number Theorem-Fullerton College, July 29, 2010 (More details, more pictures)

(posted April10) The Riemann Hypothesis-Fullerton College, September 14, 2010 (Part I, pp. 1-73, is the July 29 talk repeated; Part II, pp. 74-142 is the September 14 talk)

SUMMARY for April 7, 2015. (Page numbers refer to the Marina High School Lecture-the first reference above) Prime Number Theorem (p. 14); Riemann's legacy (p. 22); Historical Remarks (pp. 28-31); two summaries (pp. 34-35); Infinitude of the primes (Theorem 4, p. 38); Arbitrary Large blocks containing no primes (Theorem 5, p. 39); Finding pairs of primes in infinitely many intervals of size 70,000,000 (see the first two references below)

A picture is worth a thousand words (Theorem 5)

A picture is worth a thousand words (Theorem 5 and Zhang's Theorem)

REFERENCES for April 7, 2015 (Warning: Only the first two are "readable"; you should ignore the other five)

Solving a Riddle of Primes (Article in the New York Times 2013)

The Pursuit of Beauty (Article in the New Yorker Magazine 2013)

Yitang Zhang, Bounded Gaps between Primes, Annals of Mathematics 2014 (Mathematical research paper 2014)

Yitang Zhang, Bounded Gaps between Primes, Annals of Mathematics 2014 (Review of this Mathematical research paper 2014)

Maynard, James Small gaps between primes. Ann. of Math. (2) 181 (2015), no. 1, 38-413. (Review of this Mathematical research paper 2015)

Maynard, James Bounded length intervals containing two primes and an almost-prime. Bull. Lond. Math. Soc. 45 (2013), no. 4, 753-764 (Review of this Mathematical research paper 2013)

Pintz, Janos The bounded gap conjecture and bounds between consecutive Goldbach numbers. Acta Arith. 155 (2012), no. 4, 397-405. (Review of this Mathematical research paper 2012)

SUMMARY for April 14, 2015. (Page numbers refer to the Marina High School Lecture) The Zeta function (pp18-21); Riemann Hypothesis restated (p. 23); Back to the Zeta function (pp25-27); Complex Numbers (pp32-33); Fundamental Theorem of Arithmetic (FTA) (p.36); Application of FTA: the four square theorem; THE FOLLOWING TWO ITEMS WERE SCHEDULED BUT NOT INCLUDED BECAUSE OF THE TECHNICAL SNAFU:Zhang's work revisited; Lecture of Ronald Graham 2014 (You might be interested in looking at pp. 7-28,66-76,81-89,141-147,165-177,221-225,229-234 of Professor Graham's lecture)

A picture is worth a thousand words (The Riemann Hypothesis)

A picture is worth a thousand words (The four squares theorem)

The four square theorem-Reduction to primes (here)

Power Point Presentation of Professor Graham's lecture (Courtesy of Professor Graham) pps file

A Web Site about Professor Graham click here

SUMMARY for April 21, 2015. Song of Pythagoras; Proof of Pythagoras; Pythagorean Triples; It takes 4 to do Fermat's Last Tango; The saga of Andrew Wiles

"The square of the hypotenuse of a right triangle, is equal to the sum of the squares of the two adjacent sides" (Danny Kaye) click here

A picture is worth a thousand words (Proof of Pythagorean Theorem)

Finding all Pythagorean Triples (Proof of Fermat's Last Theorem (n=4))

The Saga of Andrew Wiles, according to the New York Times (here)

SUMMARY for April 28, 2015. Rational and irrational numbers (Marina High School pp 40-42); Right triangles with integer area and rational sides (Devlin pp 190-194); Primes in an arithmetic progression-variations on Euclid's second theorem-NOT DISCUSSED YET (Hardy and Wright p 13); Pythagorean Triples Revisited (vanderPoorten p.3-NOT DISCUSSED YET); Zeta function revisited (Devlin pp 59-62-NOT DISCUSSED YET)

Half the base times the height (Devlin, Millennium Problems pp190-194)

A picture is worth a thousand words (Euclid's First and Second Theorems)

A picture is worth a thousand words (An old Greek problem)

A picture is worth a thousand words (The answer to an old Greek problem)

Primes in certain arithmetical progressions (Hardy and Wright, Number Theory p. 13)

A picture is worth a thousand words (Constructing Pythagorean Triples)

Finding all Pythagorean Triples (vanderPoorten, Fermat's Last Theorem, pp1-6)

Zeta function revisited (Devlin, Millennium Problems pp59-62)

BOOK REFERENCES

Alf van der Poorten, Notes on Fermat's Last Theorem 1996 Ayala Science Library

Alf van der Poorten, Notes on Fermat's Last Theorem 1996 Amazon.com

G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers Ayala Science Library

G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers Amazon.com

Simon Singh, Fermat's Enigma. The epic quest to solve the world's greatest mathamatical problem 1997 Ayala Science Library

Simon Singh, Fermat's Enigma. The epic quest to solve the world's greatest mathamatical problem 1997 Amazon.com

SUMMARY for May 5, 2015. Primes in an arithmetic progression-variations on Euclid's second theorem (A picture is worth a thousand words-Arithmetic progressions & Dirichlet's theorem); Clock Arithmetic (Devlin pp.193-196); Millennium Problem #2 (Devlin pp196-201 and A picture is worth a thousand words-Millennium Problem #2) Review of conics (Stewart pp214-215)-NOT DISCUSSED; Variation on Fermat's Last Theorem (Stewart pp 230-231)-NOT DISCUSSED

A picture is worth a thousand words (Arithmetic Progressions)

A picture is worth a thousand words (Dirichlet's Theorem)

Counting by the Clock (Devlin, Millennium Problems pp193-196)

How to count an infinite set (Devlin, Millennium Problems pp196-199)

A picture is worth a thousand words (Millennium Problem #2-Birch and Swinnerton-Dyer Conjecture)

Millennium Problem #2 (Devlin, Millennium Problems pp197-201)

Review of Conics; Elliptic curves revisited (Stewart, Fermat's Last Theorem pp214-215)

Three fourth powers sum to a fourth power (Stewart, Fermat's Last Theorem pp230-231)

BOOK REFERENCE

Ian Stewart, David Tall, Algebraic Number theory and Fermat's Last Theorem, 2002 Ayala Science Library

Ian Stewart, David Tall, Algebraic Number theory and Fermat's Last Theorem, 2002 Amazon.com

SUMMARY for May 12, 2015. OLD BUSINESS: constructing Pythagorean triples (PT), finding all PT, Fermat's last theorem, n=4, (infinite descent); NEW BUSINESS: 4 color problem-3 colors not enough (Devlin,Golden Age,pp148-152); Networks (Devlin,Golden Age,pp152-157); Euler's formula (Devlin,Golden Age,pp158-160); de Morgan's theorem and the 5 color theorem-NOT DISCUSSED (Devlin,Golden Age,pp161-166)

3 colors not enough (Devlin, Golden Age of Mathematics, pp148-152)

networks (Devlin, Golden Age of Mathematics, pp152-157)

Euler's theorem (Devlin, Golden Age of Mathematics, pp158-162)

the 5 color theorem (Devlin, Golden Age of Mathematics, pp162-166)

Exercise-Martin Gardner's Map (Devlin, Golden Age of Mathematics, p153)

A picture is worth a thousand words (Fermat's Last Theorem if n=4)

A picture is worth a thousand words (Euler's theorem: V-E+F=1)

BOOK REFERENCE

Keith Devlin, Mathematics, the new golden age 1988 Ayala Science Library

Keith Devlin, Mathematics, the new golden age 1999 (revised edition) Amazon.com

TWO NOTEWORTHY WEBSITES

The Story of Mathematics click here

Milestones in the history of Mathematics click here

SUMMARY for May 19, 2015. THE POINCARE CONJECTURE: Millennium Problem #3 (Devlin, Millennium Problems Chapter 5)

"Cliff Notes" for Millennium Problems by Devlin, Chapter 5 (pdffile)

The London underground, a wiring diagram, Map of Konigsberg, The ring puzzle using surgery, Mobius band, Klein bottle, Nonorientability of the Mobius band, Handles and Crosscaps, The ring puzzle without surgery (Devlin, pp 163,170,175,178,179,181,183,187)

The London Underground (pdffile)

A wiring diagram p. 163

The bridges of Konigsberg p. 170

Ring Puzzle using surgery p. 175

Mobius band p. 178

Klein bottle p. 179

Klein bottle (You-Tube)

Nonorientability of the Mobius band p. 181

Handles and Crosscaps p. 183

Ring Puzzle without surgery (non invasive?) p. 187

Articles in New York times

Russian Reports He Has Solved a Celebrated Math Problem New York Times April 18, 2003

Elusive Proof, Elusive Prover: A New Mathematical Mystery New York Times August 15, 2006

Ask Science: Poincare's Conjecture New York Times August 18, 2006

Highest Honor in Mathematics is Refused New York Times August 22, 2006

Grigori Perelman's Beautiful Mind New York Times December 10, 2009

A Math Problem Solver Declines a $1 Million Prize New York Times July 1, 2010

Article in the New Yorker magazine and beyond (controversy?)

Manifold Destiny. A legendary problem and the battle over who solved it. The New Yorker August 28, 2006

New Yorker on Perelman and Poincare Controversy click here

Poincare Conjecture: Controversy and Eccentricity January 8, 2010 click here

A busy web page

Hamilton, Perelman and the Poincare Conjecture click here

PLAN for May 26, 2015. THE CLASSIFICATION OF ATOMIC SYMMETRIES (FINITE SIMPLE GROUPS) (Devlin, Mathematics, The Golden Age Chapter 5)

"Cliff Notes" for Mathematics, the Golden Age, Chapter 5 (pdffile)

The Platonic Solids

The Platonic Solids click here

Symmetry Groups of the Platonic Solids click here

Web sites on Symmetry and Group Theory

An Enormous Theorem click here

Symmetry and Groups click here

The Evolution of Group Theory (22pp) click here

Symmetry Groups and Chemistry

Symmetry and Chemistry (26pp) click here

Symmetry and Group Theory in Chemistry click here

Symmetry in Art and Group Theory in Music

Symmetry Groups in the Alhambra click here

Mathematics and Group Theory in Music (33pp) click here

Borrowed from Freshman Seminar LOVE AND MATH Spring 2014

"Cliff Notes" for Love and Math (by Frenkel) Chapters 1 to 8 (See Chapter 2 for some thoughts about Symmetry, and Chapter 7 for a hint about how Galois used symmetry (pdffile)

"Cliff Notes" for Symmetry and the Monster (by Ronan) chapter 2 (The tale of Evariste Galois) (pdffile)

"Cliff Notes" for Symmetry (by du Sautoy), Chapter 1 (See the section called Symmetry Seekers)(pdffile)

SUMMARY FOR JUNE 2, 2015--Term Papers; some book references

Bernard Russo---Autobiographical Sketch and Reflections on "Millennium Problems"; the freshman seminar (Click Here)

"The Number Mysteries-A Mathematical Odyssey through everyday life," by Marcus du Sautoy, 2011. (not in Ayala Science Library) (Amazon.com---cheap)

"Why Beauty is Truth-A History of Symmetry," by Ian Stewart, 2007. (not in Ayala Science Library) (Amazon.com---cheap)

OTHER BOOKS BY IAN STEWART (WOW!)

Books by Ian Stewart in the Ayala Science Library (Especially, 5,7,8,11,12,15,16,17,20,24,25,27,29) (Wow!) (click here)