Goldsmiths’ 2014 Mathematics

Friday 25th July

Open Problems in Mathematics

Dr Thomas Prellberg

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What is Mathematics?

What we’d like our prospective students to know:

Mathematics is not:

just “doing things with numbers and letters and other symbols”

just a collection of facts and rote recipes

just computational and arithmetic skills – although learning the times table by heart is a very important skill in primary school

Mathematics is:

a way of thinking

the language of science and a language in its own right

a creative discipline

a source of pleasure and wonder

a means of problem solving

Mathematical problems

Some million dollar problems

Examples of solved and open problems

Seven Million Dollars Prize Money

7 Prize Problems, selected by Clay Mathematics Institute in 2000

http://www.claymath.org/millennium-problems/millennium-prize-problems

The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of June 2014, six of the problems remain unsolved. A correct solution to any of the problems results in a US $1,000,000 prize (sometimes called a Millennium Prize) being awarded by the institute. The Poincaré conjecture was solved by Grigori Perelman, but he declined the award in 2010.

http://en.wikipedia.org/wiki/Millennium_Prize_Problems

The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Providence, Rhode Island. CMI’s scientific activities are managed from the President’s office in Oxford, United Kingdom. The Institute is “dedicated to increasing and disseminating mathematical knowledge.” It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998 through the sponsorship of Boston businessman Landon T. Clay. Harvard mathematician Arthur Jaffe was the first president of CMI.

http://en.wikipedia.org/wiki/Clay_Mathematics_Institute

The problems are:

Birch and Swinnerton – Dyer Conjecture; Hodge Conjecture; Navier-Stokes Equations; P vs NP; Poincare´ Conjecture (which has actually been solved recently); Riemann Hypothesis; Yang-Mills Theory

In mathematics, the Birch and Swinnerton-Dyer conjecture is an open problem in the field of number theory. It is widely recognized as one of the most challenging mathematical problems; the conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay Mathematics Institute, which has offered a $1,000,000 prize for the first correct proof. It is named after mathematicians Bryan Birch and Peter Swinnerton-Dyer who developed the conjecture during the first half of the 1960s with the help of machine computation. As of 2014, only special cases of the conjecture have been proven correct.

The conjecture relates arithmetic data associated to an elliptic curve E over a number field K to the behaviour of the Hasse–Weil L-function L(E, s) of E at s = 1. More specifically, it is conjectured that the rank of the abelian group E(K) of points of E is the order of the zero of L(E, s) at s = 1, and the first non-zero coefficient in the Taylor expansion of L(E, s) at s = 1 is given by more refined arithmetic data attached to E over K (Wiles 2006).

http://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture

The Hodge conjecture is a major unsolved problem in algebraic geometry that relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they are sums of Poincaré duals of the homology classes of subvarieties. It was formulated by the Scottish mathematician William Vallance Douglas Hodge as a result of a work in between 1930 and 1940 to enrich the description of de Rham cohomology to include extra structure that is present in the case of complex algebraic varieties. It received little attention before Hodge presented it in an address during the 1950 International Congress of Mathematicians, held in Cambridge, Massachusetts, U.S.

http://en.wikipedia.org/wiki/Hodge_conjecture

In physics, the Navier–Stokes equations [navˈjeː stəʊks], named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton’s second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term – hence describing viscous flow.

The equations are useful because they describe the physics of many things of academic and economic interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing. The Navier–Stokes equations in their full and simplified forms help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell’s equations they can be used to model and study magnetohydrodynamics.

The Navier–Stokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, given their wide range of practical uses, it has not yet been proven that in three dimensions solutions always exist (existence), or that if they do exist, then they do not contain any singularity (they are smooth). These are called the Navier–Stokes existence and smoothness problems.

http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations

The P versus NP problem is a major unsolved problem in computer science. Informally, it asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer. It was introduced in 1971 by Stephen Cook in his seminal paper “The complexity of theorem proving procedures” and is considered by many to be the most important open problem in the field.

http://en.wikipedia.org/wiki/P_versus_NP_problem

In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture that the nontrivial zeros of the Riemann zeta function all have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.

The Riemann hypothesis implies results about the distribution of prime numbers. Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics (Bombieri 2000). The Riemann hypothesis, along with the Goldbach conjecture, is part of Hilbert’s eighth problem in David Hilbert’s list of 23 unsolved problems.

http://en.wikipedia.org/wiki/Riemann_hypothesis

Yang–Mills theory is a gauge theory based on the SU(N) group, or more generally any compact, semi-simple Lie group. Yang–Mills theory seeks to describe the behavior of elementary particles using these non-Abelian Lie groups and is at the core of the unification of the electromagnetic force and weak (i.e. U(1) × SU(2)) as well as Quantum Chromodynamics, the theory of the strong force (based on SU(3)). Thus it forms the basis of our understanding of particle physics, the Standard Model.

http://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory

The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. It was developed throughout the latter half of the 20th century, as a collaborative effort of scientists around the world. The current formulation was finalized in the mid-1970s upon experimental confirmation of the existence of quarks. Since then, discoveries of the top quark (1995), the tau neutrino (2000), and more recently the Higgs boson (2013), have given further credence to the Standard Model. Because of its success in explaining a wide variety of experimental results, the Standard Model is sometimes regarded as a “theory of almost everything”.

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The Standard Model of elementary particles, with the three generations of matter, gauge bosons in the fourth column, and the Higgs boson in the fifth.

http://en.wikipedia.org/wiki/Standard_Model

These are hard problems so it might be easier to rob a bank…

Solved Problems in Mathematics

Some recently proved problems:

Fermat’s last theorem (1637, proved 1994): If an integer n is greater than 2, then the equation

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has no solutions in non-zero integers a, b, and c.

For n = 2, this is of course possible, for example

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http://en.wikipedia.org/wiki/Fermat’s_Last_Theorem

Pierre de Fermat (17 August 1601 or 1607 – 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician.

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http://en.wikipedia.org/wiki/Pierre_de_Fermat

The four colour theorem (1852, proved 1976): Given any plane separated into regions, such as a political map of the states of a country, the regions may be coloured using no more than four colours.

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http://en.wikipedia.org/wiki/Four_color_theorem

Open Problems in Mathematics

Some unsolved problems:

Goldbach’s conjecture (1742): every even integer greater than 2 can be written as the sum of two primes.

For example, 18 = 5 + 13 = 7 + 11.

Goldbach’s ternary conjecture, proved 2013: every odd integer greater than 5 can be written as the sum of three primes.

http://en.wikipedia.org/wiki/Goldbach’s_conjecture

Christian Goldbach (March 18, 1690 – November 20, 1764) was a German mathematician who also studied law. He is remembered today for Goldbach’s conjecture.

http://en.wikipedia.org/wiki/Christian_Goldbach

The twin prime conjecture (300 BC): there are infinitely many primes p such that p + 2 is also prime.

For example, 17 and 19 are twin primes.

http://en.wikipedia.org/wiki/Twin_prime

How many different Sudoku squares of size n × n are there?

6, 670, 903, 752, 021, 936, 960

valid 9 × 9 Sudoku squares. The problem is to find a formula for general n.

For more information see

http://en.wikipedia.org/wiki/Unsolved problems in mathematics 

Does a computer proof actually be a maths proof?

Physics ideas can be superceded by others but Maths is different

“The history of mathematics is a history of horrendously difficult problems being solved by young people too ignorant to know that they were impossible.”

Freeman Dyson, “Birds and Frogs”, AMS Einstein Lecture 2008

Freeman John Dyson FRS (born December 15, 1923) is an English-born American theoretical physicist and mathematician, famous for his work in quantum electrodynamics, solid-state physics, astronomy and nuclear engineering. Dyson is a member of the Board of Sponsors of the Bulletin of the Atomic Scientists.

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http://en.wikipedia.org/wiki/Freeman_Dyson

A personal example

Chord Diagrams and Walks in Wedges

How many ways are there to draw n chords connecting 2n points such that there are precisely m crossings?

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Example: how many ways are there to draw 3 chords with precisely 2 crossings?

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A general formula was given by Touchard in 1956

Jacques Touchard (1885 – 1968) was a French mathematician. In 1953, he proved that an odd perfect number must be of the form 12k + 1 or 36k + 9. In combinatorics and probability theory, he introduced the Touchard polynomials. He is also known for his solution to the ménage problem of counting seating arrangements in which men and women alternate and are not seated next to their spouses.

http://en.wikipedia.org/wiki/Jacques_Touchard

Walks in Wedges

How many ways are there to draw partially directed walks in a wedge starting at the origin, having m up-steps and ending at position (n, −n)?

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Example: how many ways are there to draw walks with 2 up-steps ending at position (3, −3)?

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A general formula was given in 2007 – identical to the 1956 formula!

Finding a direct proof?

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Walks in wedges and chord diagrams

The number of chord diagrams with n chords and m crossings is in bijection with the number of walks in a wedge with m up-steps and ending at (n, −n)

Posed as an open problem at a conference in China in 2007, two different proofs given in 2008 and 2011

Work is important in statistical mechanics and the configuration of polymers

In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other

http://en.wikipedia.org/wiki/Bijection,_injection_and_surjection

The 3n + 1 Problem

Statement of the Problem

Consider the following operation on an arbitrary positive integer:

If the number is even, divide it by two.

If the number is odd, triple it and add one.

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Form a sequence by performing this operation repeatedly, beginning with any positive integer.

Example: n = 6 produces the sequence

6, 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1

The Collatz conjecture is:

This process will eventually reach the number 1, regardless of which positive integer is chosen initially.

http://en.wikipedia.org/wiki/Collatz_conjecture

Some Examples

Examples:

n = 11 produces the sequence

11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

n = 27 produces the sequence

27,82,41,124,62,31,94,47,142,71,214,107,322,161,484,242,121,

364,182,91,274,137,412,206,103,310,155,466,233,700,350,175,

526,263,790,395,1186,593,1780,890,445,1336,668,334,167,502,

251,754,377,1132,566,283,850,425,1276,638,319,958,479,1438,

719,2158,1079,3238,1619,4858,2429,7288,3644,1822,911,2734,

1367,4102,2051,6154,3077,9232,4616,2308,1154,577,1732,866,

433,1300,650,325,976,488,244,122,61,184,92,46,23,70,35,106,

53,160,80,40,20,10,5,16,8,4,2,1

Graphing the Sequences

A graph of the sequence obtained from n = 27

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This sequence takes 111 steps, climbing to over 9000 before descending to 1.

Supporting Arguments for the Conjecture

Experimental evidence:

The conjecture has been checked by computer for all starting values up to

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A probabilistic argument:

One can show that each odd number in a sequence is on average 3/4 of the previous one, so every sequence should decrease in the long run.

This not a proof because Collatz sequences are not produced by random events

Iterating on Real Numbers

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Reduce an orbit by replacing 3n + 1 with (3n + 1)/2:

10, 5, 16, 8, 4, 2, 1, 4, 2, 1, . . . shortens to 10, 5, 8, 4, 2, 1, 2, 1, . .

The function graphed is given by

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A “cobweb” plot

A cobweb plot, or Verhulst diagram is a visual tool used in the dynamical systems field of mathematics to investigate the qualitative behaviour of one-dimensional iterated functions, such as the logistic map. Using a cobweb plot, it is possible to infer the long term status of an initial condition under repeated application of a map.

http://en.wikipedia.org/wiki/Cobweb_plot

Iterating on Complex Numbers

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3n + 1 on the Ulam Spiral

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The Ulam spiral, or prime spiral (in other languages also called the Ulam Cloth) is a simple method of visualizing the prime numbers that reveals the apparent tendency of certain quadratic polynomials to generate unusually large numbers of primes. It was discovered by the mathematician Stanislaw Ulam in 1963, while he was doodling during the presentation of a “long and very boring paper” at a scientific meeting. Shortly afterwards, in an early application of computer graphics, Ulam with collaborators Myron Stein and Mark Wells used MANIAC II at Los Alamos Scientific Laboratory to produce pictures of the spiral for numbers up to 65,000. In March of the following year, Martin Gardner wrote about the Ulam spiral in his Mathematical Games column; the Ulam spiral featured on the front cover of the issue of Scientific American in which the column appeared.

http://en.wikipedia.org/wiki/Ulam_spiral

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Stanisław Marcin Ulam (13 April 1909 – 13 May 1984) was a renowned Polish-American mathematician. He participated in America’s Manhattan Project, originated the Teller–Ulam design of thermonuclear weapons, invented the Monte Carlo method of computation, and suggested nuclear pulse propulsion. In pure and applied mathematics, he produced many results, proved many theorems, and proposed several conjectures.

http://en.wikipedia.org/wiki/Stanislaw_Ulam

”Mathematics is not yet ready for such problems.”

Paul Erd˝os, 1913 – 1996

Paul Erdős (26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was the most prolific mathematician of the 20th century, but also known for his social practice of mathematics (more than 500 collaborators) and eccentric lifestyle (Time Magazine called him The Oddball’s Oddball). Erdős pursued problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory.

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http://en.wikipedia.org/wiki/Paul_Erd%C5%91s

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