In 2000, the Clay Mathematics Institute announced the Millennium Prize problems. These were a collection of seven of the most important maths problems that remain unsolved.
Reflecting the importance of the problems, the Institute offered a $US1 million ($1.3 million) prize to anyone who could provide a rigorous, peer-reviewed solution to any of the problems.
While one of the problems, the Poincare Conjecture, was famously solved in 2006 (with the mathematician who solved it, Grigori Perelman, equally famously turning down both the million dollar prize and the coveted Fields Medal), the other six problems remain unsolved.
Here are the six maths problems so important that solving any one of them is worth a cool million.P vs NP
Some problems are easy, and some problems are hard.
In the world of maths and computer science, there are a lot of problems that we know how to program a computer to solve “quickly” – basic arithmetic, sorting a list, searching through a data table. These problems can be solved in “polynomial time,” abbreviated as “P.” It means the number of steps it takes to add two numbers, or to sort a list, grows manageably with the size of the numbers or the length of the list.
But there’s another group of problems for which it’s easy to check whether or not a possible solution to the problem is correct, but we don’t know how to efficiently find a solution. Finding the prime factors of a large number is such a problem – if I have a list of possible factors, I can multiply them together and see if I get back my original number. But there is no known way to quickly find the factors of an arbitrary large number. Indeed, the security of the Internet relies on this fact.
For historical and technical reasons, problems where we can quickly check a possible solution are said to be solvable in “nondeterministic polynomial time,” or “NP.”
Any problem in P is automatically in NP – if I can solve a problem quickly, I can just as quickly check a possible solution simply by actually solving the problem and seeing if the answer matches my possible solution. The essence of the P vs NP question is whether or not the reverse is true: If I have an efficient way to check solutions to a problem, is there an efficient way to actually find those solutions?
Most mathematicians and computer scientists believe the answer is no. An algorithm that could solve NP problems in polynomial time would have mind-blowing implications throughout most of maths, science, and technology, and those implications are so out-of-this-world that they suggest reason to doubt that this is possible.
Of course, proving that no such algorithm exists is itself an incredibly daunting task. Being able to definitively make such a statement about these kinds of problems would likely require a much deeper understanding of the nature of information and computation than we currently have, and would almost certainly have profound and far-reaching consequences.
Read the Clay Mathematics Institute’s official description of P vs NP here.The Navier-Stokes equations
It’s surprisingly difficult to explain what happens when you stir cream into your morning coffee.
The Navier-Stokes equations are the fluid-dynamics version of Newton’s three laws of motion. They describe how the flow of a liquid or a gas will evolve under various conditions. Just as Newton’s second law gives a description of how an object’s velocity will change under the influence of an outside force, the Navier-Stokes equations describe how the speed of a fluid’s flow will change under internal forces like pressure and viscosity, as well as outside forces like gravity.
The Navier-Stokes equations are a system of differential equations. Differential equations describe how a particular quantity changes over time, given some initial starting conditions, and they are useful in describing all sorts of physical systems. In the case of the Navier-Stokes equations, we start with some initial fluid flow, and the differential equations describe how that flow evolves.
Solving a differential equation means finding some mathematical formula to determine what your quantity of interest actually will be at any particular time, based on the equations that describe how the quantity changes. Many physical systems described by differential equations, like a vibrating guitar string, or the flow of heat from a hot object to a cold object, have well-known solutions of this type.
The Navier-Stokes equations, however, are harder. Mathematically, the tools used to solve other differential equations have not proven as useful here. Physically, fluids can exhibit chaotic and turbulent behaviour: Smoke coming off a candle or cigarette tends to initially flow smoothly and predictably, but quickly devolves into unpredictable vortices and whorls.
It’s possible that this kind of turbulent and chaotic behaviour means that the Navier-Stokes equations can’t actually be solved exactly in all cases. It might be possible to construct some idealised mathematical fluid that, following the equations, eventually becomes infinitely turbulent.
Anyone who can construct a way to solve the Navier-Stokes equations in all cases, or show an example where the equations cannot be solved, would win the Millennium Prize for this problem.
Read the Clay Mathematics Institute’s official description of the Navier-Stokes equations here.Yang-Mills theory and the quantum mass gap
Maths and physics have always had a mutually beneficial relationship. Developments in mathematics have often opened new approaches to physical theory, and new discoveries in physics spur deeper investigations into their underlying mathematical explanations.
Quantum mechanics has been, arguably, the most successful physical theory in history. Matter and energy behave very differently at the scale of atoms and subatomic particles, and one of the great achievements of the 20th century was developing a theoretical and experimental understanding of that behaviour.
One of the major underpinnings of modern quantum mechanics is Yang-Mills theory, which describes the quantum behaviour of electromagnetism and the weak and strong nuclear forces in terms of mathematical structures that arise in studying geometric symmetries. The predictions of Yang-Mills theory have been verified by countless experiments, and the theory is an important part of our understanding of how atoms are put together.
Despite that physical success, the theoretical mathematical underpinnings of the theory remain unclear. One particular problem of interest is the “mass gap,” which requires that certain subatomic particles that are in some ways analogous to massless photons instead actually have a positive mass. The mass gap is an important part of why nuclear forces are extremely strong relative to electromagnetism and gravity, but have extremely short ranges.
The Millennium Prize problem, then, is to show a general mathematical theory behind the physical Yang-Mills theory, and to have a good mathematical explanation for the mass gap.
Read the Clay Mathematics Institute’s official description of the Yang-Mills theory and mass gap problem here.The Riemann Hypothesis
Going back to ancient times, the prime numbers – numbers divisible only by themselves and 1 – have been an object of fascination to mathematicians. On a fundamental level, the primes are the “building blocks” of the whole numbers, as any whole number can be uniquely broken down into a product of prime numbers.
Given the centrality of the prime numbers to mathematics, questions about how primes are distributed along the number line – that is, how far away prime numbers are from each other – are active areas of interest.
By the 19th century, mathematicians had discovered various formulas that give an approximate idea of the average distance between primes. What remains unknown, however, is how close to that average the true distribution of primes stays – that is, whether there are parts of the number line where there are “too many” or “too few” primes according to those average formulas.
The Riemann Hypothesis limits that possibility by establishing bounds on how far from average the distribution of prime numbers can stray. The hypothesis is equivalent to, and usually stated in terms of, whether or not the solutions to an equation based on a mathematical construct called the “Riemann zeta function” all lie along a particular line in the complex number plane. Indeed, the study of functions like the zeta function has become its own area of mathematical interest, making the Riemann Hypothesis and related problems all the more important.
Like several of the Millennium Prize problems, there is significant evidence suggesting that the Riemann Hypothesis is true, but a rigorous proof remains elusive. To date, computational methods have found that around 10 trillion solutions to the zeta function equation fall along the required line, with no counter-examples found.
Of course, from a mathematical perspective, 10 trillion examples of a hypothesis being true absolutely does not substitute for a full proof of that hypothesis, leaving the Riemann Hypothesis one of the open Millennium Prize problems.
Read the Clay Mathematics Institute’s official description of the Riemann Hypothesis here.The Birch and Swinnerton-Dyer conjecture
One of the oldest and broadest objects of mathematical study are the diophantine equations, or polynomial equations for which we want to find whole-number solutions. A classic example many might remember from high school geometry are the Pythagorean triples, or sets of three integers that satisfy the Pythagorean theorem x2 + y2 = z2.
In recent years, algebraists have particularly studied elliptic curves, which are defined by a particular type of diophantine equation. These curves have important applications in number theory and cryptography, and finding whole-number or rational solutions to them is a major area of study.
One of the most stunning mathematical developments of the last few decades was Andrew Wiles’ proof of the classic Fermat’s Last Theorem, stating that higher-power versions of Pythagorean triples don’t exist. Wiles’ proof of that theorem was a consequence of a broader development of the theory of elliptic curves.
The Birch and Swinnerton-Dyer conjecture provides an extra set of analytical tools in understanding the solutions to equations defined by elliptic curves.
Read the Clay Mathematics Institute’s official description of the Birch and Swinnerton-Dyer conjecture here.The Hodge conjecture
The mathematical discipline of algebraic geometry is, broadly speaking, the study of the higher-dimensional shapes that can be defined algebraically as the solution sets to algebraic equations.
As an extremely simple example, you may recall from high school algebra that the equation y = x2 results in a parabolic curve when the solutions to that equation are drawn out on a piece of graph paper. Algebraic geometry deals with the higher-dimensional analogues of that kind of curve when one considers systems of multiple equations, equations with more variables, and equations over the complex number plane, rather than the real numbers.
The 20th century saw a flourishing of sophisticated techniques to understand the curves, surfaces, and hyper-surfaces that are the subjects of algebraic geometry. The difficult-to-imagine shapes can be made more tractable through complicated computational tools.
The Hodge conjecture suggests that certain types of geometric structures have a particularly useful algebraic counterpart that can be used to better study and classify these shapes.
Read the Clay Mathematics Institute’s official description of the Hodge conjecture here.
This story Administrator ready to work first appeared on Nanjing Night Net.
Newcastle City Council interim CEO Jeremy Bath at the Stewart Avenue building where Newcastle City Council administration will be based. Photo: Simon McCarthyNewcastle City Council chief executive officer Jeremy Bath is no fan of the word “council” and wants itscrapped from the local government area’s title.
Mr Bath, who was appointed last week after acting in an interim capacity since May, told the Newcastle Herald that the council’s planned move to the west end in 2019 could provide an opportunity for a name change.
“I don’t like the word council. I think it’s tired;it’s outdated.I think it stands for a level of bureaucracy and red tape that we’re very much committed to getting away from,” he said.
“I’m far more comfortable in telling people that I’m the CEO of the City of Newcastle rather than I’m the CEO of Newcastle City Council.
“I’m not unique in thinking that way. You’ve only got to look at other major councils, such as the City of Sydney, who came to that realisation 10 years ago.”
Some local government areasin Australia, including City of Melbourne and City of Gold Coast, brand themselves without the word “council” but also refer to themselves as councils.
The City of Perth and City of Sydney generally eschew the term “council”, but Brisbane City Council and Wollongong City Council embraceit.
Newcastle City Council was known as City of Newcastle from 1949 to 1993 before adopting its present official name, although it still uses the former title in its logo.
Mr Bath said the move to a new building in Stewart Avenue was primarily about bringing council employees under one roof in better office accommodation.But he said the location shift anddropping “council” from his organisation’s titlewould reflect its agenda of transforming Newcastle into a more modern city.
“When you survey people, their feelings about council are very tired, they’re very negative,” he said.
“As we change and present aculture of this organisation that’s far more receptive to community needs, and, I guess, that’s far more bold in its thinking and its bravery in its decision making, we need to communicate that change to people.
“My personal view is that the term council is very much a ’70s term.”
The council will have to change its address on digital and printed letterheads when it moves, which could provide an opportunity for rebranding.
“It will, but right now that’s not a focus,” Mr Bath said.
“It’s certainly something I’m thinking about, but I don’t have a timetable, and that’s certainly something I need to have a discussion with the elected councillors about, and I haven’t done that.”
Good Food. 14th of November 2017. Labld in Marrickville. Photo: Dominic LorrimerFirst-home owners – whether single, a couple, or a young family – don’t necessarily need a property with a SMEG kitchen and marble benchtops; but they do want to get into the market in a location that will see their home grow in value, and offer a good lifestyle.
For many, even a fixer-upper has been out of reach for the last four years. Fortunately for first home buyers, new developments are springing up everywhere in and around Sydney.
Buyer’s Agent Peter Kelaher, owner of PK properties Sydney, believes the next three years will be an exciting time for first-home buyers. “There’s an oversupply of new units coming onto the market, so developers will be forced by the banks to meet the market price wise ??? it will be very interesting,” says Kelaher.
Even now, first-home buyers can find a sweet deal in Sydney in a great neighbourhood. Here are our top five picks.
608/39-47 Orara Street, Waitara NSW Price guide: $690,000 to $720,000
Waitara is the most liveable suburb on the leafy green upper north shore, according to the 2016 Domain’s Liveability study, and has plenty to offer first-home buyers. Just over 70 per cent of long-term residents are families, meaning primary schools and even local shops put on regular events, and plenty of community support is on offer for parents.
Peter Kelaher has been selling in the area for 20 years. “Waitara has really good access to all the private schools, great rail access to the north shore, out to Parramatta and up to the central coast, and it’s a five-minute drive to Hornsby Westfield and shopping district. New unit properties offer bushland and district views,” he says.
Kelaher says Waitara is on the verge of big things. Residential development in Waitara is peaking, making it a great spot to invest: “Basically there’s no more land to build units on up there, so once everything is sucked up you are in a position where you can’t have anything else built up against you.”
For those first-home owners who want to be close to the action, Inner West gem Marrickville offers easy access to the city via train, plus loads of action close to home, with its multicultural heritage of Vietnamese and Greek families who still run friendly groceries, and a new wave of hip cafes, wine bars and restaurants.
The Property Sellers’ Kate Webster is a local Marrickville resident and sells a lot of homes to young families. “There are slightly larger blocks and more freestanding homes [than other inner west suburbs], leafy broad streets and wide, stroller friendly footpaths,” she says.
Good Food. 14th of November 2017. Labld in Marrickville. Photo: Dominic Lorrimer Photo: Dominic Lorrimer
“I moved here when I had my first child,” says Webster; “I bought a two bedroom freestanding house for the same price I sold my one bedroom Bondi flat and never looked back. My kids are now 20 and 19, they love the area and they do not want to move.”
Botany in south eastern Sydney might just be the city’s best kept secret. Adjacent to the two kilometre bayside suburb of Brighton-le-Sands, most of the former industrial suburb is now residential with both older homes and new unit complexes.
Just 10 kilometres from the city, five minutes from Maroubra and nudging the inner west, Botany attracts buyers who want the eastern suburbs lifestyle without the price tag.
Ausin Group’s Natalie Stathis is selling spacious new units in at Pemberton on the park development and feels that the suburb is becoming the next Newtown.
Botany Bay. Photo: Ben Rushton/Fairfax Media.
“When I was a kid growing up in the eastern suburbs, you wouldn’t have thought to move to Botany, but now it’s a no brainer!” she says; “Why not? With the proximity to the CBD and airport and Sydney’s best university [UNSW] plus good high schools like Sydney boys and Sydney girls, Botany is really undervalued.”
“The suburb is changing really quickly” says Stathis “but it’s so friendly, it’s really still like a small town.”
4. North Parramatta
52/34 Albert Street, North Parramatta NSW Price guide: $650,000-$700,000
Parramatta, once a farming settlement and now the geographical heart of Sydney is the cultural centre of the west, with sensational arts and dining, excellent infrastructure including train and ferry access to Sydney and a thriving multicultural community.
Deanna Martinez is a mother of three and finds Parramatta to be a family friendly hub: “We’re half an hour from the city. My family has a farm in Mudgee so we go out there a lot. There’s a lot on for families in our area as well a lot of kid friendly places we go to, with a cr??che where the mums can sit down and have a coffee,” she says.
Across the river, leafier North Parramatta offers first-home buyers the chance to invest in a little more space. “North Parra” is packed with heritage listed buildings, has lovely parks and is connected to Parramatta CBD by Church Street and its eclectic dining scene.
32 Valder Ave, Richmond NSW Price guide: $665,000 to $695,000
Leafy Richmond on the Hawkesbury River at the foot of the Blue Mountains offers a taste of bush country life, a little more than an hour to Sydney and just under an hour from Parramatta.
Home to an air force base and university campus, buyers’ agent Peter Kelaher says an increasing amount of land is becoming available around Richmond, “The government has been going nuts with regards to land releases and people are able to get themselves a very nice sized block for a reasonable price,” he says.
US immigrant Heather Torrey was living in Pyrmont with water views but when she started a family she decided to move.
“The rent continued to climb while we were spending at least six hours a week driving out to Richmond to ride our horses during the polo season,” says Torrey. “For the same weekly cost, we moved from our 3 bedroom, 175 square-metre apartment rental to a 5 bedroom 315 square-metre Richmond home. Day care went from $120 per day to $84. We were welcomed to the neighbourhood with a fruit basket.”
Torrey says the village of Richmond is just as convenient as Pyrmont: “Shopping, cafes and public transport are still a 10-minute walk. We don’t have bridge views anymore, but we can see the stars at night and have the freedom to make our space our home.”
This story Administrator ready to work first appeared on Nanjing Night Net.
SHOWTIME: Drive-In owner Scott Seddon in a classic 1950 MG, ready for the 50th anniversary screening of To Sir With Love. Picture: Max Mason-Hubers.Heddon Greta Drive-In is usually closed for four weeks’ maintenance at this time of year, but will reopen on Wednesday for aspecial event.
The night will mark 50 years since the drive-in first opened, and it will celebrate with a screening of the film shown on its opening night –the classic British dramaTo Sir, With Love.
Owner Scott Seddon said he wanted to make the night about more than just the drive-in, so he sought a local charity to partner with for the event.
The nightwill benefit Richmond Vale Railway Museum, which wasdevastated by a bushfirein September.
The blazeleft the museum with an estimated $1 million damage bill, withtwo kilometres of track,stainless steel passenger cars,a 100-year-old break van andat least 30 hoppers among the losses.
The Rebuilding Richi appeal was establishedto raise moneyfor therestoration of the site to operation and exhibits for display.
About $26,000 has already been raised through corporate and individual donations, online fundraising, a trivia night and guessing competition.
It’s a small portion of what will be required to get the museum fully operational again –bridge repairs alone are estimated at $50,000.
Museum volunteer Graham Smith said they were thrilled to have the drive-in’s support.
“It’s tremendous –it came to us out of the blue, and we really appreciate Scott’s generosity,” he said.
Coincidentally,Richmond Main mine ceased shipping coal in July 1967, just five months beforethe drive-in opened.
The drive-in closed twice in the 1980s and ‘90s, before MrSeddon bought and re-opened the complex in 1996.
It is now one of only two drive-in movie theatres still operating in NSW–the other is in Blacktown.
Mr Seddon said the drive-in offers a chance for moviegoers to “experience something different”.
He said the opening of the Hunter Expressway has brought a lot more young people to the drive-in.
The 50thanniversary screening ofTo Sir, With Lovewill be held on Wednesday at 8.30pm.
Admission will be $30 per car, with proceeds going to theRebuilding Richi Appeal.
The drive-in will reopenon Boxing Day.
This story Administrator ready to work first appeared on Nanjing Night Net.
Professor Trevor Waring with Everymind’s current director Jaelea Skehan.
I am feeling particularly reflective and grateful this week as Everymind marks its 25th anniversary. I have been privileged to be part of that journey for 16 years, with five years as the director.
Everymind, formerly the Hunter Institute of Mental Health, was established by the Hunter New England Local Health District in 1992 as a solution to the education and training needs of the mental health workforce and the broader community in the Hunter.
Led by well-known psychologist, and my first mentor, Professor Trevor Waring, the institute was self-funding from the first year and broadened its scope to deliver state-wide and national programs within its first five years.
The growth and reputation of the institute expanded under Trevor Hazell’s leadershipbetween 2004 and 2012, as the flagship prevention programs working with the media, universities, schools, families and the community gained traction.In 2017, we have more than 40 staff designing and delivering cutting-edge programs in suicide prevention, innovation in workplace and family mental health, and we are set to co-lead a nationally funded Prevention Hub.
I am really proud of the work we have done, and are doing, at Everymind. This includes two decades of internationally recognised work with media in Australia to improve the reporting and portrayal of suicide and mental illness as part of the Mindframe Initiative.
The institute has pioneered pre-service training approaches for teachers and early childhood workers, and have developed practical community and family programs like Partners in Depression and Conversations Matter.We have also led improvements in workplace mental health approaches, with some of our current priorities focused on the mental health and wellbeing of small business owners, early career teachers, and medical professionals.
Everymind is one of a consortium of services locally that is leading the first state trial of Lifespan – an integrated all of system and all of community approach to suicide prevention. Plus we are working with partners such as Hunter Primary Care to support the development and evaluation of the Way Back Support Service in Newcastle, and a complementary family support program for those impacted by a suicide attempt.
I have often been asked, how easy is it to run a nationally focused institute out of Newcastle? The answer – very easily. Not only does Newcastle have a track record for research and innovation, I believe that it is a unique point of difference that Everymindworks from a regional base.
With a reform agenda that is advocating for national leadership and regional implementation, I believe it is a strength to understand how national or state programs work at the local level, and how prevention-focused work can best support and integrate with the service system. I will often sit in a national meeting when strategies are being discussed and think about whether that would work for Hunter communities.
As part of our 25 year celebrations, I am delighted to announce a new annual event for the region as a legacy to Professor Trevor Waring, the founding director of our institute.Everymind will partner with Hunter New England Health and the University of Newcastle to deliver an Expert in Residence Series and a Trevor Waring Memorial Lecture annually from 2018. The event will acknowledge ProfessorWaring’scontribution, and I couldn’t think of a better way to mark a significant anniversary.