25.01.2011

A solution that counts: long-standing mathematical conjecture finally proved

A conjecture presented in 1985 - the Andrews and Robbins conjecture - has recently been proved for the first time. It is thus clear that the structure which goes by the name of "totally symmetric plane partitions" can be described using a single formula. Producing the proof required vast computer resources and was only possible after the formula had been prepared for computer-assisted calculation.

This finding by a Austrian Science Fund FWF supported research group based in Linz, Austria will be published in the Proceedings of the National Academy of Sciences today. The proof means that the last of a long list of famous mathematical conjectures relating to plane partitions has finally been proved.

Even mathematicians play with building blocks. At least if they are interested in so-called plane partitions, which are visualized with columns of "building blocks" (cubes) on a surface resembling a chessboard. When "building" such plane partitions, the mathematicians must adhere to certain rules: No column may be higher than the width of the surface, or than another column behind it or left of it. The question of how many column permutations may be built on a given surface area is easily answered, thanks to a specific formula. However, it becomes trickier if the permutations must follow stipulated symmetries, or if, instead of counting the permutations, you wish to count its constituents. Although formulas have been designed to do this too, the crux of the matter is that not all of these formulas have really been proved to be accurate. It is only conjectured.

... more about:

»Calculates »Computation »Conjecture »Correct »FWF »Science TV »Symbolic »Symbolic Computation »building block

»Calculates »Computation »Conjecture »Correct »FWF »Science TV »Symbolic »Symbolic Computation »building block

The proof that one of these formulas is correct has now been found by Dr. Christoph Koutschan and Dr. Manuel Kauers from the Research Institute for Symbolic Computation of the Johannes Kepler University Linz, Austria, in cooperation with Prof. Doron Zeilberger from the United States. It is a formula that is employed for calculating the individual components in totally symmetric plane partitions. Dr. Koutschan comments on the special method they used to find the proof: "We let the computer do the work! In some areas of mathematics this has long been a matter of routine." The underlying principle of such computer-assisted proof is simple. In order to prove A=B, the computer calculates an adjoint equation U=V with the following two properties: "If U=V is true, then A=B is also true" and "it is easy to verify that U=V".

Although it may sound easy, it represents a great challenge, according to Dr. Koutschan: "This method does not work for every equation. The most important step was for us to convert the Andrews-Robbins conjecture into a suitable form for the computer to be able to prove it." The fact that the adjoint equation was really somewhat more complex than "U=V", is illustrated by its size: if it were printed, it would cover approximately 1 million A4 pages, which makes it probably the longest equation ever used in a mathematical proof.

STANLEY`S LISTIn the end, the work that was spent on the "formulation" was well worth it. With the proof of the Andrews and Robbins conjecture, the scientists have managed to prove the last of a number of famous conjectures, which were presented by US mathematician Richard Stanley at a historic conference in Montreal in 1985. In the years following the conference, all of these conjectures were proved except for the Andrews and Robbins conjecture. Dr. Kauers comments: "As the last remaining item on Stanley s list, this conjecture attracted the attention of many experts. Still, it remained unproved for almost thirty years. The proof was finally obtained with an automatic method, which goes to show that modern computer programs can crack mathematical problems where traditional mathematicians fall short."

Granted, such successful results are still an exception. However, this FWF project underscores the potential of computer-based proof. Given the great pace at which computer performance is advancing, such methods will perhaps one day even offer answers to the great unsolved questions in mathematics.

Image and text will be available from Tuesday, 25th January 2011, 9 am CET onwards:http://www.fwf.ac.at/en/public_relations/press/pv201101-2en.html

Original publication: A proof of George Andrews` and David Robbins` q-TSPP conjecture. C. Koutschan, M. Kauers, D. Zeilberger. DOI: 10.1073/pnas.1019186108

Scientific contact:PD Dr. Manuel Kauers

Johannes Kepler University

Research Institute for Symbolic Computation Altenberger Straße 69 4040 Linz, Austria

T: +43/732/2468 9958

E: Manuel.Kauers@risc.jku.at

Austrian Science Fund (FWF)

Mag. Stefan Bernhardt

Haus der Forschung

Sensengasse 1

1090 Vienna, Austria

T: +43/1/5056740-8111

E: stefan.bernhardt@fwf.ac.at

W: http.//www.fwf.ac.at

Copy Editing & Distribution

PR&D - Public Relations for Research & Education Mariannengasse 8 1090 Vienna, Austria

T: +43 / 1 / 505 70 44

E: contact@prd.at

W: http://www.prd.at

Jacqueline Bogdanovic | PR&D

Further information:

http://www.fwf.ac.at

**Further reports about:**
> Calculates
> Computation
> Conjecture
> Correct
> FWF
> Science TV
> Symbolic
> Symbolic Computation
> building block

Cloud technology: Dynamic certificates make cloud service providers more secure

15.01.2018 | Technische Universität München

New discovery could improve brain-like memory and computing

10.01.2018 | University of Minnesota

On the way to an intelligent laboratory, physicists from Innsbruck and Vienna present an artificial agent that autonomously designs quantum experiments. In initial experiments, the system has independently (re)discovered experimental techniques that are nowadays standard in modern quantum optical laboratories. This shows how machines could play a more creative role in research in the future.

We carry smartphones in our pockets, the streets are dotted with semi-autonomous cars, but in the research laboratory experiments are still being designed by...

What enables electrons to be transferred swiftly, for example during photosynthesis? An interdisciplinary team of researchers has worked out the details of how...

For the first time, scientists have precisely measured the effective electrical charge of a single molecule in solution. This fundamental insight of an SNSF Professor could also pave the way for future medical diagnostics.

Electrical charge is one of the key properties that allows molecules to interact. Life itself depends on this phenomenon: many biological processes involve...

At the JEC World Composite Show in Paris in March 2018, the Fraunhofer Institute for Laser Technology ILT will be focusing on the latest trends and innovations in laser machining of composites. Among other things, researchers at the booth shared with the Aachen Center for Integrative Lightweight Production (AZL) will demonstrate how lasers can be used for joining, structuring, cutting and drilling composite materials.

No other industry has attracted as much public attention to composite materials as the automotive industry, which along with the aerospace industry is a driver...

Scientists at Tokyo Institute of Technology (Tokyo Tech) and Tohoku University have developed high-quality GFO epitaxial films and systematically investigated their ferroelectric and ferromagnetic properties. They also demonstrated the room-temperature magnetocapacitance effects of these GFO thin films.

Multiferroic materials show magnetically driven ferroelectricity. They are attracting increasing attention because of their fascinating properties such as...

Anzeige

Anzeige

Event News

08.01.2018 | Event News

See, understand and experience the work of the future

11.12.2017 | Event News

Innovative strategies to tackle parasitic worms

08.12.2017 | Event News

Latest News

19.01.2018 | Materials Sciences

How cancer metastasis happens: Researchers reveal a key mechanism

19.01.2018 | Health and Medicine

Meteoritic stardust unlocks timing of supernova dust formation

19.01.2018 | Physics and Astronomy

VideoLinks

NASA | A Year in the Life of Earth's CO2

NASA Computer Model Provides a New Portrait of Carbon Dioxide

Black Holes Come to the Big Screen

The new movie "Interstellar" explores a longstanding fascination, but UA astrophysicists are using cutting-edge technology to go one better.

NASA's Swift Mission Observes Mega Flares from a Mini Star

NASA's Swift satellite detected the strongest, hottest, and longest-lasting sequence of stellar flares ever seen from a nearby red dwarf star.

NASA | Global Hawks Soar into Storms

NASA's airborne Hurricane and Severe Storm Sentinel or HS3 mission, will revisit the Atlantic Ocean for the third year in a row.

Baffin Island - Disappearing ice caps

Giff Miller, geologist and paleoclima-tologist, is walking the margins of melting glaciers on Baffin Island, Nunavut, Canada.

The Infrasound Network and how it works

The CTBTO uses infrasound stations to monitor the Earth mainly for atmospheric explosions.

B2B-VideoLinks

Special emitters for optimal energy efficiency

Heraeus special emitters promote both: energy production and energy saving

Gascatalytical infrared heat ...

... can save time, space and money by drying coatings with infrared heat

Efficient reduction of odour and grease with Heraeus UV solutions

Kitchen exhaust air cleaning with UV in gastronomy

Drying and curing of paints on glass and ceramics

Bright and brilliant paints on glass and ceramics require safe solutions for drying and curing.

JULABO World of Temperature

Explore the World of Temperature with JULABO - Superior Temperature Technology for a Better Life.

Acoustic Wave Separation: How It Works

In this animated video, see how Acoustic Wave Separation technology works in full detail.

Infrared Heat for printed electronics

Drying and sintering of printed electronics by specialty light sources from Heraeus

All about Data Logger, how to use

Wolfgang Rudolph explains: all information worth knowing about the data logger and the practical test by means of a drone