Exploring the Mathematical Universe

A team of mathematicians from 12 countries has begun charting the terrain of rich, new mathematical worlds. The mathematical universe is filled with both familiar and exotic items. The “L-functions and Modular Forms Database,” abbreviated LMFDB—a sophisticated web interface that allows both experts and amateurs to easily navigate its contents.

According to Benedict Gross, an emeritus professor of mathematics at Harvard University, “Number theory is a subject that is as old as written history itself. Throughout its development, numerical computations have proved critical to discoveries, including the prime number theorem, and more recently, the conjecture of Birch and Swinnerton-Dyer on elliptic curves. The LMFDB pulls together all of the amazing computations that have been done with these objects.

Prime numbers have fascinated mathematicians throughout the ages. The distribution of primes is believed to be random, but proving this remains beyond the grasp of mathematicians to date. Under the Riemann hypothesis, the distribution of primes is intimately related to the Riemann zeta function, which is the simplest example of an L-­function. The LMFDB contains more than twenty million L-­functions, each of which has an analogous Riemann hypothesis that is believed to govern the distribution of wide range of more exotic mathematical objects. Patterns found in the study of these L-­functions also arise in complex quantum systems, and there is a conjectured to be direct connection to quantum physics.

A recent contribution by Andrew Sutherland at MIT used 72,000 cores of Google’s Compute Engine to complete in one weekend a tabulation that would have taken more than a century on a single computer. The application of large-scale cloud computing to research in pure mathematics is just one of the ways in which the project is pushing forward the frontier of mathematics.

Reference: https://www.sciencedaily.com/releases/2016/05/160510084152.htm

 

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Supercomputing For a Superproblem: A Computational Journey into Pure Mathematics

One of the most reputable and respected mathematician known to have solved one of the subject’s most challenging problems has published his latest work as a University of Leicester research report.

This follows the visit that famed mathematician Yuri Matiyasevich made to the Department of Mathematics where he talked about his pioneering work. He visited UK by invitation of the Isaac Newton Institute for Mathematical Sciences.

In 1900, twenty-three unsolved mathematical problems, known as Hilbert’s Problems, were compiled as a definitive list by mathematician David Hilbert.

A century later, the seven most important unsolved mathematical problems to date, known as the ‘Millennium Problems’, were listed by the Clay Mathematics Institute. Solving one of these Millennium Problems has a reward of US $1,000,000, and so far only one has been resolved, namely the famous Poincare Conjecture, which only recently was verified by G. Perelman.

Yuri Matiyasevich found a negative solution to one of Hilbert’s problems. Now, he’s working on the more challenging of maths problems — and the only one that appears on both lists — Riemann’s zeta function hypothesis.

Professor Alexander Gorban, from the University of Leicester, said: “His visit was a great event for our mathematics and computer science departments.

“Matiyasevich has now published a paper through the University that regards the zeros of Riemann Zeta Function (RZF). This is a mathematical function which has been studied for over a hundred years.

“There is previous evidence of famous pure mathematical problems using massive computations. Unfortunately, the Riemann hypothesis is not reduced to a finite problem and, therefore, the computations can disprove but cannot prove it. Computations here provide the tools for guessing and disproving the guesses only.”

Reference: https://www.sciencedaily.com/releases/2012/11/121106125558.htm

 

Computers Unlock More Secrets of the Mysterious Indus Valley Script

Lots of artefacts left by an urban civilization living on what is now the border between Pakistan and India, have been discovered. Now a team of Indian and American researchers are using mathematics and computer science to try to piece together information about the still-unknown script.

The team used computers to extract patterns in ancient Indus symbols. The study shows distinct patterns in the symbols’ placement in sequences and creates a statistical model for the unknown language.

Despite dozens of attempts, nobody has yet interpreted the Indus script. The symbols are found on tiny seals, tablets and amulets, left by people inhabiting the Indus Valley from about 2600 to 1900 B.C. Each artefact is inscribed with a sequence that is typically five to six symbols long.

The new study shows that the order of symbols is meaningful; taking one symbol from a sequence found on an artefact and changing its position produces a new sequence that has a much lower probability of belonging to the hypothetical language.

Seals with sequences of Indus symbols have been found as far away as West Asia, specifically Mesopotamia and site of modern-day Iraq. The statistical results showed that the West-Asian sequences are ordered differently from sequences on artifacts found in the Indus valley. This supports earlier theories that the script may have been used by Indus traders in West Asia to represent different information compared to the Indus region.

They used a Markov model, a statistical method that estimates the likelihood of a future event based on past patterns.

One application described in the paper uses the statistical model to fill in missing symbols on damaged archaeological artifacts. Such filled-in texts can increase the pool of data available for deciphering the writings of ancient civilizations.

Reference: https://www.sciencedaily.com/releases/2009/08/090803185836.htm

 

Computer Scientist Reveals the Math and Science behind Blockbuster Movies

It’s clear that the computer-generated special effects in Pirates of the Caribbean and others breathe life to such fantasies. Amazingly, the amount of math and science behind such blockbusters baffles even the adept scientist.

Computer graphics (CG) experts used to have to make a Catch-22 decision. They could run inferior algorithms on many processors or run the best algorithm on only one processor. The problem is that many algorithms do not scale well to larger numbers of processors. But about a year and a half ago Fedkiw, who has consulted for ILM for six years, figured out how to run a star algorithm on many processors, resulting in special effects unprecedented in their realism.

He designs new algorithms for diverse applications such as computational fluid dynamics and solid mechanics, computer graphics, computer vision and computational biomechanics. The algorithms may rotate objects, simulate textures, generate reflections or mimic collisions. Or they may mathematically stitch together slices of a falling water drop, rising smoke wisp or flickering flame to weave realism into CG images.

Fedkiw received screen credits for his work on Poseidon, on Terminator 3: Rise of the Machines for the liquid terminator and the nuclear explosions, and on Star Wars: Episode III—Revenge of the Sith for explosions in space battle scenes.

Most of Fedkiw’s students double-major in math and computer science. “Graphics itself is a bit less important, and many of them don’t take their first graphics class until their junior or senior year of college.

Fedkiw’s favorite movie employing CG is Revenge of the Sith. “When I watched the first [Star Wars film] at 9 years old, I never dreamed that I’d eventually be helping to make the last one.” He says.

Reference: https://news.stanford.edu/news/2007/april4/fed-040407.html