29 Jul Major Mathematical Achievments of the Recent Years

In the last few years, math has seen a constant flow of answers (sometimes partial ones) to difficult mathematical questions that been a source of confusion for mathematicians for ages. From solving ancient problems to identifying new breakthroughs, here are several stunning major mathematical highlights.

The Sphere Packing Problem is Solved in Higher Dimensions

The Ukrainian mathematician, Maryna Viazovska has found two high-dimensional solutions to the long-standing “sphere packing” problem. In dimensions 8 and 24, she has demonstrated the existence of two highly-symmetrical configurations that pack spheres in the most compact way.

Mathematicians have accumulated convincing evidence that E8, as well as the Leech lattice (clarification on these confusing terms is available at StudyCrumb), are the most effective packings of spheres in their dimensions. However, until recently, this evidence was still not enough. For over ten years, the missing component for the proof has been well-known – the “auxiliary” function that can calculate the highest allowed sphere density – but it constantly escaped identification.

In a new paper published online on the 14th of March in 2016, Maryna Viazovska, then a postdoc at the Berlin Mathematical School, has discovered this missing element in dimension 8. Her research is based on the theory of modular structures, which provide powerful mathematical functions that can reveal massive amounts of data. In this case, identifying the most appropriate modular form enabled Viazovska to demonstrate, in just 23 pages, the fact that E8 is the most efficient eight-dimensional packing. 

Progress on the Riemann Hypothesis

The Riemann Hypothesis is naturally regarded as the largest and most elusive problem in modern math. It has been in existence since 1859 and is linked to the way prime numbers function and a variety of other mathematical branches.

Four mathematicians, Ken Ono, Don Zagier, Larry Rolen, and Michael Griffin, have delivered a remarkable result believed to be on the road toward proving the most renowned of mathematical mysteries remaining unsolved, that is The Riemann hypothesis.

In a fascinating research paper, they have revived an old logic, that was thought to be entirely out of use, initially invented in the late 1960s by Johan Jensen and George Polya. Their line of thought relies upon “Jensen polynomials.”

Ken Ono emphasizes that he, along with the other authors, did not develop any new methods or mathematical terms. In fact, the benefit of their work is its plainness (the paper is just eight pages in total!).

The Collatz Conjecture 

Another one of the most challenging open math issues got closer to a solution in 2019. Auspicious results announced by the renowned productive mathematician Terrence Tao (whose body of work you can get acquainted with at studybounty.com) stunned math society.

At its core, Tao’s research suggests that any counterexamples to Collatz Conjecture are going to be extremely uncommon. There’s a deeper significance to the rarity we’re discussing, but it’s, obviously, far from being nonexistent.

Even with Dr. Tao’s most recent insights, the issue is not yet closed and could require years, or even decades, to fully unravel. 

The Sensitivity Conjecture 

Presented in 1994, the Sensitivity Conjecture has been an extremely crucial and formidable open issue in the field of theoretical computer science for over three decades. The conjecture was solved in 2019 thanks to Professor Hao Huang. In the timespan of chaotic few weeks after the announcement, scientists recapped Dr. Huang’s findings in one page of amazingness.

The Sensitivity Conjecture refers to boolean data that converts information into a true-false (1-0) binary. Boolean functions are essential for the field of complexity theory and also for the creation of circuits and chips for digital computers.

Huang came up with an algebraic method to prove the theory. “I expect this technique might also have some prospect to be utilized in other combinatorial and complexity problems significant to computer science,” Huang declares. 

A Breakthrough in Ramsey Theory

In Ramsey Theory, mathematicians search for patterns that can be predicted within huge quantities of chaos. One particular issue from 1969 was solved; the researchers explained the answer using an interesting metaphor of an “ever-winning lottery ticket .”

Are there lottery tickets that never lose? That’s the popular take on a mathematical puzzle that was made in the year 1969 by English mathematician Adrian Mathias within the realm of set theory, a subject that deals with mathematical infinity.

The mystery remained throughout the 1970s, 1980s, and 1990s while set theorists from all over the world attempted to resolve it. Professor Asger Dag Tornquist was introduced to the issue in 2002 as he was completing his dissertation at the University of California. In the course of five years study, The researcher found that no complete coincidence could exist.