A milestone in Kyota, Masaki Kashiwara’s house of mathematician, is a river Kamo. In some points, residents leave the stones that allow the river to move away from bridges. If you take them closer to these stones, you can see how the waters are formed and small edits around them. Describing this flow of a liquid is not easy. You need to solve complicated equations known for centuries, but today they have a lot of mysteries posed today: equations always have the solution? How can they be calculated? And what properties do they have? Mathematicians seem to have achieved a limit with their trading tools. To move forward, the new toolbox is required. The Japanese mathematician Masaki Kashiwara developed a question box in the 1970s.
Kashiwarak Algebra studied proven exams: the function of the functions, borders and other concepts in the underlying theory, created a completely new branch of mathematics: algebraic analysis. This led to important advances in several areas. For example, Kashiwarak XX. At the beginning of the 19th century, David Hilbert managed to resolve one of the problems created by mathematician and developed new techniques used in modern physics.
Kashwara has been amazing theorem with methods that no one could imagine. “Read the recently press release from the Norwegian Academy of Science and Letters.
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Kashiwara was born near Tokyo in 1947. At a young age he found a fan of math through the young puzzles known from the young tsurukamezan. These puzzles involve calculating the number of cranes and turtles: Consider x Heads and y The legs are visible. How many cranes and turtles are there? Kashiwara’s parents did not have much exposure to the abstract subject, but the young massaki enjoyed solving this problem using algebraic methods.
Here is an example: each crane and turtle has two and four legs respectively (y) – And they both have only one head (x). To calculate the number of cranes (k) and turtles (extermance), these equations must be solved: 2k + 4extermance = y and k + extermance = x. For example, if 16 legs and five heads are seen, there must be two cranes and three turtles.
Kashiwa realized that he had generalized such questions. He highlighted his achievements at school. When he met Mikio Sato Mathematics late, when he was a student at the University of Tokyo Sato, he offered it to this way to solve this problem. Kashiwara was at the right place at the right time: Sato and his colleagues were developing a completely new branch of mathematics, which combines two different areas: analysis and algebra.
Nothing is stationary
Kashiwara worked with his tutor in differential equations. Everything in our world is in motion; Nothing continues to be constantly. A giant mountain range like Himalayas also grows or chopping over time. Such changes can be mathematically expressed with the support of derivatives. It is based on equations with all physics derivatives, the so-called differential equations. These can be used to describe the speed of living things, the Moon’s route or the flow of the river Kamo.
Although differential equations can be written quickly, they are much more difficult to solve. In some special cases, the solution is known. At other times, however, there is no clear problem whether it can be solved. One of the most important unresolved problems in Mathematics is described whether they describe the behavior of whiteboard fluids. always solution. Despite the centuries in the field of study, many of the most serious problems remain unresolved.
When you are stuck with a problem, it sometimes helps to look at a different perspective. It is often helpful to go back and study the problem from a distance. In this case, specific details can be faded, but the overall structure of the subject becomes visible. This approach can not only help with practical daily problems, but also in mathematics.
A Japanese research team aimed at Sato won a similar view. The team wanted to study differential equations from another perspective. To do this, researchers left the study area and turned it into algebra. Algebra is much more abstract: the focus is not necessarily mathematical objects – in this case, equations and their derivatives, but rather behavior. As in physics, examining a new particle examining the interplay of different particles, the interplay of different equations should reveal new views. That is the idea below the algebraic study.
Therefore, instead of choosing the exact differential equation and examined in detail, Sato and his colleagues focused on the full class of these equations. In addition, differential equations allowed not only on the plane, but also moving in curved surfaces, as if trying to describe the river on a strange planet. This approach seems quite complex, but it opens completely new. This allows the general properties to derive in the class of differential equations that are not exposed to individual equations.
In the late 1960s, Sato organized a weekly seminar in which the participants worked together to develop the concepts of the new theory. Among all the experts was Kashiwara, then an eagerly attended a young student.
With a fast lane Process-Modules
In 1970 the master’s thesis was subject to Kashwaran Sato. Its task was to develop algebraic tools for analyzing objects. Then he was only 23 years old, he entered the so-called Kashwara Process-Modules that allow you to extract valuable information from differential equations. Process-Dermellations, such as equation solutions “to determine whether they are specialties, to determine whether the regions that are supposed to have infinite values. The modules can also be used to calculate the equations.
The results of the master’s thesis of Kashiwara created an area of algebraic analysis. He wrote in his Japanese research, however, was 25 years before he returned English and therefore was accessible to a wider audience.
After graduating, Kashiwara went to the University of Kyoto, where he continued to collaborate with Sato and won his doctorate. In doing so, he developed new methods established in his master’s thesis. “From 1970 to 1980, Kashiwara solved almost all basic questions Process-Modulu theory “, remembered by Pierre Schapira France 2008 Prepint paper It was based on 2007 speech. After completing the doctorate, Kashiwara put the position at Nagoya University, conducted research at the Massachusetts Institute Institute of Technology and returned to Japan to accept a teacher at Kyoto University in 1978.
With the help Process-Modules, Kashiwara solved one of the most important problems in 1980 in 1980, in 1980, one of the most important problems in the International Mathematics Congress in Paris. Hilbertek XX. It took into account the 21 trials for 19th-century research, 21. The problems deal with differential equations. The German mathematician always wanted to find a differential equation that he had his solutions with a curved surface. Kashiwara proved, in fact, in these cases, that it can be calculated proper differential equation.
Process-Modules have led to progress in various mathematics. But they are proving Helpful in physics. In 2023 Max Planck Mathematics Mathematics Mathematics in Mathematics in Leipzig, Germany and other experts used in sciences Process-Modules to evaluate quantum “comprehensive pathways”. These are the calculation of processes in particle accelerators, such as when the two protons collide, creating some new particles. They are very complex, can be seen as a solution for differential equations, which is why the methods of algebraic analysis can help determine their properties.
Symmetries and quantum groups
Kashiwara also had a great influence in other areas of mathematics. One of these is the theory of representation, used to describe symmetries. An object is symmetrical if the same after certain transformations (such as rotations or reflections). For example, an equilateral triangle can be rotated by 120 degree multiples without changing its shape. Delegation theories allow experts to calculate the transformations of symmetry: What happens, for example, if you make a rotation of 270 degrees with a reflection y-axis? Such questions can be answered in particular if you represent symmetry transformations using matrices: combining transformations corresponds to the multiplication of the corresponding matrices.
However, the correct representations cannot be found for all types of symmetry. During his work, many Kashiwa focused on continuous symmetry, known as a group of liars in Mathematics. He made significant progress in researching representations.
He also explored discrete “quantum groups” that are not constant. Such quantum groups have an important role in quantum physics. In the microscopic level, most quantities appear only in small parts; The world seems to be quantified in the smallest scale. To describe the symmetries of quantified quantities, Kashiwara presented the concept of crystals. These allow them to represent quantum groups through networks. This provides tremendous advantages through combinary reflections on answering questions about the theory of theory (organizing objects in a finite set), they are generally much easier. Since these concepts have proven in mathematics and physics.
“Masaki Kashiwar has been re-enriched by the Algewaric Analysis and the Theory of Norway. The mathematician has already been awarded the life of the Mathematician life. The Abel Award is modeled by Nobel Prizes, which do not include mathematics, and 7.5 million Norwegian Kroner (approximately $ 710,000).
78-year-olds do not seem to retire: It still publishes new research discoveries and trying to enrich math with new stones oppressed.
