Sep 28, 2025
3 min read
Using the teenage concept, the group of advances begins to answer the questions picked at the start of mathematics.
The new mathematics refresh the old geometric problems
In the third century BC PAG, asked how many circumcision could be drawn by one point.
The question requesting a set of geometric conditions, is the most favorite of ancient Greek.Sheldon Katz.
As math continued, the desired items to calculate mathematicians became more complex.It became the field of their own study, known as the geometry of mathematics.
It seemed that mathematicians could come up with geometry problems read.But by the mid-20th century, mathematicians had begun to lose interest.Geometers have moved beyond the specific number related problems and focused on more general abstractions and deeper truths.
Which can change now.Mathematicians have learned how to apply the theory of decades to the questions listed.Researchers offer a solution not only to the original problem, but also to the problem version in an infinite exotic numerical system.
The idea facilitated the revival of the geomet field, which operates and coordinates the number of other learning, including algebra, the most important functions, far away from them.
At the same time, these results come when questions.At the same time, the numbers you are looking for, but it also provides more information that they will fight to respond.
The key word has adapted the new talent of talent to participate.
All the geometric problems listed are essentially dependent on the calculation of objects in space.But even the simplest examples can quickly become complicated.
Think about two circles a distance from each other on a piece of paper.How many lines can you draw each circle more than once?The answer is four:
You can expel these clubs or reduce one size or reduce a size or reduce the half-rated, and then you can draw a line that affects only one time.
In the absence of such pain.Only three configurations had to be taken into account in this example, but often it is very difficult for investigators to resolve each possible case.
In practice, mathematicians try to write the geometric limitations of the problem as a collection of equations and then recognize how many solutions they satisfy at the same time.Although they know that the number of sole is not always stable, there is nothing in the nature of the equations they write and if it is a new configuration.
An exception is - when the issue is defined in the form of complex figures.The most common number of statistics, and the image of the math.
If you are looking for circles and lines in the above samples and lines, if you are looking for the number of complex solutions in your equations, you always get four in response in response to any time.
By about 1900, mathematicians developed techniques to solve any problem of geometry in a complex area.
But the method is no longer suitable when the calculation just wants to get, enter, total loud noises for geometric problems in geometric problem.a or number of numbers.
At the same time, the mysterious, and the simplest methods of investigating the differences between mathematicians and other numbers that are encountered at the time of their entire numbers.
Among them was the great math of Hilbert.
In the 1970s and 1970s, his arlender Grotherine and set up a refugee problem that caused Hilbert's current problem with Gegebraic region.Setz said: Geometry wireless never been a light fire that Hilbert had thought;Some investigation strings have sunken a Manohematicas route.
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A few years later, however, the development of strings theory temporarily provided a list of geometry second wind.Many problems of strings theory can be formulated in terms of counting: chains theoreticians wanted to find the number of separate types of geometry of the list "again became very fashionable," said Katz.
But it was a short period of day.After the question of nationality, accepted.Myths are still included in another number of numbers.
This was so, while Kirsten and Jessen's mathematics and Jesse Casa came to sudden awareness: this detailed geometry can provide an accurate in -depth understanding that Galbert hoped for.
Puzzle-fuzzle views
Cass and Wickelgren met at the end of the 2000s and soon became normal co -operators.Their behaviors were no longer different in many ways.Vickelgren was warm, but with moderation and intentional.When I asked her to verify that I was released a statement, she would give her a moment. The other side, Coss is afraid.He is easily enthusiastic and speaks at a quick pace.
But Kass and Wicklegren worked well and shared many interests - including a love of expanding geometry reached in other areas.
In 2015, the cat moved to Atlanta, where Wichgren lived, and decided to turn to her last obsession with her: she wanted to review the questions listed in limited digital systems, which is a long -abandoned endeavor.
He brought many free ideas and old documents that seemed relevant."I realized it was a kind of cake in the sky," Casa said."She explained to me very politely that all my answers were stupid."
In the same 1977 paper that Sharold Levit and David Eisenbudh was used by the lawsuit.
Eisenbood and lr did felt a hidden in a simple disappear of number.This answer is placed in a form mode: Number of negative terms.
This was Wickelgren's lamp.In the decades since Eisenbud and Levine published their proofs, mathematicians created a seemingly cloudy structure called horses theory.This structure, which addressed the solutions to equations as special mathematical spaces and studied the relationships between them, was complex and powerful.Relationships using specific types of square forms.
When he listened to Kass, Wickelgren immediately realized that Eisenbud and Levine came in one of these forms.Mathematics made the motivational homotopy theory without realizing it - and they gave it the answer they applied.
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A deeper view
Remember that geometric problems usually include the number of solutions that meet the collection of equations.Space.
This way, the problems that are modified to use can use homotophy theory that has an incentive with it.This allows them to calculate two power styles.Now they have to know that these two images concern their original problems.
While working on complex numbers, they know, everything should be done is to calculate the number of variables that differ in the form of a quadrat containing.In the amount of solutions for an unfavorable geometry problem.
So they went to other numerical systems.For real numbers, it has become a little more complicated.Once calculated the square form in this approach, they had to look at its signature.And the signature did not give the precise answer: it gave a minimum for what the answer could be.I, for any problem of enumerative geometry that involves real numbers, had a way to calculate a smaller limit, a good starting point.
But many works, they all had they are knocked for the other square systems, and they said while that predator of that gamous declaration.
In 2017, Cass and Wickelgren performed for one of the most popular theories in the most popular geometry: Cubic surface was at maximum 27 lines.Using new ways, the answer, answer 27 in complex number.
Mathematicians are the first time to say something important about calculating complex and real numbers.
"It's not just about real or complex numbers," Wicelgren said."They are just a special result that is valid in any system."
And that was just the beginning.
Fresh
TheIn WICKELLGRAN, ccy and other issues were casual and other issues, which included the information in the region.
"All geometric structures are responsible for people's integers," said Mark Levin, a mathematician at the University of Duisburg-Eske, who fed the same ideas."Now you get what you give you a square form in response."
Mathematics have made great progress from the original work of Casser and Vikelgrain when it comes to their understanding in different digital systems.
"Now," said ArvIndins Asocation of Southern California University, try to get information from breeding forms that attract young mathematics and added. "
Such a dispute is extraordinary in today's abstract mathematical landscape."Mathematics is still a high level in abstraction, and then, sometimes I feel about what I don't know anymore, who I am talking about," Sabrina Pauley, who was the first graduate student of Wickergen and is now a professor at the Technical University of Germany's Drumstad.Earth
Victgen, Cas, Levin and others have recently been used to re -examine these issues related to their methods of strings - but in many new systems and settings.
In all cases, math can see the new way, line, around the enormicer geometrical shutded to me in the enumeric killed in the enumeric command."This is the key of mathematics on the paper."
