The mathematical work that shakes the world.

"As a graduate student, Maryam Mirzakhani (center) transformed the field of hyperbolic geometry. But she died at age 40 before she could answer many of the questions that interested her. The mathematicians Laura Monk (left) and Nalini Anantharaman are now picking up her work where she left off."  (QuantaMagazine, Years After the Early Death of a Math Genius, Her Ideas Gain New Life)

Happy International Women's Day. Today, I will handle one work of a female mathematician who died too early. Have any of you ever heard the name  Maryam Mirzakhani? That mathematician died of cancer at the age of 40. The work of  Maryam Mirzakhani handled hyperbolic surfaces.  She was the first woman who win the Fields medal. The highest recognition in mathematics. 

"Her earliest work dealt with “hyperbolic” surfaces. On such a surface, parallel lines arc away from each other rather than staying the same distance apart, and at every point, the surface curves in two opposing directions like a saddle. Although we can picture the surface of a sphere or doughnut, hyperbolic surfaces have such strange geometric properties that they’re impossible to visualize. But they’re also important to understand because such surfaces are ubiquitous in mathematics and even string theory." (QuantaMagazine, Years After the Early Death of a Math Genius, Her Ideas Gain New Life) 

(QuantaMagazine, Years After the Early Death of a Math Genius, Her Ideas Gain New Life)

The ability to model those surfaces makes it possible to model the black holes. The black holes and wormholes can be a series of hyperbolic surfaces. Also, molecular, atomic, and subatomic bonds are the things that act like wormholes. There are many useful things in that kind of mathematical formula from information to engineering. 

Or, we can think that those things are hyperbolic curves that are rolled into the tunnel-shaped structures. The hyperbolic surfaces and their interconnections can also shape the connections between black holes. And maybe those things can help researchers to make the models of the entropy in the black holes. The geodesic that illuminates the surface is the thing that can used to model the time loops. The time loop is the thing that forms an energy wave that travels back in time. 

(QuantaMagazine, Years After the Early Death of a Math Genius, Her Ideas Gain New Life)

The reason why time travels in a certain direction is this. The past is in higher energy levels than in the future. That means information can travel to the past if the event's energy level is high enough. The time loop means that there is a thing that raises the energy level in the future higher than in the past. When information travels in a certain moment in time, outcoming effects will pump energy to them. 

Then that energy causes time dilation that pushes information back in time. The problem is that the entropy mixes that information to a way, that denies us to understand it. The thing is that. The size of the quantum system determines entropy. But another thing is that the limited systems have limited entropy. Theoretically is possible to see things that happened in the past or even in the future. 

But entropy causes a disorder in the information. That disorder makes information nonunderstandable by sorting it into a new order. The entropy is like whirls that disturb the information form. But if the entropy is limited that means it's possible to calculate those whirls backward. The system can put the pieces of information into the original form. That thing is possible only if the system calculator knows every movement and part of the system. 

This kind of work will make fundamental things in chemistry, information technology, and maybe many other types of engineering and scientific work. The ability to model surfaces and curves makes it possible to create models about Hall fields. And that helps the system control information in the quantum tunnels. 

https://www.quantamagazine.org/years-after-the-early-death-of-a-math-genius-her-ideas-gain-new-life-20250303/

https://en.wikipedia.org/wiki/Maryam_Mirzakhani

The robot as intelligent as humans can be closer than we dare to think.

Artificial muscles and synthetic organic intelligence can be tools that make it possible to create robots. More intelligent than humans. The artificial muscles are tools that mimic human body movements. Those muscles can be protein fibers that are made to mimic the muscles. Those fibers can change their length. When something like electric impulses impacts those fibers. 

Those systems are called "dry" solutions. The wet solution includes small bags, filled with electrolytes.  When electricity is conducted to electrodes that are at both ends of the bag,  electricity pulls electrolytes to that side of the structure. Another way is to use cloned muscle tissue and 3D printers to make that kind of muscle. 

When we think about computers and computer-controlled systems those artificial muscles require nutrients. Those muscles can be put on the metallic- or carbon fiber skeleton. There must be some kind of bone marrow. That makes blood to those artificial muscles. The brains for that kind of system can be computers. Or they can be so-called synthetic organic intelligence. Synthetic tissues require nutrients. 

Synthetic organic intelligence means the cloned neurons. These are 3D bioprinted things that can think like human brains. Researchers hacked the DNA that makes our brains unique. Then the system injects this special DNA into the mini-brains that are created for medical research. The system can grow those brains under the class domes where the system feeds those cells. 

Then the system can drive memories to those cells using the microchip and then the brain and its substitence will transfer to a robot body. That kind of robot can be our worst nightmare or our salvation. The robot that is more intelligent than humans was theory a couple of years ago. But bio- and genetic engineering and highly advanced AI-based computing are making those visions closer than ever before.

https://scitechdaily.com/the-secret-to-human-intelligence-scientists-uncover-dna-that-supercharged-our-brains/ 

The function: that revolutionizes calculus.

"Plot of Weierstrass function over the interval [−2, 2]. Like some other fractals, the function exhibits self-similarity: every zoom (red circle) is similar to the global plot." (Wikipedia, Weierstrass function)

In the late 19th century (1872) mathematician Karl Weierstrass introduced a formula that is one of the most revolutionary in the world of mathematics. "In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is also an example of a fractal curve" (Wikipedia, Weierstrass function) The term fractal curve. 

This means that when we zoom that curve or otherwise use more and more accurate numbers and shorter number distances between the integers we can find more and more roughness in that interesting function. When we make Weierstrass function calculations and curve for the answers we can get quite a smooth curve. But when we start to use decimal numbers that turns the curve rough. The fact is that there is a limitless number of decimal numbers between two characters. 

"Animation based on the increasing of the b value from 0.1 to 5. (Wikipedia, Weierstrass function)

We can continue all decimal numbers as much as we want. So there is an unlimited number of decimal numbers between for example 0,1 and 0,11. That means we can put unlimited numbers between those numbers that seem to be very close in the number line. But we can put 0,1001 or 0,1009 or any number series between 0,1 and 0,11 and the only thing that we must be sure of is that the number series that we give is smaller than 0,11. (0,11>x). 

So Weierstrass formula is the thing that can introduce the situation that the little difference might be a big thing if we look at it in the right scale. The Weiserstrass formula was revolutionized in the 19th. Century. And sometimes people say that this thing is an early fractal. When we research this function we can see that there is a repeating form that is similar to small and large areas or periods. 

"Euler diagram for P, NP, NP-complete, and NP-hard set of problems (excluding the empty language and its complement, which belong to P but are not NP-complete)". (Wikipedia, P versus NP problem). 

"Diagram of complexity classes provided that P ≠ NP. The existence of problems within NP but outside both P and NP-complete, under that assumption, was established by Ladner's theorem. "(Wikipedia, P versus NP problem). 

Weierstrass formula and P=NP. 

Normally P=NP is introduced as the computational or virtual problem. That means the P=NP or P≠NP is used to introduce similarity between P and NP. The non-proven question is: is it always that P=NP or P≠NP? And is that universal? But the P=NP is much more interesting and large-size complexity than other things. 

The Weierstrass formula has a connection with an unproven mathematical millennium problem called P=NP. Can the P-level be the same as the NP level? And can the P level interact with the NP level? That thing is one of the key questions of computing. 

The P=NP means that if we have two identical systems whose size is different when we move something in a smaller system same thing moves in a larger scale system. The P=NP means that when something happens at the P level same thing happens at the NP level. But the major problem is this. In the real world, the P cannot interact directly with the NP. So there forms a medium between those layers or spaces. And the P and NP don't exist in pure forms. The world is full of layers and spaces. Those spaces disturb information that travels between P and NP. 

That thing can cause the idea about the P=NP. The P=NP means that if something changes in the P-level same thing happens in the NP  level. So, P=NP means that we move something in the microcosmos same thing moves in the microcosmos. So the P=NP should be universal. That formula should have an effect between virtual and "real" (material) systems. But the thing is that the energy level that the virtual system can give to the material system is so weak that the virtual system cannot affect the material or physical systems. 

The P=NP problem is the thing that originates in Buddha, Siddharta Gautama. The Buddha introduced the microcosmos is connected with the macrocosmos. So when something happens in the microcosmos that thing reflects immediately into the microcosmos. The fact is that the P and NP levels are not determined and we can say that abstract, virtual mind, or imagination is the P level. 

And physical world is the NP level. If that thing is real, that means the telekinesis is possible. But the fact is that there is needed much higher or stronger force to make the visible interaction between those virtual and physical worlds possible. The P=NP means that if we move something in the macro cosmos we move the same thing in the microcosmos. As I wrote before the P=NP means that all cosmoses or layers. Without depending on their size have a twin in the microcosmos. 

https://www.quantamagazine.org/the-jagged-monstrous-function-that-broke-calculus-20250123/

https://en.wikipedia.org/wiki/NP-intermediate

https://en.wikipedia.org/wiki/P_versus_NP_problem