Mathematician's Exploration of Unsolved Mathematical Problems

A mathematician, a Fields Medalist, is focused on solving complex mathematical problems and expanding the understanding of mathematical phenomena.

### Fields Medalist Terence Tao: The Quest for Mathematical Enigmas and Interdisciplinary Discoveries
* 🏅 Terence Tao, a Fields Medalist under 40, has been **tackling mathematical challenges** for over 10 years.
* 🧠 Mathematicians are not only good at **solving problems** but even better at **finding problems**.
* 🌌 The number of **new types of phenomena and problems that humans cannot understand** is increasing over time.
* ❓ "The reason for approaching these problems is that they have not been solved despite **many people trying for a long time**."
* 🏆 Professor Tao has solved **11 difficult problems** even before obtaining a doctorate.
* 📈 He is currently studying the **eigenvalues of matrices** to understand the **structure of spaces** formed by matrices.
* 🎨 "By discovering the relationship between **two different mathematical fields or objects**, we can approach it."

Yes, I am a mathematician who researches mathematics by commuting between Princeton University in the United States and Korea Advanced Institute of Science and Technology. Have you ever heard of a math prodigy like me appearing? It seems not. There are many math underachievers around me, including myself, but there are no mathematicians. Professor, yes. First of all, I am now... the Fields Medalist, and we have been trying to collaborate with Professor Heo Jun-hee, and finally, he agreed to work together. Thank you once again for calling me. Thank you for inviting me. Yes, let me introduce myself a bit. The Fields Medal is awarded by the International Mathematical Union every four years to mathematicians under 40 who have made outstanding research achievements. Since there is no Nobel Prize in mathematics, it is called the Nobel Prize of mathematics. Over the past 10 years, I have been tackling mathematical challenges.

Professor Heo Jun is the first Korean to be awarded for his contribution to solving 11 problems. This is important. British mathematician Andrew Wiles, who solved Fermat's Last Theorem that remained unsolved for 300 years, received the Wolf Prize. I'm sorry. Professor, from the perspective of someone who has studied a little, Pythagoras' theorem, what Fermat wanted to solve was not about not cleaning the desk, right? Professor, anyway, there are things like this. There are countless. There are still many. There are a lot. One common misunderstanding about mathematics is that even though there are many challenges, people can solve them one by one.

Now, many of the problems seem to have been resolved, right? So, what should I do now? People often think like this. However, mathematicians are not only good at solving problems but even better at finding problems. So, as time goes by, the number of new types of phenomena and problems that humans cannot understand is increasing. It's intriguing, isn't it? Why do we need to solve them? It's a common and natural question. But why? Why do we need to solve these problems? Well, that's a really common and natural question. The reason for approaching these problems, firstly, is that these problems have not been solved despite many people trying for a long time, that's why they are called problems. At first glance, it seems quite natural to be able to approach and understand them.

Even simple propositions, the fact that we cannot understand why they are that way, is more of a fundamental flaw in the way we think rather than showing the potential for further development. That's why we focus on enigmas. Oh, it seems like that was the way to go. In my opinion, no, I think it's something like this, but it doesn't seem like we can solve it, right? Now, there are propositions that we understand and propositions that we do not understand. For Fermat's Last Theorem, some things don't seem that far apart from here to there, but the reason we can't go this way is that there are some mountain ranges, so to speak, blocking the way from a logical point of view. Therefore, our brains work in a certain way to solve enigmas.

By leaving, we gradually come to understand better. First of all, let's slowly approach this professor. Most mathematicians find it challenging to solve a single problem in their lifetime, as mentioned earlier. However, this professor has solved 11 more difficult problems even before obtaining a doctorate. It is said that it is like grabbing a tennis racket at 18 and winning Wimbledon at 20. Let me introduce the Riemann Hypothesis. He solved the problem in 45 years, and the Hodge Conjecture in 41 years. There's also the Dicca Papa Dimat Conjecture. I've never heard of this before. What is it exactly? The name itself is really difficult. Could you briefly explain just one of the problems the professor solved?

I think this will be too difficult, so there's nothing simple I can explain. I'm sorry. If you want me to explain, well, how long does it take for you to explain this? But if you really want to talk about math, you might end up making slightly inaccurate expressions in the process of explaining it in an easy-to-understand way. But mathematicians have a professional deformation and are not good at tolerating inaccurate expressions. It has to be accurate. Everything in the world cannot be summarized in three lines, right? So, Professor, I have only one request. What is the invariant of a coloring in a graph? I can't go in. No, no, I can't go in. That's it. I can't go in. [MUSIC] Right now, he's looking at the eigenvalues of matrices, and by looking at them, he explains all about Laurentian matrices.

I'm trying to understand the structure of spaces formed by matrices. First of all, I used the term space. So, now I'm kind of studying the shapes, appearance, a discipline of studying the shape is called geometry. However, now in understanding the shape of that space, I drew this type of picture. This is commonly researched by combinatorialists, called a graph, studying which vertex is connected to which vertex. By discovering the relationship between two different mathematical fields or two mathematical objects, we can approach it. It's like saying, I thought this person was friends with that person, but they were actually just acquaintances. Isn't it fascinating? Do you still feel that way? Yes, exactly. This and that seem unrelated, but in fact, it may be a bias that we humans have.

If we have such thoughts, it's because the frameworks of geometry and combinatorics, which used to be useful at some point, have already been established. That's why we perceive things that seem unrelated to these frameworks as irrelevant. By breaking the existing frameworks and creating a larger framework that can now encompass everything, we continue to discover things that have existed since ancient times. There is a unique pleasure in discovering them, isn't there?

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A Mathematician's Journey: From Underachiever to Fields Medalist

A mathematician who commutes between the US and South Korea has become a Fields Medalist, the equivalent of a Nobel Prize in mathematics.

He is now collaborating with the first Korean mathematician to solve 11 important problems.

The journey of tackling mathematical challenges and breaking common misunderstandings about the field continues for mathematicians worldwide.

The Role of Mathematicians in Solving Unresolved Problems

Mathematicians are not only good at solving problems but also excel at finding new ones, as the number of unexplained phenomena increases over time.

The need to solve these problems arises from their unsolved nature despite many attempts, highlighting a fundamental flaw in human thinking and the potential for further development.

The presence of "mountain ranges" of obstacles in the logical path to understanding certain phenomena drives mathematicians to focus on solving these enigmas.

The Professor Who Solved 14 Legendary Math Problems

A mathematician solved 14 difficult problems before getting a doctorate, including the Riemann Hypothesis and the Hodge Conjecture.

One of the lesser-known problems he solved is the Dicca Papa Dimat Conjecture.

The professor is currently focused on explaining Laurentian matrices and the invariant of coloring in a graph.

Exploring the Relationship Between Geometry, Graphs, and Mathematical Frameworks

The study of the shape of matrix spaces involves the discipline of geometry and is commonly researched by combinatorialists through the use of graphs.

Discovering the relationship between these mathematical fields allows for a broader understanding of the space's shape.

Breaking existing frameworks and creating a larger, more encompassing framework is essential for perceiving the connections that have existed since ancient times.