Inspiring Minds: The Role of Mathematics in Critical Thinking

0:00:01 - Announcer

You are listening to the National University Podcast. 

0:00:10 - Kimberly King

Hello, I'm Kimberly King. Welcome to the National University Podcast, where we offer a holistic approach to student support, well-being and success- the whole human education. We put passion into practice by offering accessible, achievable higher education to lifelong learners. Today we are talking about the power of mathematics and, according to a recent article in the New York Times, learning mathematics is both crucial to the learning development of the 21st century students. So as not to be imposed upon learners too heavily. So, learning mathematics develops problem-solving skills which combine logic and reasoning in students as they grow. 

We're going to be having a great conversation about the power of mathematics coming up on today's show. On today's episode, we're talking about the power of mathematics, and joining us is National University's Dr. Igor Subbotin, and he earned his PhD in mathematics at the Mathematics Institute of the National Academy of Sciences of Ukraine. Before joining National University, he taught mathematics at the most prestigious university in Ukraine, Kiev Polytech Institute. At National University, Dr. Subbotin regularly teaches different mathematics classes and supervises mathematics courses. Dr. Subbotin's main area of research is algebra. His list of publications include more than 170 research articles in algebra published in major mathematics journals around the globe, and he has had the privilege to collaborate with several world-class mathematicians from different countries. He also authored more than 50 articles in mathematics education, dedicated mostly to the theoretical base of some topics of high school and college mathematics, and he's published several books. We welcome you to the podcast, Dr. Subbotin. How are you? 

0:02:07 - Igor Subbotin

I'm fine. Thank you very much for inviting me. I'm really happy to be with you, thank you. 

0:02:13 - Kimberly King

Thank you. Why don't you fill our audience in a little bit on your mission and your work before we get to today's show topic? 

0:02:22 - Igor Subbotin

It's very easy to talk about things that you love it. I love mathematics and felt a love in mathematics a long time ago. I continue loving it, the same kind of, let's say, powers that used to be when I was very young and I love my students and the love to my students even grows. Comparison that I was young, because I got experience and understand people better. I love my university and I have been working for National University for 30 years already and university growth- I looked, I was part of the university development and growth. It was small at that time when I came and now it's a big university with some traditions, prestige, some kind of place in the American higher education and I'm happy to work with this and I'm happy to continue development of the mathematics education in our country, in different countries, and Europe, that I participated in many different collaborations with many different scientists and promote some mathematics- new ideas and also disseminate these ideas, which is extremely important. Thank you for inviting me. 

0:03:51 - Kimberly King

Absolutely. I love more than anything, I can hear your passion for teaching and really helping your students understand the joy that you have for mathematics, and so today we're talking about the power of mathematics. And so, Doctor, why is mathematics so useful? 

0:04:10 - Igor Subbotin

I will answer for this just bringing the citation from Joseph Louis LaGrange, one of the bright stars on the mathematics horizon. He was born in the middle of the 18th century, I mean, started to work in the middle of the 18th century in Turin, Italy, but he is a French mathematician actually, after all, and he was a key figure in many different mathematics development of that time- special calculus, differential equations. It was a time very in, and he said like that, mathematics as the queen and the servant of all sciences. Mathematics is a queen and the servant. 

I know that some other people say that mathematics, that science, became a science only if the science used mathematics. Start to use mathematics, it's number one. And also I would like to repeat the word attributed to Galileo Galilei, who said that mathematics is a language in which God speaks to us. God could be just changed to nature, but the meaning is the same. Our nature, our God, will speak to us through the mathematics language, mathematics language. And why mathematics is so remarkably useful in every single human activity, not only in science, not only in physics, everywhere, everywhere. Number one, philosophically talking about everything. Every single event has some kind of qualities. How can we assess this quality? How we will talk about that? 

First of all, we try to measure this in some way- qualitative measurement- also appeals to quantitative measurement. There is no quality without quantity. There is no quantity without quality. There are two structures that inter-influence each other. It's number one. Why mathematics has so much power and why it's so useful? Because the main idea of mathematics is to strip off the second line, details, to look at the stems, not on the leaf and the stems ignoring by some details. When you start to study some physics or chemistry, some kind of events or what happened with them in this specific event, you just miss so many details, so many details- you don't know to what you need to concentrate your attention. In this case, mathematics helps you to strip off of this mail by focusing on the most essential aspects. 

Mathematic enables a comprehensive understanding of complex natural processes. It distills vast amounts of information, stripping away irrelevant details to emphasize what truly matters, what truly matters. So mathematics is not a calculation. Mathematics is not only geometry. Mathematics is a way of thinking. This is exactly what we call critical thinking of the highest level of development. That's why mathematics is so powerful. And this mathematics is powerful not only in science. It's powerful in any kind of human activity. And I will tell you what is mathematics' role in this human activity. This is the main thing. 

0:08:20 - Kimberly King

Well, I love that you explain mathematics as a language and again your passion comes through. In fact, last time I interviewed you I kept saying, as soon as we were through- I wish you were my mathematics professor, because you share such a passion and you make it easy for others to understand, and that is truly a gift. So thank you. Why is mathematics so universal? 

0:08:50 - Igor Subbotin

The idea of universality of mathematics- it belongs to its own structure. What is mathematics study? What is mathematics study? The main process in every single thing, event, and you know, sometime it always amazed me and not only me, maybe, it's amazed many different people for sure, that when some science or some human activity thing face some needs of mathematical analysis, which means qualitative or quantitative analysis, the appropriate corresponding series already exists. You don't have to create something new, it's already there. What does it mean philosophically? For me, it means that this is some kind of answer for the very deep questions that everybody who is doing mathematics asks themselves. What we are doing? Creating new mathematical ideas? Or we are discovering this mathematics world, like a known country? So maybe all this idea exists already and we are just discovering them, like in physics, like in chemistry, like in any other thing, or we are creating these ideas. I believe it's my opinion and not everybody shares this opinion that everything already exists. We are just not inventing, we are exploring these ideas. So what does it mean? So, for example, when we are talking we'll talk today a lot about my favorite area of mathematic- algebra. This is not the same algebra that we are talking about during high school mathematics. No, it's a totally different subject. I will talk about this today. 

But the power of algebra based on the idea of isomorphism. Isomorphism, what is that? It means if you have two different structures consisting of two of different subjects, objects of different objects, and you find out some kind of one-to-one relations between these two structures, in which it doesn't matter what you're doing in one subject, in one object, work with the same kind of result for the second object. It's in the isomorphism. You can start to study one area and after that, all the rules that you will come to will work for another way. This is the power of mathematics. Simple examples- you can use the same kind of linear equation to describe many different things, many different things. In mechanic, in accounting, in, let's say, the different disciplines that are directed to the, for example, structural things, some like concrete structures and so on. 

It's a very simple example. Idea of isomorphism this is algebraic ideas that just came to our attention, I believe, not so long time ago, maybe 200 years ago, no more. But mathematics, mathematics use it for long period of time, many, many, many years, without understanding what is that. I believe that algebra, which is the most abstract subject in mathematics, could be a wonderful illustration how this idea works. The idea of isomorphism is crucial in explaining how mathematical concepts can be applied across diverse fields. For instance, in algebra, the same equation can be used to solve problem various areas, showcasing the universality of mathematical principles. So we will talk about this today. 

0:13:13 - Kimberly King

So interesting- yeah, go ahead. 

0:13:16 - Igor Subbotin

Let me add a little bit about your remark about the study of mathematics, how the teacher role is important in that. I would like to assure our future students, or some people, that we have right now at National University, that main idea for selecting faculty for mathematics department for classes, staffing them to the classes, is the idea how these people really feel about the subject, if they're really motivated to bring their knowledge and their passion in the classroom and they really understand with whom they're dealing with. Because it's a very different approach in our study when we come to the class of the elementary teacher future elementary teacher or to some art designers, all of us are very passionate about the subject. All of us understand our role at the university and how to treat students in the right way. 

I believe not only me, it's statistical knowledge that most of the hate to mathematics born in the elementary school classrooms, where some teachers hate it and don't understand it enough. That's why our mission, our mission- and we teach elementary teachers also- to bring the light of mathematics understanding to them, to build up the respect to the subject, respect to the teaching. And that's why I believe that major, I would say almost all our students are successful, not because we are not keeping rigor. We are keeping high rigor in our classes, our classes. But we are doing our best not only to fill out students like a job but to light them as a torch in mathematics. Sorry for interruption, but it's an important point that I would like to mention, answering for your remark.

0:15:38 - Kimberly King

I'm glad you did, because it is true that when we're learning, I mean it's almost like now you're playing catch up to get these students to love and have that affair- a love affair- with mathematics and that understanding, and it really does need to start at a younger level, just so that you know we can continue to move forward and grow. So thank you for taking that moment out to explain that, because it really does truly show, and I think we're doing a disservice, you know, for those teachers that are in place and either don't have that love, that understanding, that passion, and then they're, you know, not necessarily bringing up our kids, our children, to love it like you do. So it's good, thank you. Can you discuss abstract algebra and how it's stated and its applications? 

0:16:31 - Igor Subbotin

Most of our students who are not math majors, they will not study abstract algebra in the university course. They will just, I believe, will be starting studying, some of them, calculus. Some of them will study just college algebra. Some of them will study linear algebra at most, like computer science people. Abstract algebra, this is only for math majors and this is very interesting to trace the genesis of algebra, how it became totally different from other areas of mathematic language, developed language, develop understanding and what is the most important- at the end I will show this- how this may be one of the most abstract, without any, sometimes, visualization ability, subject became the most useful and most applied. It's interesting. So if I will start about talking about algebra, you immediately just come to the original. 

Algebra can be found in the mathematics of ancient civilizations, particularly Babylonian people and ancient Greece, of course, with Euclid, with his famous book Elements. Do you know that the book Elements of Euclid was it's about 2,300 years ago published? by Euclid, and Euclid is a very mystical figure in mathematics because there is not any portrait of Euclid that exists. Most of the mathematician portraits we have it came to us from the anthropology but not Euclid’s picture and according to his, let's say his in quotation input, in mathematics it's too much for one person to be so educated and so powerful. So there is some hypothesis that Euclid this is just, let's say, like nickname for the group of mathematicians of that time they put together their knowledge in the group of elements, elements. It's had in the group of elements. You now that the book elements is the second book by the amount of publications after Bible, only one book that was published more than Euclid. Why? Because during 2000 years, it has been the maybe only textbook for mathematics for our civilization, for 2000 years almost. So, Euclid, ancient Greece, so who started developing equation solving procedure and manipulating symbols to depict mathematical relationships. 

After that we jump to the golden age of Islamic, that next significant advancement which was made possible by the writing of academic-like Al-Khwarizmi. Al-Khwarizmi - listen, algorithm. Al-Khwarizmi- [laughs] it’s the same, it's the same. Algorithm come from Al-Khwarizmi name. This is some golden age of the Islamic age, something 15th, 14th, 15th century, when this very famous author wrote the book. I will read the book- “al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah”- translating is going to be the Comprehensive Book on Calculation, Completion and Balancing- Al-Jabr. Al-Jabr, this is a book we named. It was brought to Europe by mathematicians from Middle Eastern countries. So what is interesting? Definitely, this book is not absolutely just created by people from Islamic countries. It was a lot of roots in China, a lot of roots come to India and so on. 

Let me remind you my favorite words from brilliant mathematician David Gilbert, who was one of the most prominent, if not most prominent mathematicians in the world in the end of the 19th, beginning of 20th century up to the middle of 20th century. He said that for mathematics, there is no boundary in culture and race. Mathematics considered the entire intellectual world as one country. If you take mathematics, you will not find any other science that would be so internationally developed, internationally developed. And now we have many different countries that work in mathematic development. It was maybe something like 19th century. 

The most influential was French mathematics. Before it was Newton, English, England mathematics. At the same time it was German mathematics. After that, again, German mathematics became prevalent. After that, Soviet Union mathematical school became a golden, when it was golden age. It's the most powerful and most developed school. 

American mathematics. American mathematics became very, very influential and powerful, but it's mostly after 40 years of the previous century, 20th century, and I will bring you many other examples like that. Every Chinese, Chinese school. Look at China now. How many prominent fields, medal holders and mathematicians we have in China. What a genius was born in India. What a great development. 

In my own experience and seeing how the Middle East developed mathematics, about 30, 40 years ago they didn't talk about, for example, abstract algebra. Now they have a few, a few journals in the Middle East, especially in Iranian people. It doesn't matter what kind of relation we have with Iran. Mathematics is unity, it's all people work together and we develop the same subject and we work on the same field, which is extremely important. That's why mathematics is so influential. Why we are doing it? This is the reason, because everybody needs it. Everybody needs it, not only for developing some kind of new technical idea, but also for understanding the world. 

So after this Islamic time, symbolic notation was developed by a mathematician during the Renaissance time, European mathematicians during the Renaissance time. What happens at that time? People can solve quadratic equations. Long time ago, linear equation was not a problem at all, but quadratic equation long time ago. But when people face equation of the power 3, power 3 with one variable, it was a big problem. Sooner or later it was solved. Also, what does it mean solving equation? It means to get a formula that will express the solution, the roots of the equation, expressing them through the main algebraic operation addition, subtraction, multiplication, division and so on, and radicals and so on, in one formulas through the coefficient, coefficient- number of this equation, using this equation. So for cube roots it could be huge formula. After that, for four roots, people were successful. They got it. 

After they got it, I believe that the creator of this was a mathematician who was nicknamed Tartaglia. Tartaglia mean people with some kind of empire of speaking, Tartaglia in Italy. This, it was a very strange person. He was very- He was not, he didn't have really nice personality, he was, let's say, some kind of gloomy and always not happy guy. But he, according to what I read, he was the person who solved the equation of the power of four. At the same time. Cardano, Federico Cardano, who is a physicist, great physician, great engineer and great mathematician of the time, very big person at the time, star number one in Italy, have heard that Tartaglia has some formula for solving equation of the force power, certain force power. 

Why it was so important? At that time, to get position to have some money for doing mathematics was a very difficult thing. It was a very difficult thing. It was very limited opportunity, for example, to be some kind of the mathematician in the court of some kind of prince or king or duke or something like that. So people just applying for this position, they're supposed to go through the competition. Competition looks like that that the persons who would like to apply for the position, two weeks before the meeting, send to each other the list of the problems that they would like that their counterpart solved. A counterpart solved Okay, if you have a formula that nobody knows, you are a winner. It's a big chance that you solve the problems that this person sent to you. But there is no chance that somebody will solve the problem if they don't know the formula. So it was a huge privilege, it was a huge, huge benefit to having the formula. It was a big secret. Nobody knows. 

Cardano, who was a huge guy, a huge star at that time, came to this unknown guy, Tartaglia, and asked him tell me please about the formula I have heard you have it. Tell me please. And Tartaglia said okay, but don't tell anybody. For Tartaglia it was a big, big how to say honor to meet a guy like that and he told him the formula. What Cardano did- from the point of view of nowadays, he did a very honest and great thing. He published a book and the end of the book published the appendix and he said this formula was given to me by my esteemed great colleagues Tartaglia. So he didn't try to cheat, he didn't try to get his Tartaglia let's say, copyright, like we said right now. But from point of view of nowadays you can say it was a huge support to the people, to the Tartaglia from the big, big star. 

No, guys, that time it was totally different. Tartaglia lost his power to win in competition by publishing this formula. So, change time, change the vision, change the vision. So this time now the formula calls like usually called, usually called something that if you create in mathematics it's always a different person name. Coming to the history, it's Cardano formula, but definitely it was Tertaglia. In mathematics there are some kind of buyer rule that most of the invention in mathematics are called by the name of the people who didn't do this. It's even said like that Bayer rules, okay. 

So what happened after that? People started to study the equation of power of five. Next fifth power no success, no success. This way, that way, no success. Nobody can do this, anything. And until October 25, 1811, a brilliant, very unusual, mathematic star was born- Evariste Galois. Evariste Galois, French mathematician. He was born, again, in 1811. He created his main idea and write his main idea in written when he was 16 years old. 16 years old and he was killed when he was 20. Only 70 years after his death, Camille Jordan published his work in his book about matrices, and this is the beginning of modern algebra, beginning of modern algebra. 

Do you know what the Evariste Galois finds out? He finds out that any equation, polynomial equation with one variable, with a power greater than four, like five, six, 7, and so on, has no formula at all in general, cannot be found. But partial cases, please. To find some approximate solution, please, but not general. There is no, and never going to happen. It's mathematical power. You see, he proves that it and never going to happen, it's mathematical power. You see, he proves that it's never going to happen that anybody with the hugest star in the world there is no formula in general, case like that. Moreover, he finds out when, in some partial cases, this formula exists and when not. What's the condition for this. This is what. 16 years old how to say, teenager, 16 years old. 

He of course tried to find out the opinion of the people who was in power in mathematics that time. He sent his manuscript to Augustin-Louis Cauchy. If you will study calculus- I'm talking to our future audience- you will find Cauchy's name every I mean tens, dozens of times in different theorems and calculus. He was a huge star at that time it was the first quarter maybe, of the 19th century right and he published a lot of different works. He was a genius and he is a genius. And Galois sent him a manuscript and after some time asked what do you think about that? And Augustin-Louis Cauchy said all right to him, I lost it. I don't know if it was true or not. Some people said that after that, the publication of Augustin-Louis Cauchy has some influence on this paper. So it's again, humans are humans everywhere, not only in the economics and history, but also in science, even such straight science like mathematics. Human is human. 

So what can we say about this situation? And Evariste Galois continued to promote his idea. He organized special seminars for people who wanted to come there. But he also participated in the revolutionary activity and he was very active in this, and the police decided to say too active, too much active, the police in French, and they sent to him the killer. He just sent him invitation to duel because of some woman. Sorry but it's true and he just came to the duel and before the night of the duel he continued to work on mathematics and he was killed in the duel. It was a main idea. It was political kill, definitely, but he was 20. He was 20 at that time. So you know, teenagers create a new huge area of mathematics. 

It started with a name, like Galois theory and we studied Galois theory. But this is a partial idea. The idea was to study, not the numbers, not the equation- operations, operations. This is the power of abstract algebra. This is the power when the abstract algebra was born. Study operation. 

I will give you a very simple example that everybody will understand about operations and so on. You know that entire world knows chess game. In different countries we have different names, different names for the checkboard names. We have different names for the figures. We have different names for the different language, for the combinations that we consider. But we have great masters from different countries. They play the same game. They cannot speak common language, but they know the rules and the rules are the same rules. It's operations, how to operate with this special figure in this special situation. This is a rule. So algebra, abstract algebra, they don't deal with the equations, specifically. They don't deal with the numbers. They can deal with the matrices, which are big tables. They can deal with the transformation of the space of the plane, doesn't matter the idea how this object behaves under the operations. 

Under the operations. 

0:36:20 - Igor Subbotin

It's a huge step. It became a very abstract subject, a very abstract subject and it's very interesting to say the algebra started with the ideals of Galois and the idea of Galois leads us to the group theory. This group, the algebra subject, without operation group is the most possible. Group theory. But group theory was long time stay long time as a group theory of permutations. It's a special object in algebra. Only in 1920s, 1920s, great mathematician, Otto Yulyevich Schmidt. Otto Schmidt, it's a Soviet Union mathematician, Russian but definitely with the German roots. Schmidt, it's German name. Otto Yulyevich Otto, also German name. It's also another kind of brilliant guy and I'm his scientific grandson. 

0:37:26 - Kimberly King

Perfect yes. 

0:37:30 - Igor Subbotin

Why you will be amazed in a different way. In our department we have biologists, people who study biology. We have now a department called Mathematics and Natural Science. One of these professors is a prominent researcher in biology who uses a lot of statistics, and his dissertation also was supervised by the statistician mathematician statistician, because to study biology you need statistics, you need to watch. Okay, so what is interesting? When we together came four steps back, we will find out that our roots both of us, came to Carl Friedrich Gauss. I am grand-grand-grandson and he is grand-grand-grandson to Carl Friedrich Gauss. 

0:38:23 - Kimberly King

Wow, that is fascinating. My goodness Wow. 

0:38:27 - Igor Subbotin

The world is unique, it's one. So Otto Yulyevich Schmidt, Otto Schmidt was not only mathematician, with the he, by the way, first wrote the book which called Abstract Group Theory. When this object, in the group object, elements of this set, absolutely abstract, doesn't matter what the nature, only operation is important. Otto Schmitt was the one and is the one of the most famous geodesists about the science of the Earth, and he is well known also as a creator of the first scientific theory how our solar system was born. This is his name. Also, he was a guy who was the, who was the how to say director of the expedition to the North Pole on the ship Chelyuskin, and that time, in 1930s, it was the most like today, let's say, a trip to the moon. It was the same kind of importance for the entire human race. And so, by the way, some of them article in group theory and algebra written in the Chelyuskin during this expedition, written in the Chelyuskin during this expedition and signed up like ships Chelyuskin, whereas it was written Chips Chelyuskin. So he was there in the North Pole and write the book, and write the book and write the article in mathematics, people like that is a brilliant our human civilization topics. Okay. So algebra became very abstract. Nobody expected that algebra became really, really applicable, right. So because I have been working in algebra for some time, I see it in my own eye when the group theory subject just transformed to the new abstract algebra. Of course I was not born in 1920s when it came, but in 1970s I see the most peak of development on the infinite group theory, and now I see that some other subject was developed like that. Let me continue with the history and I will tell you a very exciting thing about how algebra, so abstract, became so useful and became so applicable. 

Next step was German mathematicians, like we are supposed to mention David Hilbert and his school, and also the biggest star in mathematics for all times, Emmy Noether. Emmy Noether, this is not only you know, of course. Everybody knows about Sofia Kovalevskaya. Sofia Kovalevskaya or Gepardia Alexandriyevskaya, some other woman who brings their huge input in mathematics, but Emmy Noether is number one. She was a German mathematician. In 1938, she immigrated, like many other people, from Nazi Germany to the United States and I believe she was a professor in Bryn Maur College after her death. She developed the main idea of investigating some algebraic structure like rings, fields, groups and so on, the idea of chains. I'm not going to proceed with this too far, but she gave us the instruments, the tools to open these fields of investigation. Everybody up to now work on that, everybody up to now work on this and will continue, because this is only one, let's say by now, useful tool to study infinite structure, algebraic structures. So it's very important to mention that again. 

That algebra abstract algebra, I would like to underline this was created by many different people but having their definitely first step and the most important input by the genius Evariste Galois and in particular, group theory, has made great progress in the sphere of new ideas and theorems. Many different mathematicians brought their attention and now this is a very well-developed part of algebra that have their application in physics, in cosmology, in painting, in art, in crystallography, everywhere where we have symmetry, symmetry. You know everybody what now? What's the tool to study symmetry, group theory. So it was created for solving equations, but find their application in any science that dealing with the symmetry, any kind of symmetry, geometrical symmetry or some other kind of symmetry, elementary particle symmetry, cosmology, and so on and so on. But I would like to finish with the development of how it was developed and I'm ready for your next question because I will continue this forever, definitely. 

0:44:26 - Kimberly King

It's really fascinating, though, to hear the history and to hear all of the countries that have been involved with the very beginning, the establishment and the beginning of mathematics. So this is quite fascinating. We do have to take a quick break, doctor, if you don't mind, and we'll be right back with you in just a moment. Don't go away and hold onto those thoughts, stay with us. You Thank you. And now back to our interview with National University's math professor, Dr. Igor Subbotin, and we're discussing the power of mathematics and, doctor, this has been so interesting, just hearing the history of it and how it is all the nations working together, as we were just discussing, without a particular agenda other than the love of mathematics. The answers, we say numbers don't lie, and so, with everybody working together, it is universal. Can you talk about the bridge between high abstraction and realm? 

0:46:14 - Igor Subbotin

I would be happy to do it. I will give you some kind of examples that I faced myself lately. 

Okay, this is again for our future students. It's very important that the people who will teach you the subject be active in this subject, be professional in this subject, not just read the book and explain book to you, but do something by their hands in order to develop and to bring very modest input, but input in the object. I will give you some interesting story about that that based on my own experience lately, I have a very old friend not old person, old friend to me which we are friends, we have our friendship, for you cannot believe more than 50 years and we keep our collaboration, maybe the same amount of years because we are from the same school. Our supervisor was a brilliant star who was one of the founders of Infinite Group Theory, Sergei Chernikov. He is a huge international star and we are proud to be his students. I was lucky to work under his supervision and in his seminar since 1967 to 1987 when he passed away. I was lucky and my friends that I'm talking about, I can give you his name. Somebody can go to Google to look and find out who is it. It's Leonid Kurdachenko. It's Ukrainian mathematician, very, very, very famous mathematician in our area, abstract algebra, distinguished professor, and so on and so on and my close friends. We worked together for many years and we both have our roots in working in abstract algebra, in group theory, and we've been witnesses and we are witnesses about the time when the group theory and pheningo theory was very powerful in new development field and so on. But lately we found out, by some different reasons, some people came to work in so-called braces theory. Braces, not the stomatological braces, not the braces that you use like parentheses, some kind of parentheses. Algebra brace is totally different. It's algebraic structure, new algebraic structure. What is it about? Taking example from physical algebraic equation, we have behavior of particles and waves which make possible to predict what will happen next, to explain nature of phenomena. Again, this is about symmetry, for example, biological sign, complete genetic interaction, dynamics and so on. Algebra is resonant also to artistic work, but lately it's happened like that. Let me give you the exact what I would like to say. The theory of braces, the theory of braces is very young, very young, it's just right now. 

It has its roots in addressing the Young-Baxter equation, a fundamental concept with profound amplification in both pure mathematics and physics. It originated from the groundbreaking work of the Nobel Prize winner, physicist Young, of the Nobel Prize winner, physicist Young, in the realm of statistical mechanics and, independently, in the contribution of Baxter to the 8-vertex model. It came from the knot theory and came from the statistical mechanics, its quantum theory. This theory holds substantial significance across diverse domains such as knot theory, braid theory, operator theory, hoppe, algebra, quantum groups, Tremont Foyle and monodromy of the differential equation. This is a fundamental equation in mathematics and physics that arises in the study of central algebraic structure, arises the study of central algebraic structure and it came to our attention thanks to the works in some first time in 1960s. But about the theory itself is thanks to the work of other mathematician, is became popular since 2008,9. 

Many people just came to study this because it has huge application. What is it about? It's about to find the properties of the solution of this equation. Solution of this equation is not numbers, it's matrices. It's matrices, different kinds of matrices, special subjects in algebra. We don't have a general formula or a general approach to solving this equation. We don't have it, but it's very important to us not waiting until maybe we'll never have it, who knows? 

I gave you today an example of a various Galois equation. There is no formula for the solution of the fifth power equation. There is no formula. We are doing this approximately. We have many methods to solve approximately, which is enough for us. For the Young-Baxter equation. We have some situation, but study the properties of solutions as very, very important for many different disciplines and the people start to study this kind of equation, study the properties of this equation, even though we don't have these solutions. We don't have solutions, but we study properties of the solution and they find out that this just could be studied with the approach of abstract algebra. The properties of this solution could be described with the help of the new algebraic structure, old algebraic structure groups, fields, rings and so on. It's classical structures already. This is absolutely young, absolutely young and new. This is braces. Braces like a fusion of two groups together. It's very difficult to describe on the fingers but it's very important. 

Solution of the Young-Baxter equation, known as the Young-Baxter matrices or R matrices, have found numerous applications beyond their original context. People start to study this. My friend, Leonid Kordachenko, just tried to apply the ideas that we had developed in the group theory, not we. Saying we, I mean all mathematics community, not myself, separate, okay. And he was so kind, he involved me to this and we together started to work on some specific points of it me to this and we together start to work on some specific points of it and find out that, you know, the idea of group theory works there. Works there, not different results different, the same approach, the same approach, the same idea how to mine this, but totally different results. Totally different results. But its results are natural and ideas natural. So we start to work on this and let's say we work with success and we are doing this for the last two years and we published already a few articles about that and we also developed some and delivered some talks in different conferences and was very welcome in the community of the people who work in this area and very famous and algebraic for there. Why? Because of applications. 

But what is interesting, even though this is absolutely new structure in algebra, not like we used to study and we will study in our algebra course in our university- New structure, absolutely new. The approaches that we use, the approaches work there in the same way. So the ideas work there. So what I would like to say- It exists. When physicists need it, mathematicians said welcome, we have it. We have it. We were just a little bit adjusted and the lock will be open. It's the power of mathematics. That's why we need to study that. That's why everybody likes to study mathematics around the globe. That's why there's no difference to us, to our colleagues, whoever, wherever they live, whether it's race, whether it's nationality, whether it's language, we don't care. We are one community. We united the globe, we united the human nation together like nobody else. 

0:56:03 - Kimberly King

Well, I love that and I wish that we can continue to just be united as a nation and not get politics involved in everything. Mathematics, there is a universal answer and that's just beautiful. And speaking of the language, how are language and mathematics alike and what are the differences? I know you've been talking a little bit about this, but I just I really do love that you've talked about the history of it in a universal manner, but talk about the reasons why mathematics and language are alike and different. 

0:56:36 - Igor Subbotin

Mathematics and language. You know, English is not my native language, as you may have already seen, and my third language that I use, and analyzing my experience in writing mathematics and analyzing my experience in writing mathematics in Russian, in Ukrainian, in English, I can find out some very interesting things of the cross influence of the language and mathematics. As I told you before, mathematics is a language, Mathematics is a language. So, in my my opinion, it looks like that we have a box which called mathematics. On the input, we have a regular language. We translate this language in mathematics language using symbols, place this in the box and forget about everything. Forget about the, what we are dealing with. We are just using the rules automatically, like in algebra, solving equation. We don't care what the A, what the B, what the C. We have a formula. We substitute number to the formula, get the result, go back, output and translate the solution to the common language. This is how it works, right In reality. But it's very interesting that English, in my opinion. 

I am not a polyglot. I don't have too many languages in my, let's say, possession, but I'm really good at Russian and I'm not bad at Ukrainian. I can express myself in English, but English is a very interesting language. It's totally different than the language that I started to use before. English is a beautiful language because it's close to mathematics nature. It's a very straightforward language. Everything is structural. For example, a Russian can say something like this is a beautiful girl and this girl is beautiful. I can say beautiful, this girl in Russian. It will be the same meaning In English. No, in the order in the sentence. It's very, very important. So it became English, close to mathematics. 

Also, I find out when I translate my articles. For example, I need to write an abstract. For some international journal, it's going to be in two languages, for example Russian and English. I look in English. I have, let's say, 75% of the amount of sentences written in Russian words. It's shorter, straightforward. Also, in English we prefer something like short sentences. In Russia, for example, in Leo Tolstoy, War and Peace, you will find War and Peace. You will find something like two pages languages by the same Charles Dickens. I found out that the same in Charles Dickens' writing huge sentences. But for mathematics, English- English maybe the most, in my experience, the most close to express their idea, knowledge, very easy to understand, very easy to write, mathematics, much easier to write in any other languages, it's number one. So language and mathematics, while seemingly distant, share common futures and serve as conveyors of thoughts, ideas and concepts. 

Symbolic system, use words and structure to represent ideas, allowing for the exchange of information without requiring a deep understanding of the subject matter, which never happened in mathematics. Language, much more rich structure. You can explain something that you don't understand for yourself. For example, I talked today about Young-Baxter equation. I don't want to pretend that I understand this equation absolutely clearly like physicists no way. I look at this from one part, from the algebraic approach, and also I'm not far to be a full understanding of this. Okay, but I can express my opinion about, I can express my opinion about, I can express my approach and so on. 

It's a language, mathematics. You cannot do it. It's only one subject that always answers for the question why, why? And this is very important. That's why I love mathematics. But definitely language and mathematics have a lot in common, because there is no mathematics without language and I believe that we cannot express any of our thoughts without language. Mathematics is a kind of shortness and compact. It's some kind of observations that they make, that when you have an information, some piece of information and you just really study this. It became very small in your brain and take only one small cell. When you need to go back, you just open it up again in the big structure. Mathematics does the same in language so-called word problems. It's interesting. 

1:02:13 - Kimberly King

It is so interesting also the way you have explained that English has mathematical you know, and when you compare it to Russian or Ukrainian, I mean I can't even imagine. I have heard that English is one of the hardest languages to speak and to learn, which I don't believe. That, because I think Russian would be just off the top. I can't even imagine. So kudos to you for being so proficient and putting this all together. I think it's so fascinating and I love interviewing you every time we have you on. So thank you for your time today. This has been wonderful that you've shared your knowledge today. This has been wonderful that you've shared your knowledge, and if you want more information, you can visit National University's website. It's nu.edu. Thank you, doctor, so very much for your time. 

1:03:01 - Igor Subbotin

Thank you, Kimberly. I really appreciate it. I'm always happy to meet with you and I was happy to work with you as a one team to promote my favorite subject mathematics. 

1:03:15 - Kimberly King

We need mathematicians. Thank you so much. 

1:03:17 - Igor Subbotin

Thank you, thank you. 

1:03:22 - Kimberly King

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