The Interconnected Web of Mathematical Knowledge with Professor Andrei V. Tetenov

The Interconnected Web of Mathematical Knowledge

When we begin our conversation with Professor Andrei, the first thing that strikes us is his philosophical approach to mathematics.

"You see, all this kind of research, they are just like many branches of one tree. They are all connected, they have some visible and invisible connections.”

His perspective on fractal geometry is refreshingly holistic. Rather than viewing it as an isolated mathematical discipline, he sees it as fundamentally interconnected with virtually every area of human knowledge.

"So it is not very easy to separate fractal geometry from other sciences because it had roots in many disciplines. Some ideas of fractal geometry we can find in ancient Greece. The others are in medieval mathematics and many works of classic mathematicians have very good fractal things."

"For example, if you take the Weierstrass function, Weierstrass function was invented by Weierstrass, but it is in some sense an unsolved problem of fractal geometry because for most parameters the dimension still is unknown."

The Weierstrass function shows that fractal geometry cannot be neatly separated from the rest of mathematics - or even from other sciences. Still an active site of research, its mysteries reveal how fractal geometry is less a closed-off branch than a web of connections reaching across disciplines. Of course, it is more complex.

"This influence of mathematical research has many stages. So one thing influences another, like a chain of dominoes," he adds.

The Complexities and the Passion

One of Professor Andrei's most significant research areas involves the study of self-similar sets and the Hausdorff dimensions of their extreme points. When we delve into this work. His passion for the subject is apparent.

"At first we considered fractals in the beginning of 2000s. Most fractals in plane, could be placed in some kind of a polygon - so for example, Sierpinski is a triangle, Sierpinski carpets is square. So there was a question whether the convex hull of a fractal is a polygon."

Whether the boundary of a fractal forms a polygon led Professor Andrei into deep mathematical territory.

"So I studied these convex hulls and then found that it may be, it is polygon in the sense that the measure of the sets of extreme points of a convex cone in a plane is zero. But in some cases, in good cases, its dimension is zero."

What emerges from this discussion is a picture of mathematical research where questions open up into landscapes of unsolved problems. The set of extreme points, he explains, "is a totally disconnected set," which adds another layer of complexity to their study.

Interconnectedness of the Discipline

Our discussion turns to Professor Andrei's work on the rigidity of self-affine arcs, we encounter more of his research. Professor Andrei’s explanation of the weak separation property reveals the connections between different areas of fractal theory.

"You see in the theory of self-similar sets, there is a so-called weak separation property. And this weak separation property applies only for self-similar sets. And it is used for finding the dimension of self-similar sets, maybe to prove that dimension is equal to similarity dimension or that the measure in the Hausdorff dimension is positive and finite."

The beauty of the approach becomes clear as he explains how this property extends beyond its original context: "But this application of this property, especially when we consider its violation, is much, much... It has some geometrical and topological character."

The geometric intuition he provides is particularly elegant: "If you apply it to self-similar curve, then you will see that it repeats with some minimal period... Because, otherwise, if it can be shifted along itself by a similarity, then it becomes some kind of periodic curve. And if we can shift it to an arbitrarily small distance along itself, then it becomes a line." "But this property can be extended to self-affine curves.”

On Fractal Tilings

Professor Andrei credits much of this work to his colleague: "So it is not so much my work as the work of Dmitry Mekhontsev who studied, he made this IFS style program for studying of the tiles and he was studying the tiles very long ago - since 2001 maybe. And they made different programs and then he had success."

"He made a program which can find millions of tiles. You need only to look through them.” He adds that merely looking at the parameters will help you find the special tiles.

Professor Andrei was also very enthusiastic to show us what he means in a computer program - it helped us remarkably with visualizing the variety.

Mathematical Collaboration

Professor Andrei also reflected on the collaborations in his mathematical research journey. His most fruitful collaboration was with "Professor Saif. He is now 76 years old. And for a short time he had an excursion to fractal geometry. And then he returned again to quasi-symmetricity. It was maybe 2001, and he gave me advice to look at fractal geometry."

This advice proved transformative: "At that moment, I heard his question about convex hulls. So this was a problem which was quite easy to solve at first level and quite different to solve at further levels because I still don't understand how the convex hull of self-similar sets in dimension 3 look like. Because these convex hull are very complicated. So there are a lot of open questions in that area. He proceeded to give us many intricate examples which he has worked on.” His interaction with Professor Saif continues all the time - they discuss the questions of complex analysis and more.

His collaboration with Professor Chand has been equally productive: "There is very good collaboration with Professor Chand. It began, I suppose, in 2013, it's 12 years." Also interesting is his account of how a simple question from Professor Chand led to years of deep thinking: "Chand asked me for a counter example. Is it possible to construct a similar curve in R3, in three-dimensional space, which doesn't satisfy open set condition?"

The solution took five years to develop: "So I understood how to do this, because some idea appeared. I suppose I visited him in 2010. And suddenly in 2015 I found the answer after five years of thinking, so it was not the most intense thinking of all time, but just some kind of backwards work of thought."

The Philosophy of Mathematical Rigor and Innovation

We asked Professor Andrei to address the relationship between mathematical rigor and innovation. Here is what he had to say,

"There is no separation between mathematical rigor and mathematical innovation."

"Because if you read some, for example, a classical book of topology, be it Bourbaki or be it Kuratowski, then the spirit of innovation rests everywhere here. But at the same time, Bourbaki's book is the masterpiece of mathematical rigor. Because it has very good educational sense in that sense that they always show the shortest and most optimal way for proving even the simplest things.”

He adds that a very firm feeling of self-assurance is formed when one has read the book, understood it, and then seen the exercises with counter examples. One goes back to fill the gaps in their understanding.

"Without good mathematical rigor, you cannot do any kind of mathematical innovation."

The Mystery of Mathematical Intuition

Professor Andrei’s personal anecdotes reveal the almost mystical aspects of mathematical discovery.

"There are different ways of mathematical thinking. Sometimes you need to solve the problem, and if you really begin to solve the problem, then you cannot stop from solving it. You can fall asleep, then wake up and think a little more, and suddenly you wake up and the problem is solved, because it was solved while you were sleeping,” he laughs.

"Some flashes of intuition are unpredictable."

During his student days, traveling from Novosibirsk to Novosibirsk University: "There was one place when I was passing it and a good idea came to me. I caught an idea how to construct a very special example of Kleinian groups."

"Sometimes an idea comes to me when I am travelling -- when it is snowing and dark. And suddenly I understood how to solve the main problem in my PhD thesis. It was just a moment. I remember where I stood, what I had seen, and it was just a moment.”

He shared with us an anecdote about another mathematician: "There was one very good mathematician. He had some psychological problems, but he was a very good mathematician. He was thinking about a problem... And he saw in a dream that he sits in his office and one woman comes to him and says, what to do with that and that. And he said to her, do this and this, and wrote it on a paper. Yes, and when he woke up, this 'do this and this' was just a solution to his problem."

Guidance for the Next Generation

We asked Professor Andrei advice for mathematicians entering fractal geometry, "Is it something very simple or something very complicated?”

“Because each one has his own type of thinking, own tempo of development”, he replied, “I cannot advise anyone what is good and what is bad. I can only encourage."

And on the role of visualization in understanding fractal geometry?

"In some sense it helps. You see, you get some feeling what happens, how it looks like."

The value of experimentation, even failed experiments, is something he emphasizes strongly: "It's very encouraging to make an experiment. Even if it doesn't show anything, it shows something. So I made lots of experiments. Some experiments were useful, some were pragmatically useless. Sometimes I got very nice, wonderful pictures, yes, which I could not understand.”

One particular picture captures his sense of wonder: "For example, this one. I don't know what it is. I don't know what it is, but it looks very nice. There were some mistakes inserted in the code and this led to this very nice picture. I'm looking at it and thinking, what is this?"

The next major frontier in fractal geometry research? How does the Professor think the field will evolve in the coming years?

“It is not very easy to imagine all things can turn to an unpredictable direction.”

“These things have many, many outcomes. Some outcomes are in science, some outcomes in visual arts, aesthetics, something maybe in understanding human consciousness."

“So it’s just a superficial observation that fractal geometry is something different from our usual life."

A Personal Journey Through Mathematics

The most touching part of our conversation comes when Professor Andrei reflected on his personal journey into mathematics. His path was evident from an early age:

"As I was told by my parents, it was clear because there was some kind of psychological attachment to numbers, computation, etc. At a very early age I was writing not letters but numbers."

His transition from Kleinian groups to fractal geometry in the early 2000s represents a case study in how mathematical careers evolve. The transition was motivated partly by practical considerations:

"There were some very difficult unsolved problems in the groups which I was solving. Before 2000, I had very low access to mathematical papers."

"Really I began to study when I was just beginning to study. We read some works of Thurston... But many of the things were unavailable. So we had some very serious gaps in our knowledge of current results. So this was the situation in 80s and 90s."

When he moved to fractal geometry, "And then when I came to a new area, then it was much easier. It was much easier to get new results and questions."

But his previous experience wasn't wasted: "But my experience in Kleinian groups, of course, I considered the limit sets of Kleinian groups which are diverse fractals. So I was working fractal geometry but I didn't know that."

The Counter-Intuitive Nature of Mathematical Discovery

When we suggest that fractal geometry has particularly counter-intuitive approaches, his response is characteristically nuanced.

"I cannot say this because general topology also contains a lot of complicated approaches. Mathematical paradoxes appeared all the time. From very ancient times we had a lot of paradoxes. For example, the proof of irrationality of square root of 2 is also, at some moment was such a counter example. But then it was firmly established in human minds."

"You see, science develops like, for example, when you come to some uninhabited place and begin to build something. So when you build, then you have some roads and streets and everything looks nice.”

"But on the border of this area, there are problems. There is nothing in this inhabited place."

The Continuing Journey

As our much-appreciated conversation drew to a close, Professor Andrei reflected on the questions we've explored together. The conversation ended, but the questions it raised will continue to resonate. In the world of fractal geometry that Professor Andrei has opened up for us, every ending is also a new beginning, every conclusion a hand-hold to explore further.