Interview with Yulij Ilyashenko
Date: 10:30 02/10/2020
This is an interview with  Yulij Sergeevich Ilyashenko. He is mainly known for his 
contribution to Hilbert's sixteenth problem and holomorphic foliations. 
The interview will be around the following questions: 

1. Please tell us about your first mathematical
experiences: They were fostered by which people and who were your first mathematics teachers?

2. What is your first feeling of discovery in mathematics? 

3. I did my Ph.D. thesis based on your 1969 article "The origin of limit Cycles under perturbation of equation...". Please, tell
us how you started  to write this article?

4. In the same article, I learned that the zeros of Abelian integrals are responsible for producing limit cycles. Can you tell us about 
the origin of infinitesimal Hilbert's sixteenth problem and the shift from counting limit cycles to counting zeros of Abelian integrals?

5. Tell us about the Petrovsky-Landis 1955 article in which they claimed that have found an upper bound for the number of 
limit cycles of planar differential equations. What is the reason why in the area of limit cycles, we have many  failed or wrong arguments?  

6. Tell us about your proof of finiteness theorems for limit cycles. How many years it took to complete the proof? 
When did you know about Jean Écalle's proof?


7. What is your vision for the final solution of uniform boundedness for limit cycles? Which tools must be developed yet?

8. Do you think any arithmetic will be involved? For instance, studying polynomial vector fields by doing modulo primes?

9  What do you think of studying polynomial differential equations in a purely algebraic-geometric framework with less dynamics involved?

10 After the comlexification of planar differential equations under the name holomorphic foliations, many other problems and conjectures 
such as the minimal set problem have arisen, and now there are many people working on these problems without interests on limit cycles. 
What do you think about this process of getting away from the origin of mathematical  objects? 

11. Many interesting differential equations which come from nature have real numbers as time parameter! What is a complex time for you?

12 If you were to start a new career, what you would choose in mathematics or even science or art?





What is your favorite planar differential equations and why?