'I was influenced by Russian mathematics'

Daodao Yang shared his experience of studying at the HSE master’s programme in mathematics.

1. Tell about yourself. Where are you from? Where have you been studying before?

I am from Dali, a tourist attraction in southwest of China. I studied Communication Engineering at the Beijing University of Posts and Telecommunications from 2010 to 2014. I also studied mathematics at the Peking University during the Spring 2014 semester under the Enhanced Mathematics Program for Graduate Study.

 2. Why did you decide to study at Faculty of Mathematics of Higher School of Economics?

I have been interested in topology since high school. In fact, the first topology book I read was the book written by Soviet mathematicians В.Р.Болтянски and В.А.Ефемович. That book was translated in Chinese, including a lot of interesting topics and theorems. For example, the Sierpinski triangle, the Peano curve, the Classification Theorem for Compact Surfaces and so on. Thus, since high school, I have been influenced by Russian mathematics. Before I came here, I received admissions from three programs: the Master of Higher School of Economics (HSE), the Master of New York University (NYU) and the PhD of Berlin Mathematical School (BMS). I received the admission from NYU first. But I found their Master program was quite elementary. The courses are something like Introduction to Math Analysis and Linear Algebra. When I sent them e-mail asking whether I could waive those courses and take the PhD-level courses instead, they did not agree. So I thought it was not a suitable program for me. Later, I received the admissions from HSE and BMS. Both two programs offer advanced math courses. It’s hard to choose between HSE and BMS. At that moment, HSE did not promise to provide me with full scholarship, while the admission letter of BMS included full scholarships. However, I thought a great advisor would be very important for a person like me whose undergraduate major was not math. I was interested in complex analysis, topology and algebraic geometry. After carefully reading the webpages of professors of HSE and BMS, I thought professor Victor A. Vassiliev would be the most suitable person for me.  His research areas covers a number of math branches, from knot theory, singularity theory, topology, integral geometry, partial differential equations, complex analysis, to Picard-Lefschetz theory. And from his photos in the Internet, I could feel that he is a very nice and kind person. So I decided to make a bet. I contacted professor Vassiliev, asking whether he could be my advisor. He said ‘Yes’.  Then I came.

 3. What did you like most in the Faculty of Mathematics?

 First, HSE has a number of excellent mathematicians. What is more important is that these mathematicians are very kind and they are willing to spend time with students. My advisor, professor Vassiliev, is such a mathematician. In average, we meet at least once a week (except the summer vacations). During our meeting, professor Vassiliev is very patient to answer my questions. If a notion or theorem is hard to understand, he will explain several examples to me, from the simple example to the difficult example. He emphasizes the ideas and notions more than the details of proof to theorems. With his help, I could learn very fast and understand the modern language of mathematics. Moreover, each time we discuss several hours. In total, since September of 2014, professor Vassiliev has spent more than 200 hours to teach me separately. With his careful guidance and encouragement, I develop my interest in topology and I am more confident in mathematical research than 2014, the time when I arrived in Moscow. I really appreciate him. Besides professor Vassiliev, during the two years, I also have learned a lot from professors Alexey Sossinsky, Alexey Gorodentsev, Maxim Kazarian, Sergei Lando, Serge Lvovski, Vadim Vologodsky, Vladlen Timorin and so on. All of them are very nice and friendly professors. Their teaching styles are quite different. It’s enjoyable to discuss math with them. I thank all of them. 
Second, the academic atmosphere here is quite nice. Both HSE undergraduate and graduate students are very active to discuss with professors in the class. For instance, I was impressed by my classmates Vasily Bolbachan, Aleksanrd Berdnikov and Svetlana Makarova. They usually asked a lot of interesting questions during the class. Moreover, HSE students are really hard-working. I once was shocked when seeing a third-year undergraduate student was reading a math book even when he was crossing the streets with traffic lights! This guy was my classmate for the course differential topology.  I do not know his name, but I am impressed by his wonderful comments during the lectures. It’s very exciting and nervous to study with those smart guys. I usually tell myself that I should work harder since there are so many students here that are younger than me, smarter than me and know more math than me.

  4. What was the most challenging part of your study in Moscow? 

The most challenging thing is that no instructor follows a textbook. There are some advantages and disadvantages for this.The advantages that the instructor does not follow a textbook are as follows: The instructor can design freely what to teach. And the instructor can teach what he thinks are most valuable and ignores those contents he thinks are not important. Usually, the instructor also introduces the contents related to his or her research. The disadvantages that the instructor does not follow a textbook are as follows.
Several instructors speak very fast and they do not write what they have said on the blackboard. So the students may not understand right away and might soon get confused. Fortunately, when students do not understand something, they can make an appointment with the professors to discuss the problems.

 5. How did these two years of MSc study shape your scientific interests (if they did)?

During the two years, I choose lots of courses and do two research projects. Now I know that my scientific interests are topology, knot theory, singularity theory and complex algebraic geometry.  
I choose a wide range of courses, from number theory, algebra, geometry to topology. I was particularly influenced by the following courses: the knot theory course and the topology course by professor Sossinsky, the differential topology course by professor Vologodsky, the topology course and singularity theory courses by professor Vassiliev. Though I took more algebra courses in my first semester, later I found that I was more interested in topology than algebra. I preferred concrete topological pictures to abstract algebraic constructions.
My research projects are about Picard–Lefschetz theory. This theory combines both algebraic topology and complex algebraic geometry.  I do research on the fundamental group of the complement of the discriminant of an algebraic hypersurface and the monodromy action of this fundamental group on several kinds of relative homology groups associated to the algebraic hypersurface and a generic hyperplane. During the research, I meet a lot of fascinating geometry pictures, for instance: the classification for algebraic hypersurfaces, the transversal and asymptotic cases for Picard–Lefschetz formulas, the vanishing cycles and Milnor fibers.

 6. Did anything surprise you in the mathematical education process at HSE?

Yes. At very young age, in fact, since the first year study at HSE, students have been doing research under academic supervisions of excellent mathematicians. Furthermore, a number of young students can make very significant achievements rather than simple computations for some practical examples or explanations for certain new theories. For instance, since first year undergraduate, Aleksanrd Berdnikov has written several term papers under the supervision of professor Vassiliev. His research topics included The Schwartz Genus of a Fiber Bundle and Topological Complexity of Roots of Real Polynomial Systems.  Another example is Svetlana Makarova. During undergraduate study, she has done research on S-Λ Duality and Covariant Serre–Grothendieck Duality under the supervision of professor L. Positselski. I think that there are just few schools in the world that undergraduate students could receive such high-level research trainings. Actually, this is an excellent Russian tradition. We know that Arnold solved the Hilbert's thirteenth problem at the age of 19. Usually, people would emphasize the extraordinary math talent of Arnold and ignore an important fact that Arnold was a student of Kolmogorov. What’s more, Kolmogorov had made contributions to the Hilbert's thirteenth problem before Arnold solved the problem. Arnold then expanded on his work. Thus I do not think Arnold could solve the Hilbert's thirteenth problem at the age of 19 if he was not a student of Kolmogorov. Similarly, at the second year of undergraduate, Vassiliev began to do research under the supervision of Arnold. Later, at the age of 23, Vassiliev solved a very important problem in singularity theory, which was suggested by Arnold.
HSE math department offers a lot of advanced courses. For example, the differential topology course by Vologodsky included the Pontryagin-Thom Construction and the proof to h-cobordism theorem. The knot theory course by Sossinsky included the Jones polynomial, Vassiliev invariants and Kontsevich integral. And the singularity theory course by Vassiliev included the Thom Polynomials and the Smale-Hirsch-Gromov h-principle.
In my opinion, the math education and research training methods of HSE are one of the most successful in the world. I think before the year 2059, HSE’s alumni could receive the Fields Medal.

 7. What are you going to do after graduation? 

I will go to the Indiana University Bloomington for a PhD degree in Mathematics.  

 8. Will you visit Moscow again in future?

Yes, I am willing to come here again to discuss with professor Vassiliev and other mathematicians. I am still interested in singularity theory, Picard–Lefschetz theory and the spectral sequences developed by professor Vassiliev. Two years are still not enough for me to understand the whole theories and to learn the powerful tools. In the following several years, I plan to visit Moscow approximately from May 9 to July 7 (during the summer vacations of Indiana University). I think it is very valuable to come here again for research.