# After Turning Down Offers in New York and Berlin, Chinese Student Pursues Mathematics Training at HSE

For DaoDao Yang, a native of Dali, China, choosing a programme in mathematics came down to the decision of where he felt most challenged and most comfortable. This ultimately meant turning down offers at New York University and the Berlin Mathematical School to accept an offer from the Master’s programme at the Higher School of Economics. In an extensive interview with the HSE news service, Dao Dao recently spoke about what led him to choose HSE for his postgraduate study; his impressions of the programme, its professors and his fellow students; and his future plans.

**— Where did you study previously, and what brought you to the Faculty of Mathematics at HSE?**

— 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.

I have been interested in topology since high school. In fact, the first topology book I read was by the Soviet mathematicians Boltyansky and Efemovich. Their book was translated into Chinese, and contained a number of interesting topics and theorems, for example, the Sierpinski triangle, the Peano curve, the Classification Theorem for Compact Surfaces, and so on. Ever since high school, I have been influenced by Russian mathematics.

Before I came here, I received admissions offers from three programmes: the Master’s at the Higher School of Economics, the Master’s at New York University (NYU) and the PhD programme at the Berlin Mathematical School (BMS). I found the programme at NYU to be too elementary (they did not agree to waiving basic courses), and while the admissions offer from BMS included full scholarships, I thought having 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 at HSE and BMS, I thought that Professor Viktor Vassiliev would be the most suitable person for me. I contacted him, asking whether he could be my advisor. He said ‘yes’, so I came and joined Master's programme in Mathematics.

HSE has a number of excellent mathematicians. What is more important is that they are very kind and they are willing to spend time with students.

**— What did you like most about the Faculty of Mathematics?**

— First, HSE has a number of excellent mathematicians. What is more important is that they are very kind and they are willing to spend time with students. My advisor, Professor Vassiliev, is such a mathematician. On average, we meet at least once a week, and during our meetings, he is very patient in answering my questions. If a notion or theorem is hard to understand, he will explain several examples to me, from a simple one to a difficult one. He emphasizes the ideas and notions more than the details to prove the theorems.

Besides Professor Vassiliev, during the two years, I also have learned a lot from Alexey Sossinsky, Alexey Gorodentsev, Maxim Kazarian, Sergei Lando, Serge Lvovski, Vadim Vologodsky, Vladlen Timorin and others. All of them are very nice and friendly professors. Their teaching styles are quite different. It’s enjoyable to discuss math with them.

Second, the academic atmosphere here is quite nice. I have been impressed by my classmates; they usually ask a lot of interesting questions during class. Moreover, HSE students are really hard-working. I once was shocked when seeing a third-year undergraduate student reading a math book even when he was crossing the street!

**— What has been 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 are that the instructor can freely design what to teach and ignore content he thinks is not important. Usually, the instructor also introduces the contents related to his or her research. The disadvantages are that several instructors speak very fast and do not write on the blackboard what they have said. So the students may not understand right away and might get confused.

Fortunately, when students do not understand something, they can make an appointment with professors to discuss the problems.

**— Has anything surprised you at HSE?**

— Yes. Beginning at very young age, in fact, from their first year study at HSE, students do research under the supervision of excellent mathematicians. I think that there are just a few schools in the world where undergraduate students could receive such high-level research training.

Actually, this is an excellent Russian tradition. We know that Vladimir Arnold solved 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 Andrey Kolmogorov. What’s more, Kolmogorov had made contributions to Hilbert's thirteenth problem before Arnold solved it. Arnold then expanded on his work. I don’t think Arnold could have solved Hilbert's thirteenth problem at the age of 19 if he had not been a student of Kolmogorov. Similarly, in his second year as an 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 had been proposed by Arnold.

**— What are your thoughts on the courses offered in the Faculty of Mathematics?**

— HSE’s math faculty offers a lot of advanced courses. For example, the differential topology course by Vologodsky includes the Pontryagin-Thom Construction and the proof to h-cobordism theorem. The knot theory course by Sossinsky includes the Jones polynomial, Vassiliev invariants and Kontsevich integral. And the singularity theory course by Vassiliev includes the Thom Polynomials and the Smale-Hirsch-Gromov h-principle.

In my opinion, the math education and research training methods used at HSE are some of the most successful in the world. I think HSE alumni could eventually win the Fields Medal.

**— How have these two years of study shaped your research interests, if at all?**

— During the two years, I have completed two research projects, and I now know that my research interests cover topology, knot theory, singularity theory and complex algebraic geometry.

I have taken a wide range of courses, from number theory, algebra and geometry to topology. I was particularly influenced by the knot theory and topology courses by Professor Sossinsky, the differential topology course by Professor Vologodsky, and the topology and singularity theory courses by Professor Vassiliev.

My research projects concern Picard–Lefschetz theory, which combines both algebraic topology and complex algebraic geometry. During my research, I encounter a number of fascinating geometry pictures, for example, the classification of algebraic hypersurfaces, the transversal and asymptotic cases for Picard–Lefschetz formulas, the vanishing cycles and Milnor fibres.

**— What are your plans following graduation from HSE, and will you be returning to Moscow?**

— I will go to Indiana University Bloomington to pursue a PhD in Mathematics. I hope to come here again to continue discussions 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 not enough for me to understand the entire theories and learn the powerful tools.

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Daodao Yang

Master's student at the HSE Faculty of Mathematics