Трапезникова Ольга Юрьевна, Факультет математики (Москва)

Благодаря своему трудолюбию и математическому таланту, Оле с блеском удаётся разбираться в, на первый взгляд, самых непростых вопросах. Особенно здорово ей удаётся классическая алгебраическая геометрия, которой она замечательно занимается под руководством В. Жгуна.

Сергей Яковенко, профессор института Вайцмана (Израиль) просил поместить следующий развернутый комментарий:

Dear colleagues,
This letter is to express my full support to the nomination of Olya Trapeznikova for the prize. Olga was selected for the Kupcinet International Summer School for undergraduate students in the Weizmann Institute, a highly competitive program lasting 2 months in summer (in particular, based on her excellent grades and very warm recommendations). Quite obviously, one does not expect real results (usually
in the area not very familiar to a student) in so short time, yet from time to time there are pleasant surprises. Olga chose me as a mentor and I gave her a rather old paper by M. Kashiwara which puzzled me for quite some time. The Kashiwara theorem describes the monodromy group of local systems (at regular meromorphic connections), more specifically, the monodromy along "small loops" (supported by singular holomorphic curves). The main claim is that this if the monodromy operators have only roots of unity as eigenvalues for small loops centered at codimension 1 component of the singular locus, then all small
loops (no matter how degenerate are the supporting curves) retain this property.

Usually results of this type are proved using the desingularization technique, using a sequence of blow-up transformations reducing the situation to the simplest case where the singular locus is a normal crossing. However, the original proof by Kashiwara (very complicated and difficult to read) clearly followed a
different line. The goal of the project was an attempt to find an alternative proof of the Kashiwara theorem using the desingularization scheme. A simple one-step examples suggested that this was possible, yet in an unexpected way the difficulties mounted when studying more involved examples.

It turned out into a challenge (we almost daily met with Olga and discussed the difficulties and insights. To make a long story short, we succeeded in finding the desingularization-based proof of the Kashiwara theorem, yet it features an argument which I did not see previously in any such proofs. After the desingularization tree is constructed, usually the proofs of required properties go by induction from the leaves of the tree all the way down to its root. In this case it turned out that one has to study a certain function which is harmonic at the tree (i.e., its value at any vertex is the mean of values in all adjacent vertices) and the conclusion follows from a discrete analog of the uniqueness for a Dirichlet boundary value problem. Thus it was not just the tree-like structure of the desingularization scheme, but the global properties of this tree, which are central for the proof.

We are now in the process of writing up our findings (which also imply a certain generalization of the Kashiwara theorem for monodromy operators having only modulus one eigenvalues) and plan to submit them for publication in a good journal. In my 25-years-old practice of working with summer students, this is
only the fourth case where the "toy" summer project apparently will evolve into a fully edged research paper.
It was a real pleasure to work with with Olga. She is very modest and shy, yet her proficiency in mathematics is surprising for an undergraduate student. She is keen, assiduous and curious in the best sense of these words.

I cannot think of a better candidate for the HSE Students' prize. This silver nestling will soon spread golden wings.

Sincerely yours, Sergei Yakovenko