The combinatorial Mandelbrot set as the quotient of the space of geolaminations

We interpret the combinatorial Mandelbrot set in terms of \it{quadratic laminations} (equivalence relations $\sim$ on the unit circle invariant under $\sigma_2$). To each lamination we associate a particular {\em geolamination} (the collection $\mathcal{L}_\sim$ of points of the circle and edges of convex hulls of $\sim$-equivalence classes) so that the closure of the set of all of them is a compact metric space with the Hausdorff metric. Two such geolaminations are said to be {\em minor equivalent} if their {\em minors} (images of their longest chords) intersect. We show that the corresponding quotient space of this topological space is homeomorphic to the boundary of the combinatorial Mandelbrot set. To each equivalence class of these geolaminations we associate a unique lamination and its topological polynomial so that this interpretation can be viewed as a way to endow the space of all quadratic topological polynomials with a suitable topology.Comment: 28 pages; in the new version a few typos are corrected; to appear in Contemporary Mathematic

Schubert calculus and Gelfand-Zetlin polytopes

We describe a new approach to the Schubert calculus on complete flag varieties using the volume polynomial associated with Gelfand-Zetlin polytopes. This approach allows us to compute the intersection products of Schubert cycles by intersecting faces of a polytope.Comment: 33 pages, 4 figures, introduction rewritten, Section 4 restructured, typos correcte

Symmetric cubic polynomials

We describe a model $\mathcal{M}_3^{comb}$ for the boundary of the connectedness locus $\mathcal{M}^{sy}_3$ of the parameter space of cubic symmetric polynomials $p_c(z)=z^3-3c^2z$. We show that there exists a monotone continuous function $\pi:\partial \mathcal{M}_c^{sy}\to \mathcal{M}_3^{comb}$ which is a homeomorphism if $\mathcal{M}^{sy}_3$ is locally connected.Comment: 34 pages, 3 figure

Creating world class department

В статье рассказывается о становлении факультета математики НИУ ВШЭ. Факультет был создан в 2008 г. и к настоящему времени, по оценке ряда ведущих экспертов, является одним из мировых лидеров подготовки исследователей в области математики и ее приложений. В статье описаны стоящие перед факультетом новые задачи и способы, которые предполагается использовать для их решения, принципы, на которых строится факультет, и применяемые на нем методы управления учебным процессом и исследованиями.The article describes the creation of faculty of mathematics at NRU HSE. It was founded in 2008 and by now has become one of the international leaders in training researchers in the field of mathematics and its applications. The article describes new tasks for the department and ways of dealing with them, principles, on which the department is built and methods of teaching process and research management used there