Abstracts of lectures

In the context of a decline in the incidence of coronavirus in many countries, the accumulated data allow presenting an analysis in a wide range of values ​​from the beginning of the epidemic to its end. We will discuss mathematical models of the dynamics of the number of cases, based on the so-called equations of the logistic type, which are, in the simplest case, first-order differential equations studied in junior courses. Analytical solutions of such equations are given. To estimate the coefficients of these equations, data on the incidence of coronavirus from the website of the World Health Organization (12 countries were selected) were used. The ways of improving mathematical models using the methods of modern mathematics, which are developed in science (including the Master's program at the HSE), are shown.

In a first approximation, crystals can be thought of as periodic tiling of a plane (or space) by polygon molecules (or polyhedron molecules). But is it possible to pick up such polygon molecules so that they can pave the plane without "holes" and the overlap of polygons, but so that this paving is not periodic? It turns out that this is possible, and Penrose tilings are an example of such tilings. Moreover, there are such alloys (they are usually called quasicrystals) in which the molecules are actually arranged in this way. The physical and chemical properties of quasicrystals are well studied experimentally. But attempts to see how these properties follow directly from the geometry of quasicrystals (and for this it is necessary to understand what the properties of the solutions of the corresponding Schrödinger equation are) in the general case remain unsuccessful. In some special cases, however, the answer has been received. Namely, in order to describe the properties of solutions of the Schrödinger equation, it is necessary to understand the spectral properties of the corresponding Schrödinger operator, and, say, in the case of one-dimensional quasicrystals, its spectrum will be a fractal (Cantor) set. The report will be devoted to how all these concepts are related to each other and to many other areas of mathematics.

My talk  focused on rhythm-generating neural circuits that can maintain a single oscillatory regime independent of sensory feedback or, depending on the external drive, produce and switch between multiple activity patterns or gaits with distinct phase-lags of the constituent neurons. Of particular interest are the regulation, multi-functionality and interplay between their states, connectivity and the dynamics of individual neurons and synapses in such circuits.