Диссертации, представленные на защиту и подготовленные в НИУ ВШЭ

Classical Hurwitz numbers have many definitions, among them there is a topological one as thenumber of ways to glue ribbons to a collection of disks so as to obtain a given surface with boundary. A purely algebraic one as the number of sequences of transpositions with a product of a specified cyclic structure;and an algebro-geometric one as the number of ramified coverings of the sphere with a prescribed number of simple critical values and a given type of ramification over the infinity. Furthermore the generating function of the Hurwitz numbers satisfies a PDE parabolic equation called the cut-and-join equation, and as a result can be expressed via Schur polynomials. Finally the generating function of Hurwitz numbers is a tau-function for the KP hierarchy. We will present new types of Hurwitz numbers: in the first part we will describe "twisted" versions of all this: ribbons are allowed to twist (literally), transpositions double, and the ramified coverings become conjugation-equivariant. The cut-and-join is replaced by the Laplace--Beltrami equation for the parameter b=1, and an explicit formula for the twisted Hurwitz numbers uses zonal polynomials instead of Schur. In the second part we will show that there is a parallel theory of Hurwitz numbers for the reflection groups of series B and D : transpositions are replaced by reflections, the cut-and-join for the group B is actually a tensor square of the classical cut-and-join, and the cut-and-join for the D group is a direct sum of the cut-and-join for B and the classical cut-and-join rescaled. This leads to an expression of the Hurwitz numbers for B and D via Schur polynomials. The generating function for the Hurwitz numbers for the reflection group B is a 2-parameter family of tau-functions of the KP hierarchy, independently in 2 variables, and we have a similar result for D. Finally Hurwitz numbers for the group B involve ribbon decomposition of surfaces equipped with an involution.

A classical problem going back to the XIX-th century works by A.Hurwitz et al. is to count meromorphic functions with prescribed singularities on complex curves. Most results of the thesis deal with ramifications of this problem: the curves with real structure, complicated singularities and counting with weights. These results occupy mostly Chapters 1 and 3 of the thesis. The analysis of the problems involves some results of combinatorics similar to the classical matrix-tree theorem by G.Kirchhoff (1847). The thesis contains numerous generalizations of this theorem, too, mostly found in Chapter 2.

The dissertation is dedicated to number marking across lexical classes in the East Caucasian language family. I focus on non-nominal number and specialized number marking, i.e. those instances where number is not cumulated with other categories, such as gender or definiteness. Investigating these patterns in genealogical, areal, and diachronic perspective allows me to describe specialized number marking in East Caucasian and hypothesize how number marking phenomena may arise.In particular, I investigate number marking on adjectives and verbs across East Caucasian. Then I discuss in more detail number marking in dialects of Andi.First, it turned out that what looks like number marking may instantiate different phenomena, depending on lexical class and syntactic context: genuine number (nominal and pronominal number, number expressed on modifiers, verbal number), number agreement or allocutive number marking.Second, number marking on adjectives and verbs is conditioned differently in different languages. Possible conditions include 1) lexical (some lexemes are marked for number, while others are not), 2) full / short form (for adjectives), 3) animacy of the NP head or verb’s argument, 4) TAM-form of the verb. Besides, in some languages number marking in certain lexical classes may be obligatory or impossible altogether.Third, diachronic sources of specialized number marking include nominal plural markers (with adjectives and verbs alike), distributive marking (with adjectives) and full / contrastive morphology (with adjectives). For many patterns, diachronic sources have not been identified. In several East Caucasian languages, number agreement or agreement-like phenomena may have been borrowed under contact influence.

The thesis examined how specific executive functions (i.e., updating, shifting, and inhibition) relate to emotion regulation strategies (i.e., cognitive reappraisal and expressive suppression), and to understand if executive functions working with affective information are more related to emotion regulation. Four major experiments were carried out. Overall, the results showed that the effectiveness of emotion regulation strategies manifests itself differently depending on the type of measures used during the assessment. Non-affective and affective tasks of executive functions do not differ in their relationship with emotion regulation strategies. While higher updating was related to cognitive reappraisal, inhibition was associated with the more frequent use of expressive suppression. The implications of these findings are discussed in relation to existing theories and practice.