Main Research Accomplishments and Research Plans
Valentin Vankov Iliev: Main Research Accomplishments

 

 My publications are in three mathematical areas:
1) Algebraic geometry: theory of surfaces;
2) Algebra: associative algebras, representation theory of the symmetric group;
3) Combinatorics: applications of the representation theory of the symmetric group to enumeration problems, and to some problems of isomerism in organic chemistry.

Below I give a brief description of my research in these three branches of mathematics.

1) Algebraic Geometry: Theory of Surfaces
Algebraic geometry studies the solution sets of systems of polynomial equations with coefficients in a field, which in the classical case coincides with the field of complex numbers. These solution sets are called algebraic manifolds. The most fundamental invariant of an algebraic manifold is its dimension. The 1-dimensional manifolds are called curves, the 2-dimesional - surfaces, and so on.  The theory of algebraic curves, or, which is the same, Riemannian surfaces, is a classical mathematical discipline which yields prototypes of results in higher dimensions.  The rough classification of algebraic curves is made according to their genus g, that is, the number of 'handles' of their underlying topological space.  When g = 0, we obtain the (real) two-dimensional sphere or, equivalently, the projective line.  If g = 1, we get the elliptic curves.  In case g is greater than 1 (this is so called 'general case') the classification is based on the relation between the rational maps of the curve in a projective space and the classes of divisors on that curve.  It is known that the map defined by the canonical class K is, in general, a birational isomorphism and so called projective model of the curve is defined by this map up to a projective transformation. The classification of the surfaces 'repeats' with significant complications the above picture: The surfaces which are analogues of the projective line and the elliptic curves have a complete classification. The surfaces which are analogues of the curves with genus greater than 1 are called surfaces of general type.  Two invariants are fundamental for the classification of the surfaces S of general type:  their geometric genus p(S) and the index of self-intersection $K^2$ of their canonical class K.  These invariants are related via Noether inequality

 p(S) ≤ K² + 2.            (1)

In my papers [1], [2], [3], [6] and [7] I consider so called Horikawa surfaces for which (1) becomes an equality.  The main results in [1] - [3] contain a complete description of the canonical models of Horikawa surfaces S with p(S) = 3 and K² = 3 and the assertion that the canonical model of a type II surface is deformation of canonical models of type I surfaces (the second result was proved by Horikawa when the canonical class K is ample).  The results from [3] are cited in the monograph Compact Complex Surfaces of Barth, Peters and van de Ven.  In the papers [6] and [7] are described the canonical rings of almost all other Horikawa surfaces via generators and relations.

2) Algebra: Associative Algebras and Representation Theory of the Symmetric Group.
The d-th homogeneous component of the symmetric or the anti-symmetric algebras of a module E can be obtained from d-th homogeneous component of its tensor algebra T(E) via a total symmetrization or anti-symmetrization with respect to the unit character or the signature of the symmetric group Sd, respectively.  In the paper [8] I generalize these classical constructions, starting with any one-dimensional character χd of a permutation group Wd of degree d, thus obtaining so called semi-symmetric powers [χ]d(E).  Under some natural conditions on the sequences W =(Wd) and χ = (χd), the direct sum of these semi-symmetric powers is a graded associative algebra [χ](E) which inherits its product from T(E).  If the sequence W is fixed, the sequences χ of one-dimensional characters are classified via some one-dimensional characters of a binary cyclic code.  The classification of the sequences W seems to be a very hard problem but a step forward is the description of their direct limits from [12]. The d-th homogeneous component [χ]d(-) of a semi-symmetric algebra is a homogeneous degree d polynomial functor on the category of finite dimensional spaces.  These functors (up to the language) are studied by I.Schur in his doctoral thesis from 1901.  They correspond to the finite dimensional representations of the symmetric group Sd via an equivalence of categories.  It turns out that the semi-symmetric powers [χ]d(-) correspond to the induced monomial representations of Sd (see [11]).

3) Combinatorics: Applications of the Representation Theory of the Symmetric Group to Enumeration Problems and to Some Problems of Isomerism in Organic Chemistry.
Using the main results from [11] and [9], I show in [13] that Williamson's generalization of the famous Polya's fundamental enumeration theorem is a particular case of the main equality of characteristics of corresponding objects in Schur's theory of invariant matrices.  Thus, Polya's enumeration theory can be seen as subsumed by the representation theory of the symmetric group. In [14] I generalize Redfield's Master Theorem and use this more general statement for proving a series of enumeration results on superpositions of graphs whose automorphism groups satisfy certain conditions. The aim of paper [15] is to present a generalization of Lunn-Senior's mathematical model of isomerism in organic chemistry.  The main idea of A. C. Lunn and J. K. Senior is that if the type of isomerism is fixed, a molecule with a fixed skeleton and d univalent substituents has a symmetry group G ≤ Sd which is generally not the molecule's 3-dimensional symmetry group.  The unit character of G induces a representation of the symmetric group Sd which governs the combinatorics of the isomers of the given molecule.  Lunn-Senior's thesis is that certain non-negative integers established by this representation are upper bounds of the corresponding numbers, yielded by the experiment (and often coincide with them).  Moreover, the authors define (in a particular case) a partial order among the objects of the model, such that some simple substitution reactions correspond to inequalities.  These two groups of data determine the group G, and produce so called 'type properties' of the molecule (properties which do not depend on the nature of the univalent substituents). My hypothesis is that if one replaces the unit character of G by any one-dimensional character of G (thus we count only a part of the isomers - those having a maximum property), we also get a type property of the molecule. An instance of that is the inventory of the stereoisomers called chiral pairs. The formalism can be generalized naturally and produces some preliminary chemical results.  Especially the partial order is defined and studied in the general case and indicates the possible genetic relations among the corresponding molecules.  This partial order can be used in the following way: the relation A < B between the isomers A and B is an indication of the existence of a finite sequence of simple substitution reactions

B → C1 →...→ Cr → A,

where the compounds C1,..., Cr, are intermediate stages in a synthesis of A.  Such a sequence C1,..., Cr (which is far-away of being unique), can be constructed explicitly.  The fact that the inequality
A ≤ B is not
satisfied implies that the isomer A for sure can not be obtained from the isomer B via a finite sequence of simple substitution reactions.  The partial order is tested for finding the genetic relations of the substitution derivatives of ethene.  It is also applied in the case of di-, and tri-substitution derivatives of benzene and yields the classical Körner's relations.  An important result of E. Ruch which connects the dominance order among partitions and the existence of chiral pairs is obtained as a consequence of a much more general statement.  Ruch's formulae for the number of isomers corresponding to a given partition of d are generalized. In their fundamental paper from 1929, Lunn and Senior show that: the groups of substitution isomerism and stereoisomerism of ethane can be reconstructed up to conjugation if we know the numbers of its mono-substitution, di-substituti\-on, and tri-substitution homogeneous derivatives; the groups of substitution isomerism and stereoisomerism of cyclopropane can be reconstructed up to conjugation if we know the numbers of its mono-substitution and di-substitution homogeneous derivatives.
The proofs are exhaustive quests through the list of orbit numbers for all subgroups of the symmetric group of degree 6.  In [16] and [17] we present more conceptual proofs of these statements.

Research Plans:

My future plans include algebraic description of the symmetry groups (in Lunn-Senior sense) of given skeletons and given types of isomers in organic chemistry. Here I also intend to use the computer program GAP and the data from some appropriate reference book on isomers.  Using the groups thus obtained and the data from some reference book on isomers, I intend to check if the lattice of the dominance order for different skeletons corresponds to chemical reality.  This was checked by Lunn and Senior for many cases but only when the corresponding partition of $d$ has only two non-zero components.  Upon passing this check, the dominance order can be used for the prediction of chemical reactions and for the rejection of hypotheses concerning existence of such reactions.