


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 1dimensional manifolds are called
curves, the 2dimesional  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)
twodimensional 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
selfintersection $K^2$ 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 dth
homogeneous component of the symmetric or the antisymmetric
algebras of a module E can be obtained from dth homogeneous component of its tensor algebra T(E) via a total symmetrization
or antisymmetrization 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 onedimensional character χd
of a permutation group Wd of degree d, thus obtaining so called semisymmetric
powers [χ]d(E). Under some
natural conditions on the sequences W =(Wd) and χ = (χd),
the direct sum of these semisymmetric 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 onedimensional characters are classified via some onedimensional 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 dth
homogeneous component [χ]d() of a semisymmetric 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 semisymmetric
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 LunnSenior'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 3dimensional 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. LunnSenior's thesis is that certain nonnegative 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 onedimensional 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 faraway 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 trisubstitution 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 monosubstitution, disubstituti\on, and trisubstitution
homogeneous derivatives; the groups of substitution isomerism and stereoisomerism of cyclopropane can be reconstructed up to conjugation if we know the numbers of its monosubstitution and disubstitution 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 LunnSenior 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 nonzero 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.


