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\begin{center}
{\Large\bf Summary of Research}\\[2mm]
{\Large Sergei Duzhin}\\[3mm]
{May 11, 2010}
\end{center}
Here I am giving a review of my mathematical research done between 1981 and
2010. In is divided in three parts, ordered chronologically.
\section{Geometry of differential equations}
This area is related to my Ph.D. thesis \cite{Diss} written under the
supervision of Prof. A.\,M.\,Vinogradov.
It includes papers \cite{J1}, \cite{Fol}, \cite{DT}, \cite{DL},
\cite{2x2}, \cite{VINITI1}, \cite{VINITI2} and books \cite{SCL}, \cite{JN}.
Papers \cite{J1}, \cite{Fol}, \cite{DT} are dedicated to different variants
of Vinogradov's $\mathcal{C}$-spectral sequence generated by the filtration
of the de Rham algebra on an infinitely prolongated differential equation
(treated as a submanifold in the space of infinite jets) by the powers of
the ideal of differential forms vanishing on the Cartan distribution. This
sequence contains, as its terms, the space of infinitesimal symmetries, the
space of conservation laws and other invariants of the differential
equation.
In \cite{J1}, I studied a version of the $\mathcal{C}$-spectral sequence
on the manifold of 1-jets. One of the corollaries of this study is
a criterion for a Monge--Amp\`{e}re equation to belong to the class
of Euler--Lagrange equations and; in the positive case, I found an explicit
formula for the corresponding Lagrangian.
In \cite{Fol}, I introduced a spectral sequence associated with a foliation,
arising from the filtration of the algebra of differential forms by powers
of the ideal vanishing on the leaves of the foliation.
I found its relation with the deformation cohomology of foliations
and Gelfand--Fuks cohomology of Lie algebras of vector fields.
As a direct application of the classical Vinogradov's spectral
sequence, in \cite{DT} we proved that the nonlinear partial
differential equation $u_t = u_{xxt} + u u_x$ known in physics as
``Benjamin--Bona--Mahony equation'',
has no conservation laws other than those known earlier.
In the paper \cite{DL}
we present a geometric exposition and generalization of
S.\,Lie's and E.\,Cartan's theory of explicit integration of finite
type (in particular, ordinary) differential equations.
A new result contained in this paper is a method of hunting for particular
solutions of partial differential equations via symmetry preserving
overdetermination.
My last paper pertaining to this area is \cite{2x2}.
Here I give a classification of symbols of partial differential systems
that consist of two equations imposed on two functions in two independent
variables, with respect to the natural action of the group of linear
changes of dependent and independent variables.
Apart from the listed research papers, I have coauthored
two books belonging to this area: in \cite{SCL} (English translation
\cite{SCL-e}) I wrote one and a half
chapters about symmetries of differential equations; in \cite{JN}
(English translation \cite{JN-e}) I wrote two chapters about
fibre bundles.
\section{Computer algebra}
This section includes papers \cite{CDg}, \cite{CFD}, \cite{DK1}, \cite{DK2}.
My activity in the field of computer algebra comes, primarily,
from the fact that during 15 years (1985--2000) I was employed
in the Program Systems Institute --- an institution mainly occupied
with computer science. Two of my collaborators in that institute
have defended Ph.D. theses in computer algebra under my supervision.
The paper \cite{CDg} is about Gr\"{o}bner bases, the most important notion
of constructive commutative algebra; here we prove a formula for the
greatest common divisor of several multivariate polynomials expressed
through the Gr\"{o}bner basis of the ideal they generate.
Paper \cite{CFD} discusses the interrelation of some notions which
are crucial in computational algebra, both classical and differential.
Two papers \cite{DK1}, \cite{DK2} describe the algorithms of computations
related to combinatorial algebras arising in low-dimensional topology.
\section{Vassiliev knot invariants}
This is my main field of mathematical activity since 1992;
it comprises the following pubications:
\cite{CDub}, \cite{CDlb}, \cite{Kn96}, \cite{CDL1}, \cite{CDL2}, \cite{CDL3}
\cite{CDK}, \cite{Mati}, \cite{Klein}, \cite{Skew}, \cite{CDc}, \cite{CDki},
\cite{CDuz}, \cite{Lect}, \cite{CDbook}, \cite{str_links},
\cite{toy_theory}, \cite{elsevier}, \cite{inv_vas_gus}.
I split the review into several
subsections.
\subsection{Asymptotic estimates for the number of Vassiliev invariants}
The two papers \cite{CDub} and \cite{CDlb} gave asymptotic
estimates (best at the publication time) for the number
of independent Vassiliev invariants of a given degree from above and
from below, respectively. The upper bound proved in the first paper
initiated a series of papes by other people (K.~Ng, A.~Stoimenow,
B.~Bollob\'as and O.~Riordan, D.~Zagier) who substantially improved
our asymptotic estimate. The lower bound given in \cite{CDlb} was improved,
too. O.Dasbach who did that used our method with a small (but effective)
variation. Dasbach's upper bound is beleived by many to give the
correct asymptotics for the dimension of the graded spaces of Vassiliev
invariants. The idea of this method first appeared in my talk at the
conference Knots-96 in Tokyo \cite{Kn96} in which I proved that the number
of linearly independent primitive Vassiliev
invariants is asymptotically at least quadratic in the degree.
\subsection{Combinatorial properties of Vassiliev invariants}
In the series of 3 papers \cite{CDL1}, \cite{CDL2}, \cite{CDL3} by Chmutov,
Lando and myself we studied various combinatorial properties of Vassiliev
invariants. One important thing is the introduction of the notion of
intersection graph of a chord diagram, formulating the intersection graph
conjecture and proving it for chord diagrams whose intersection graph is a
tree. The conjecture in general proved to be false, but triggered a certain
amount of useful activity in the literature. Another thing done in these
papers was the introduction of the ``forest algebra'', a subalgebra in the
Hopf algebra of chord diagrams which has one primitive generator in each
degree. This gave the first construction of an infinite series of primitive
Vassiliev invariants which are multiplicatively independent.
Another paper falling into this subcategory is \cite{CDK} where we introduce
a combinatorial algebra generated by regular trivalent graphs modulo the IHX
and antisymmetry relations. We study the properties of this algebra which is
closely related both to the space of primitive elements in the algebra of
chord diagrams and to Vogel's algebra $\Lambda$. We discuss its role in
various areas of low dimensional topology.
\subsection{Kleinian weight systems}
The most important known construction of weight systems for finite type knot
invariants is the one based on metrized Lie algebras (due to
M.\,Kontsevich). In the papers \cite{Mati} and \cite{Klein} I introduce and
study another construction that starts from a skew-symmetric polynomial in 3
variables subject to a certain relation. It is known that for analytic
functions this construction does not give any new information; however,
non-analytic functions of this type do exists and deserve further study.
Paper \cite{Skew} is dedicated to a purely algebraic problem that arose from
these investigations; it gives a criterion for a skew-symmetric function in
$n$ variables to be expressible as a determinant consisting of values of
univariate functions.
\subsection{Recent papers}
The paper \cite{toy_theory} is concerned with a variant of the theory of
finite type invariants in a sense dual to the classical one.
In \cite{str_links} we investigate the problem of detection of the
string link invertibility with the help of Vassiliev invariants.
Finally, in \cite{conw_magn} I study the interplay between the short
closure of braids, Magnus expansion (as universal Vassiliev invariant) and
the Conway polynomial of knots.
\subsection{Other}
Papers listed in this subsection are mainly expository and
compilative; however, each of them does contain some new results.
In \cite{CDc} we review previously known formulas for Arnold's
generic curve invariants adding our own observations concerning the
invariants of spherical curves.
In \cite{CDki} we give a detailed proof of Kontsevich's fundamental
theorem which states that the only relations in the graded algebra of
Vassiliev knot invariants are the 1- and 4-term relations. The paper
contains complete proofs of numerous auxilliary propositions which are
absent elsewhere in the literature.
Paper \cite{CDuz} provides introductory reading about knot theory for
university newcomers; however, it includes such advanced topics as
the Kontsevich integral and the invariants of Legendrian knots.
The text \cite{Lect} represents lecture notes of the course about
Vassiliev invariants that I gave at the University of Tokyo in 1999.
Finally, \cite{CDbook} is a draft version of the book on which we are
currently working. In the course of this work we encounter numerous
problems; we are going to dedicate some new papers to them.
\section{Popularization}
Apart from the main research topics described in the previous sections,
I have written a number of educational and popularization
texts listed in the bibliography under numbers
\cite{Orn}, \cite{Orn-j}, \cite{Orn-e}, \cite{Moore}, \cite{Maple},
\cite{SuuSemi}, \cite{CDuz}, \cite{Kvant}.
\newpage
\begin{thebibliography}{99}
\bigskip
\centerline{\bf Books and book chapters}
\medskip
\bibitem{Orn}
S.\,Duzhin, B.\,Chebotarevsky.
\textit{Ot ornamentov do differencialnykh uravnenij.}
(``From Ornaments to Differential Equations'', in Russian).
Vysheishaya Shkola, Minsk, 1988.
The subtitle of the book is ``An introduction to the theory
of transformation groups''. It was conceived as a textbook
for university newcomers and covers a wide range of
applications of group action including an elementary
account of Sophus Lie's method of finding solutions to
differential equations using symmetry groups.
\bibitem{Orn-j}
S.\,Duzhin, B.\,Chebotarevsky.
\textit{Henkangun ny\={u}mon.}
(``Introduction to transformation groups'', in Japanese).
Springer Verlag, Tokyo, 2000 (Second printing: 2002).
Online at \verb#http://www.pdmi.ras.ru/~duzhin/papers/Ornam#.
A revised and updated Japanese translation of \cite{Orn}.
\bibitem{Orn-e}
S.\,Duzhin, B.\,Chebotarevsky.
\textit{``Transformation groups for beginners''}.
AMS, Student Mathematical Library 25, 2004, 246 pp.
Online at \verb#http://www.pdmi.ras.ru/~duzhin/papers/Ornam#.
A revised and updated English translation of \cite{Orn}.
\bibitem{SCL}
\textit{Symmetries and Conservation Laws for Differential Equations of
Mathematical Physics.} Ed. by I.\,S.\,Krasilschik and A.\,M.\,Vinogradov.
Factorial publishers, Moscow, 1997 (In Russian), 461 pages.
This book was written by 9 authors, including myself (my contribution
is the initial version of one and a half chapters),
The book contains a detailed exposition of the geometric
theory of differential equations based on the fundamental notions
of differential geometry and algebraic topology such as smooth
manifolds, fiber bundles, distributions, Lie groups, homology,
spectral sequences. It contains the algorithms for handling
symmetries and conservation laws of differential equations and
numerous examples coming from mathematical physics where such
algorithms are used.
\bibitem{SCL-e}
\textit{Symmetries and Conservation Laws for Differential Equations of
Mathematical Physics.} Ed. by I.\,S.\,Krasilschik and A.\,M.\,Vinogradov.
Translations of Math. Monographs, v. 182, AMS, 1999, 333 pages.
Updated English translation of \cite{SCL}.
\bibitem{JN}
Jet Nestruev, \textit{Smooth manifolds and observables.}
MCCME, Moscow, 2000 (Russian), 300 pp.
Jet Nestruev is a pseudonym of a group of six authors including myself.
The book provides a non-standard introduction to the theory
of smooth manifolds based on the theorem (going back to the ideas of
A.\,Grothendieck) that a manifold is uniquely
determined by the algebra of smooth functions on it. The algebraic
counterparts of the basic notions and constructions of the calculus
on manifolds, such as vector fields, differential forms, fiberings
and vector bundles, are discussed. The book also contains an
introduction to the abstract differential calculus in
commutative rings.
\bibitem{JN-e}
Jet Nestruev.
\textit{Smooth manifolds and observables.}
``Graduate Texts in Mathematics'' (Springer), v.~220, 2002.
Updated English translation of \cite{JN}.
\bibitem{CDbook}
S.\,Chmutov, S.\,Duzhin, J.\,Mostovoy.
\textit{Introduction to Vassiliev knot invariants.}
Draft version (November 17, 2009, 486 pp.) is online at
\verb#http://www.pdmi.ras.ru/~duzhin/papers/cdbook/#.
An introduction to the theory of finite type (Vassiliev) knot invariants.
Detailed exposition of the combinatorial algebras related
to Vassiliev invariants, the Kontsevich integral and related topics.
To be finished in 2010.
\bigskip
\centerline{\bf Refereed academic papers}
\medskip
\bibitem{J1}
S.\,V.\,Duzhin.
\textit{The $\mathcal{C}$-spectral sequence on the manifold $J^1(M)$.}
Uspehi mat. nauk, v. 38, no. 1, p. 165--166
(Russian; English transl. in Russian Math. Surveys, 38:1, p. 179--181),
1983.
The $\mathcal{C}$-spectral sequence is an algebro-topological tool
for the study of invariants of differential equations such
as symmetries and conservation laws. Its version studied
in this paper is related to Monge--Amp\`{e}re equations (a class of
second-order nonlinear partial differential equations).
\bibitem{Fol}
S.\,V.\,Duzhin.
\textit{A spectral sequence associated with a foliation, and
Gelfand--Fuks cohomology of certain Lie algebras of vector fields.}
Uspehi mat. nauk, v. 39, no. 1, p. 135--136 (Russian; English transl. in
Russian Math. Surveys, 39:1, p. 147--148), 1984.
To each foliation on a smooth manifold $M$ we assign a spectral
sequence that converges to the de Rham cohomology of $M$.
A connection of this spectral sequence with deformation cohomology
is established. The simplest example is completely calculated:
a ``plane-parallel'' foliation of codimension 1 on a torus.
\bibitem{DT}
S.\,V.\,Duzhin, T.\,Tsujishita.
\textit{Conservation laws of the BBM equation.}
J. Phys. A: Math. Gen., v. 17, p. 3267--3276, 1984.
Using algebro-topological techniques based on Vinogradov's spectral
sequence for differential equations, we prove that the
equation $u_t = u_{xxt} + u u_x$ known in physics as
``Benjamin--Bona--Mahony equation'',
has no conservation laws other than those known earlier.
\bibitem{DL}
S.\,V.\,Duzhin, V.\,V.\,Lychagin.
\textit{Symmetries of distributions and quadrature of ordinary
differential equations.}
Acta Applicandae Mathematicae, v. 24, p. 29--57, 1991.
Online at \verb#http://www.pdmi.ras.ru/~duzhin/papers/sdqode.tex.gz#.
We present a geometric exposition and generalization of
S.\,Lie's and E.\,Cartan's theory of explicit integration of finite
type (in particular, ordinary) differential equations.
Numerous examples of how this theory works are given.
In one of these, we propose a new method of hunting for particular
solutions of partial differential equations via symmetry preserving
overdetermination.
\bibitem{CDg}
S.\,V.\,Duzhin, S.\,V.\,Chmutov.
\textit{Gaydar's formula for the greatest common divisor
of several polynomials.}
Uspekhi Mat. Nauk, v. 48, no. 2, p. 177--178
(Russian; English translation in Russian Math. Surveys 48,
no.2, 171--172), 1993.
Online at \verb#http://www.pdmi.ras.ru/~duzhin/papers/formgayd.tex.gz#.
We give a simplified proof of the formula due to E.\,Gaydar
that expresses the greatest common divisor of several multivariate
polynomials through the Gr\"{o}bner basis of the ideal they generate
in the ring of polynomials. Gr\"{o}bner basis is one of fundamental
notions of computer algebra since it allows algorithmic
computations in finitely presented commutative rings.
\bibitem{CFD}
G.\,Carr\`a-Ferro, S.\,V.\,Duzhin.
\textit{Differential-algebraic and differential-geometric approach
to the study of involutive symbols.}
In: N.\,H.\,Ibragimov et al. (eds.), Modern Group Analysis: Advanced
Analytical and Computational Methods in Mathematical Physics, 93--99,
Kluwer Academic Publishers, 1993.
Any system of partial differential equations after a finite number of
prolongations becomes either inconsistent or involutive.
The Hilbert function of any homogeneous ideal becomes polynomial
beginning from a certian value of its argument.
Any system of generators of a polynomial ideal becomes its Gr\"{o}bner
basis when multiplied by all monomials of sufficiently big degree.
We indicate some relations between the number of steps required
in these three procedures, for systems of partial differential equations
with constant coefficients.
\bibitem{CDub}
S.\,V.\,Chmutov, S.\,V.\,Duzhin.
\textit{An upper bound for the number of Vassiliev knot inva\-riants.}
Journal of Knot Theory and its Ramifications, v. 3, 141--151,
1994.
Online at \verb#http://www.pdmi.ras.ru/~duzhin/papers/cd.ps.gz#.
Using a combinatorial argument, we prove that the number
of finite type (Vassiliev) knot invariants of order $n$
is asymptotically no greater than $(n-1)!$
\bibitem{CDL1}
S.\,V.\,Chmutov, S.\,V.\,Duzhin, S.\,K.\,Lando.
\textit{Vassiliev knot invariants: I. Introduction.}
Adv. in Soviet Math., v. 21: Special issue on
Singularities and Curves, ed. V.I.Arnold,
pp. 117--126, 1994.
Online at \verb#http://www.pdmi.ras.ru/~duzhin/papers/cdl1.ps.gz#.
This paper is introductory in two ways: first, it provides an
introduction to the theory of Vassiliev knot invariants (somewhat different
from the original one), second, it serves as an introduction to the two
subsequent papers, explaining the terminology and setting the problems.
\bibitem{CDL2}
S.\,V.\,Chmutov, S.\,V.\,Duzhin, S.\,K.\,Lando.
\textit{Vassiliev knot invariants:
II. Intersection graph conjecture for trees.}
Adv. in Soviet Math., v. 21: Special issue on
Singularities and Curves, ed. V.I.Arnold,
pp. 127--134 (II), 1994.
Online at \verb#http://www.pdmi.ras.ru/~duzhin/papers/cdl2.ps.gz#.
The intersection graph of a chord diagram is the graph whose vertices
correspond to chords and edges to chord intersections. The intersection
graph conjecture claims that the intersection graph determines the chord
diagram uniquely modulo 4-term relations. The present paper contains
a proof of this conjecture in the special case when the intersection
graph is a tree.
\bibitem{CDL3}
S.\,V.\,Chmutov, S.\,V.\,Duzhin, S.\,K.\,Lando.
\textit{Vassiliev knot invariants:
III. Forest algebra and weighted graphs.}
Adv. in Soviet Math., v. 21: Special issue on
Singularities and Curves, ed. V.I.Arnold,
pp. 135--145, 1994.
Online at \verb#http://www.pdmi.ras.ru/~duzhin/papers/cd.ps.gz#.
We introduce and study the ``forest algebra'', a
subalgebra in the Hopf algebra of chord diagrams which has one
primitive generator of each degree. In particular, this shows that
the dimension of each graded component of the primitive space is
at least 1, which implies (by a Hardy--Ramanujan argument) that
the dimension of the space of Vassiliev invariants of a given degree $n$
is asymptotically no less than $\exp(C\sqrt{n})$.
\bibitem{CDc}
S.\,V.\,Chmutov, S.\,V.\,Duzhin.
\textit{Explicit formulas for Arnold's curve invariants,}
In: ``Arnold-Gelfand Mathematical Seminars: Geometry and
Singularity Theory'', Birk\-hauser, 1997, p. 123--138.
Online at \verb#http://www.pdmi.ras.ru/~duzhin/papers/cd_curves.ps.gz#.
This paper contains a review of various explicit formulas for Arnold's
generic curve invariants due to Viro, Shumakovich and Polyak and the
authors' new observations concerning the invariants of spherical
curves and curves immersed into arbitrary orientable surfaces.
\bibitem{CDK}
S.\,V.\,Duzhin, S.\,V.\,Chmutov, A.\,I.\,Kaishev.
\textit{The algebra of 3-graphs.}
Trudy Mat. Inst. Steklova, v. 221 (1998), pp. 168--196 (Russian;
English translation in Trans. Steklov Math. Inst. 221 (1988),
157--186.
Online at \verb#http://www.pdmi.ras.ru/~duzhin/papers/cdk-e.ps.gz#.
We introduce and study the structure of an algebra in the linear
space spanned by all regular 3-valent graphs with a prescribed order
of edges at every vertex, modulo certain relations.
The role of this object in various areas of low dimensional topology
is discussed.
\bibitem{DK1}
S.\,V.\,Duzhin, A.\,I.\,Kaishev.
\textit{T-system implementation of the program of computation of the sl-
and so- polynomials for 3-graphs}. (Russian).
In: ``Programmnye sistemy'' (transactions of Program Systems
Institute, Pereslavl-Zalessky), Moscow, Nauka, Fizmatlit, 1999,
p. 214--223.
We describe a set of programs implemented on a multiprocessor computer
with automatic dynamic parallelization that enabled us to find the tables
of values of certain polynomial invariants of graphs used in the previous
paper (``The algebra of 3-graphs'').
\bibitem{CDlb}
S.\,V.\,Chmutov, S.\,V.\,Duzhin.
\textit{A lower bound for the number of Vassiliev knot invariants.}
``Topology and its Applications'', 92 (1999) 201--223.
Online at \verb#http://www.pdmi.ras.ru/~duzhin/papers/lb.ps.gz#.
We prove that the number of primitive Vassiliev knot invariants of
degree $d$ grows at least as $d^{\log(d)}$ when $d$ tends to infinity.
In particular it grows faster than any polynomial in $d$. The proof
is based on the explicit construction of an ample family of linearly
independent primitive elements in the corresponding graded Hopf algebra.
\bibitem{CDki}
S.\,V.\,Duzhin, S.\,V.\,Chmutov.
\textit{The Kontsevich integral.}
Acta Appl. Mathematicae 66 (2), p. 155--190, 2001.
Online at \verb#http://www.pdmi.ras.ru/~duzhin/papers/ki.ps.gz#.
This is an expository paper whose purpose is to explain the famous
contribution of Maxim Kontsevich to the theory of finite type knot
invariants. More specifically, our aim is to present a proof of
Kontsevich's fundamental theorem which states that for each weight system
there exists a Vassiliev invariant whose symbol is exactly the given
weight system, or, in other words, that the only relations in the graded
algebra of Vassiliev knot invariants are the 1- and 4-term relations. We
also discuss the behaviour of the Kontsevich integral with respect to
mirror reflection, change of orientation and mutations. The paper
contains complete proofs of numerous auxilliary propositions which are
absent elsewhere in the literature.
\bibitem{2x2}
S.\,V.\,Duzhin.
\textit{Infinitesimal classification of systems of
two first order partial differential equations.}
Zapiski POMI (St.Petersburg), v. 279, 2001.
Online at \verb#http://www.pdmi.ras.ru/~duzhin/papers/2x2.ps.gz#.
We give a classification of symbols of partial differential systems
that consist of two equations imposed on two functions in two independent
variables, with respect to the natural action of the group of linear
changes of dependent and independent variables.
The classification turns out to be finite, consisting of 5 types in the
real case and of 4 types in the complex case.
Apart from the elliptic, parabolic and hyperbolic types
constituting the well-known classification of second order equations
for a function $u(x,y)$, in the present (real) case we also have two
non-equivalent `degenerate' types.
Bifurcation diagrams for the singular orbits in the space of symbols
imply certain properties of the singularities of linear
and quasilinear systems of the considered type.
\bibitem{Skew}
S.\,Duzhin.
\textit{Decomposable skew-symmetric functions.}
Moscow Mathematical Journal, v. 3, no. 3 (2003), p. 881--888. Online
version: \verb#http://www.pdmi.ras.ru/~duzhin/papers/skew_fun.ps.gz#.
A skew-symmetric function $F$ in $n$ variables is said to be
decomposable if it can be represented as a determinant consisting
of the values of $n$ univariate functions on $n$ variables.
We give a criterion of the decomposability in terms of an identity
imposed on the function $F$.
\bibitem{toy_theory}
S.Duzhin, J.Mostovoy. \textit{A toy theory of Vassiliev invariants}. Moscow
Mathematical Journal 6(1), p. 85-93 (2006). Online version:
\verb#http://www.pdmi.ras.ru/~duzhin/papers/toy_theory.ps.gz#.
We set up a theory of finite type invariants for smooth
hypersurfaces in $\R^n$. For $n=1,2,3$ these invariants admit
a complete description: they form a polynomial algebra on one generator.
\bibitem{str_links}
S.V.Duzhin, M.V.Karev. \textit{Detecting the orientation of string links by
finite type invariants}. Funct. Anal. Appl., v. 41, no. 3, 48--59, 2007.
Preprint version (July 1, 2005): arXiv:math.GT/0507015.
Updated Russian and English versions are available at
http://www.pdmi.ras.ru/~duzhin/papers/.
\bigskip
\centerline{\bf Preprints}
\medskip
\bibitem{conw_magn}
S.\,Duzhin, \textit{Conway polynomial and Magnus expansion}. January 2010.
arXiv.org/abs/1001.2500. Submitted to St.Petersburg Math. J.
\bigskip
\centerline{\bf Conference proceedings (refereed)}
\medskip
\bibitem{DK2}
S.\,V.\,Duzhin, A.\,I.\,Kaishev.
\textit{Calculation of central generators of the universal enveloping
algebras and Vassiliev--Kontsevich weight systems.}
Proceedings of the international workshop
``New Computer Technologies in Control Systems'' (editors:
M.\,G.\,Dmitriev, Yu.\,L.\,Sachkow). Program Systems Institute,
Pereslavl-Zalessky, August 13--16, 1995.
Vassiliev--Kontsevich weight systems provide an explicit series
of finite type knot invariants with values in the universal enveloping
algebras of Lie algebras. In this work, we describe our algorithms
to do all the necessary computations as well as the set of programs
written in various programming languages (Pascal, C, Maple, FORM)
which implement these algorithms.
\bibitem{Klein}
S.\,Duzhin.
\textit{On the Kleinian weight systems.}
``Low-Dimensional Topology of Tomorrow''.
S\={u}rikaisekikenky\={u}sho K\={o}ky\={u}roku 1272, Kyoto,
June 2002, pp 84--90.
Online at \verb#http://www.pdmi.ras.ru/~duzhin/papers/kyoto.ps.gz#.
We introduce a family of weight systems in a sense dual to the family
of Lie algebra weight systems. The basic component in our construction
is a skew-symmetric function of three variables $f(x,y,z)$ that satisfies
the following equation (we call it the \textit{Klein equation\/}):
$f(x,y,z)f(u,v,z) - f(x,u,z)f(y,v,z) + f(x,v,z)f(y,u,z) = 0$,
which is the counterpart of the Jacobi identity for the structure tensor
of a Lie algebra.
We prove that analytic Kleinian functions lead to weight systems
expressible through the classical Lie algebraic $sl_2$-weight system.
Non-analytic Kleinian functions do exist, but the nature of the
corresponding weight systems is yet unclear.
\bigskip
\centerline{\bf Unpublished, unrefereed, teaching materials etc.}
\medskip
\bibitem{VINITI1}
S.\,V.\,Duzhin. \textit{The $\mathcal{C}$-spectral sequence on finite order jet
manifolds,} VINITI deposited manuscript, 1982 (in Russian).
\bibitem{VINITI2}
S.\,V.\,Duzhin. \textit{Deformation cohomology of one-codimensional
plane-parallel foliation on a torus}, Proc. of XVII Voronezh Winter
Math. School, Voronezh, 1983 (in Russian).
\bibitem{Diss}
S.\,V.\,Duzhin. \textit{On some versions of the $\mathcal{C}$-spectral sequence}.
Ph.D. dissertation (in Russian). Moscow University, 1982.
\bibitem{Kvant}
S.\,V.\,Duzhin, V.\,N.\,Rubtsov. \textit{Geometry of the 4-dimensional cube}.
``Kvant'', 1986, no.6 (in Russian).
\bibitem{Moore}
S.\,Duzhin.
\textit{Graphes de Moore.}
Online at \verb#http://www.pdmi.ras.ru/~duzhin/papers/moore.ps.gz#.
This is the text (in French) of 2 lectures I prepared for the graduate
students of the Sorbonne University who were visiting the Independent
University of Moscow in October 1994. A Moore graph is a regular graph of
diameter 2 and degree $k$ with $k^2+1$ vertices. It is know that such graphs
may only exist for the following values of $k$: 2, 3, 7 and 57. The
structure of Moore graphs of degrees $k = 2$, 3 and 7 is known. It is not
known yet whether such a graph exists in the case $k=57$. In the present
text I am giving a simplified proof (joint with S.\,Chmutov) of the
fundamental theorem of Hoffman and Singleton, and also a new construction of
the Moore graph of degree 7 based on the notions of graph covering and
monodromy group that come from topology.
\bibitem{Maple}
S.\,Duzhin.
\textit{Mathematics with Maple.}
Available on the Web at \verb#http://www.botik.ru/~duzhin/maple/#.
A set of 10 Maple worksheets which give an introduction to this computer
algebra system and simultaneously constitute a review course of first-year
university mathematics. Prepared and tested in 1994 in the University of
Aizu (Japan). Mathematical topics discussed: Number Sequences and Limits,
Solving Equations, Derivative and Taylor series, Integral, Elementary
Geometry, Matrices and Determinants, Linear Operators, Conics and Quadrics,
Groups.
\bibitem{Kn96}
S.\,V.\,Duzhin. \textit{A quadratic lower bound for the number of primitive
Vassiliev
invariants}. Extended abstract, KNOT'96 Conference/Workshop report, Waseda
University, Tokyo, July 1996, p. 52--54.
Online at \verb#http://www.pdmi.ras.ru/~duzhin/papers/kn96.zip#.
I prove that the number of linearly independent primitive Vassiliev
invariants is asymptotically at least quadratic in the degree.
\bibitem{SuuSemi}
S.\,V.\,Duzhin. \textit{``Seminar Diary'' column in "Suugaku seminar"},
a set of 12 essays, April 1996 -- March 1997 (in Japanese). Online at
\verb#http://www.pdmi.ras.ru/~duzhin/papers/suusemi.jis.gz# (Japanese),\\
\verb#http://www.pdmi.ras.ru/~duzhin/papers/suusemi.eng.gz# (English).
My impressions about the life of mathematical seminars in Japan and
in Russia.
\bibitem{CDuz}
S.\,V.\,Duzhin. S.\,V.\,Chmutov.
\textit{Uzly i ikh invarianty.} (``Knots and their invariants'', in Russian).
Matematicheskoe Prosveschenie, no. 3, 1999, pp. 59--93.
Online at \verb#http://www.pdmi.ras.ru/~duzhin/papers/onknots.ps.gz#.
Introductory reading meant for university newcomers. The exposition starts
with the definition of a knot and goes as far as the invariants of
Legendrian knots and the Kontsevich integral.
\bibitem{Mati}
S.\,V.\,Duzhin. \textit{The Matiyasevich polynomial, four colour theorem
and weight systems}, Art of low-dimensional topology VI (ed. T.~Kohno),
Kyoto, 2000, pp. 9--14.
On-line at \verb#http://www.pdmi.ras.ru/~duzhin/papers#.
We show that a modification of the polynomial introduced by
Yu.~Ma\-ti\-yasevich for sphere triangulations, can be defined
to be an invariant of 3-graphs and thus provide a weight
system for finite type knot invariants. The relation of this
construction with the four-colour theorem is also discussed.
\bibitem{Lect}
S.\,Duzhin.
\textit{Lectures on Vassiliev knot invariants.}
``Lectures in Mathematical Sciences'', vol. 19,
The University of Tokyo, 2002. 123 pp.
Online at \verb#http://www.pdmi.ras.ru/~duzhin/Vics/vics.ps.gz#.
Lecture notes of a course for graduate students that I gave at
Tokyo University in 1999.
\bibitem{elsevier}
S.Chmutov, S.Duzhin. The Kontsevich integral. Encyclopedia of
Mathematical Physics, eds. J.-P.Francoise, G.L.Naber and S.T.Tsou. Oxford:
Elsevier, 2006 (ISBN 978-0-1251-2666-3), volume 3, pp. 231--239. Draft
version online at arXiv:math.GT/0501040.
A review article written for the Elsevier Encyclopaedia.
\bibitem{inv_vas_gus}
S.V.Duzhin. Vassiliev--Goussarov invariants, in: Mathematics of XX century,
view from St.Petersburg. Edited by A.M.Vershik. MCCME Publishers, 2010, pp.
87--116 (in Russian).
A review written for a volume of collected papers published in
St.Peteresburg.
\end{thebibliography}
\end{document}