## on Representation Theory and Dynamical Systems

### Former talks in 1999:

December, 22
S. V. Kerov
Polynomials in Young diagrams, and the Central Limit Theorem for characters of symmetric groups
Precise relations between various polynomials in a Young diagram will be presented. The relations imply a new proof of the Central Limit Theorem for the Plancherel Measures of symmetric groups. A direct connection with the Free Probability theory by Dan Voiculescu is established.

December, 1
A. G. Izergin
Integrable differential equations for correlation functions

November, 10
S.V.Fomin
Distinguishing Schubert cells

October, 6
N.N.Vassiliev
Groebner bases and standard bases in the ring of formal power series

September, 16
S.V.Kerov.
Calculations with polynomials of Young diagrams, and the Central Limit Theorem for the latter
Precise relations between various polynomials of a Young diagram will be presented. The relations imply a new proof of the Limit Shape Theorem ("the Law of Large Numbers"), and the proof of the Central Limit Theorem for the Plancherel Measure.

September, 15
V.M.Manuilov (Moscow)
Almost representations and asymptotic representations of discrete groups
For $\e>0$ an $\e$-almost representation of a finitely presented group $\Gamma$ is a map from the set of generators of $\Gamma$ into the infinite unitary group supplied with the operaotr norm, such that all relations are satisfied up to $\e$. A continuous family of $\e_t$-almost representations is called an asymptotic representation if $\lim_{t\to\infty}\e_t=0$. Our interest to this kind of representations is caused by their relation to the $K$-theory of classifying spaces \cite{CGM,mish-noor}.

If a group is such that every its $\e$-almost representation for small enough $\e$ can be included into an asymptotic representation we call this group asymptotically stable. Free, finite, abelian and some other groups are proved to be asymptotically stable. The main point of the proof are some properties of almost commuting operators.

We give also an example of a group without asymptotical stability. This example provides also an example of a group without sufficiently many asymptotic representations, namely there are elements in the $K$-group of its classifying space which cannot be obtained from asymptotic representations.

We discuss also relations between asymptotic representations of $\Gamma$ and representations of $\Gamma\times{\bf Z}$ into the Calkin algebra (Fredholm representations). An approach developed in \cite{mm} helps in better understanding of relations between the Kasparov's $KK$-bifunctor and the $E$ functor of Connes--Higson on the category of $C^*$-algebras.

\bibitem{CGM} {\sc A.~Connes, M.~Gromov, H.~Moscovici}: Conjecture de Novikov et fibr\'es presque plats. {\it C. R. Acad. Sci. Paris}, s\'erie I, {\bf 310} (1990), 273--277.

\bibitem{mm} {\sc V. M. Manuilov, A. S. Mishchenko}: Asymptotic and Fredholm representations of discrete groups. {\it Matem. Sb.} {\bf 189} (1998), No 10, 53--72.

\bibitem{mish-noor} {\sc A.~S.~Mishchenko, Noor Mohammad}: Asymptotic representations of discrete groups. Lie Groups and Lie Algebras. Their Representations, Generalizations and Applications.'' Mathematics and its Applications {\bf 433}. Kluver Acad. Publ.: Dordrecht, 1998, 299--312.

June, 23
Tropp E.A (FTI)
Towards geometrization of the special function theory

June, 16
Student's day
The talks will be given by A.Djubina, A.Gorulskii, M.Gorulskii, S.Dobrynin, A.Kalnitskii, M.Matveev

May, 19
M. Yu. Lyubich
Ergodic Theory of Real Quadratic Maps

May, 12
Yuri Matiyasevich
An algebraic approach to graph colourings
For every graph G and natural number k one can define a polynomial in many variables the coefficients of which have surprising relations with the existence and the number of colourings of graph G in k colours. Such relations are especially numerous when G is a plane trivalent graph and we look for its edge colourings in 3 colours (the existence of such colourings is known to be equivalent to the existence of 4-colouring of the corresponding map). The results are partially represented on WWW

April, 28
A. Alekseev
The center of the universal enveloping algebra and equivariant cohomology
We give a new cohomological proof of the isomorphism of the center of the universal enveloping algebra and the ring of invariant polynomials. Our proof applies if the Lie algebra possesses an invariant scalar product.

April, 21
V. Ivanov (Moscow)
Factorial analogues of Schur Q-functions

April, 14
S. V. Kerov
Statistical properties of symmetric group representations, and the spectra of random matrices. Recent progress.
The talk will survey recent progress in the theory of random Young diagrams, especially the new approach suggested by G. Olshanski, A. Okounkov and A. Borodin. It reveals the remarkable similarity between the statistical properties of big Young diagrams, and those of spectra of big Hermitian matrices. See a list of related papers .

April, 7
Yu. V. Yakubovich
Random Young diagrams of partitions on distinct summands.
The talk will be devoted to fluctuations of normed Young diagram of partitions on distinct summands around its limit shape.

March, 31
A. M. Vershik
A discussion of new results in the theory of random walks, spectra of factorizations, ets.

March, 24
Yu. V. Matiyasevich
Four Colour Theorem and Divisibility of Binomial Coefficients
Abstract
The Four Colour Theorem can be restated in terms of divisibility of certain binomial coefficients. See details in the author's paper

March, 17
A. M. Nikitin
Ihara--Selberg zeta function: a survey of rationality results.
Abstract
The talk focuses on rationality of the Ihara-Selberg zeta function for finite connected graphs. It will be shown that these zeta-functions can be considered as generating functions for the numbers of primitive cycles in finite connected graphs. A connection of determinantal presentation of zeta-function with symbolic dynamics on the graph will be considered.

March, 10
Dr S.T.Dougherty (University of Scranton, USA)
Abstract
In this talk I shall show the connections between the theory of shadows for codes and the theory for lattices. I shall give the necessary definitions in both areas. This is a pretty hot topic these days.

March, 3
E. P. Golubeva
Geodesics in the upper half-plane, and the distribution of quadratic irrationalities
Abstract
Since the nontrivial estimates of Fourier coefficients of modular forms of semiinteger weight were recently obtained, many experts believed that the problem on the distribution of quadratic irrationalities is solved in a satisfactory way. I shall show in my talk that this opinion is not correct.

January, 19
P. P. Kulish
Deformations of Harmonic Oscillator, and q-Hermite Polynomials

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