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The talk is devoted to the problem of characterization of integrals as linear functionals. The main idea goes back to Hadamard. The first well known results in this field are the F.Riesz theorem (1909) on integral presentation of bounded linear functionals by Riemann-Stiltjes integrals on the segment and the Radon theorem (1913) on integral presentation of bounded linear functionals by Lebesque integrals on a compact in Rn. After papers of I.Radon, M.Fréchet and F.Hausdorff the problem of characterization of integrals as linear functionals is used to be formulated as the problem of extension of Radon theorem from Rn on more general topological spaces with Radon measures. This problem turned out to be rather complicated. The history of its solution is long and rich. It is quite natural to call it the Riesz-Radon-Fréchet problem of characterization of integrals. The important stages of its solution are connected with names of S.Banach (1937-38), Sacks (1937-38), Kakutani (1941), P.Halmos (1950), Hewitt (1952), Edwards (1953), N.Bourbaki (1969), and others. Some essential technical tools were developed by A.D.Alexandrov (1940--43), M.Stone (1948--49), D.Fremlin (1974), and others.
In 1997 A.V.Mikhalev and V.K.Zakharov had found a solution of Riesz-Radon-Fréchet problem of characterization for integrals on an arbitrary Hausdorff topological space for nonbounded positive radom measures.
The next modern period of this problem for arbitrary Radon measures is connected mostly with results by A.V.Mikhalev, T.V.Rodionov, and V.K.Zakharov. A special attention is paid to algebraic aspects used in the proof.

\indent We consider a model of asexually reproducing individuals. The birth and death rates of the individuals are affected by a fitness parameter. The rate of mutations that cause the fitnesses to change is proportional to the population size, $N$. The mutations may be either beneficial or deleterious. In a paper by Yu, Etheridge and Cuthbertson (2009) it was shown that the average rate at which the mean fitness increases in this model is bounded below by $\text{log}^{1-\delta}N$ for any $\delta > 0$. We achieve an upper bound on the average rate at which the mean fitness increases of $O(\text{log}\hspace{1pt}N/\text{log log} \hspace{1pt}N)$.

\indent Ask a physicist. You will be told that it's a manifold with some anticommuting coordinates. I will explain this cryptic answer by addressing the following natural questions.

(1) What does this mean? That is, in what mathematical context is it meaningful?

(2) Why would anyone want to do this? That is, what applications does it have?

\indent If we have a smooth compact manifold, then its cotangent bundle has a natural symplectic form. A smooth affine variety also has a natural symplectic form. One can ask the following question: which cotangent bundles are symplectomorphic to smooth affine varieties? We construct many cotangent bundles that are not symplectomorphic to smooth affine varieties.

\indent The main tool used to distinguish these objects is called wrapped Floer cohomology.

\small
A variety of linear algebras over a field is said to be Schreier if any subalgebra of a free algebra of this variety is free. The variety of all algebras, the variety of all commutative algebras, the variety of all anti-commutative algebras, the variety of all Lie algebras, the variety of all Lie superalgebras, varieties of all Lie p-algebras and Lie p-superalgebras are the main types of Schreier varieties of algebras.
Let A(X) be the free algebra of a Schreier variety of algebras with the set X of free generators. A system of elements $u_1,…, u_n$ of A(X) is primitive if there is a set Y of free generators of the free algebra A(X) such that $u_1,…, u_n$ belong to Y.
An element u of A(X) is said to be almost primitive if u is not a primitive element of A(X), but u is a primitive element of any subalgebra of A(X) which contains it.
Algorithms to recognize primitive systems of elements of free algebras of the main types of Schreier varieties of algebras are constructed. We obtain also algorithms to construct complements of primitive systems of elements with respect to free generating sets. Series of almost primitive elements are constructed.
This talk is based on joint works with C.Champagnier, A.A.Chepovskii, A.V.Klimakov, A.V.Mikhalev, I.P.Shestakov, U.U.Umirbaev, J.-T.Yu, and A.A.Zolotykh.