The full symmetric Toda system is the dynamical system, associated with a Cartan decomposition of a real form of semisimple Lie algebra. It is known that the system is Hamiltonian and integrable. However the construction of the first integrals of this system is not an easy task, the existing constructions are rather complicated and may seem artificial. In my talk, based on the joint work with Yu. Chernyakov and A. Sorin I will describe a way to construct a large commutative family of symmetries of this system, i.e. of vector fields, that will commute with each other and with the vector field that generates the system. The construction is based on the geometric considerations and on the structure of representations of the Lie algebra.

This is an expository talk. We will discuss some recent development in Kähler-Ricci flow on Fano manifolds. Mainly, we talk about Perelman's fundamental estimates in Kähler-Ricci flow and the progress on Hamilton-Tian conjecture.

We are going to present a review on current state of the investigation, together with a list of some unsolved problems.

In this talk, I will report a recent result of mine showing that the profinite completion of the fundamental group determines finite-volume hyperbolic 3-manifolds up to finitely many possibilities. I plan to spend most of the time discussing background of that problem, so the talk should be accessible to audience with general math background.

For a connected reductive algebraic group G defined over the field of real numbers R, the group of real points G(R) is a real Lie group, not necessarily connected; look at GL_n(R) or SO_{p,q}(R) for example. A natural problem is to determine the component group of G(R). It turns out that this problem is related to another important problem in the theory of algebraic groups: to compute the Galois cohomology H^1(R,G). We give a uniform solution to both problems in terms of combinatorial data which determine the reductive group G over R, such as the affine Dynkin diagram with a nonnegative integer labeling of its vertices and the cocharacter lattice of a maximal torus equipped with an involution. Though the answer is purely algebraic and combinatorial, the proofs involve transcendental Lie-theoretic and differential-geometric methods such as the exponential mapping on algebraic tori and an action of the affine Weyl group on their Lie algebras. This is a joint work with Mikhail Borovoi.

The asymptotic geometry is important for the classification of Ricci solitons. In this talk, we will classify the asymptotic geometry of 4D steady gradient Ricci solitons whose Ricci curvature is nonnegative outside a compact set. As an application, we will show such steady Kähler-Ricci solitons must be Ricci-flat.

The notion of quasielliptic rings appeared as a result of an attempt to classify a wide class of commutative rings of operators found in the theory of integrable systems, such as rings of commuting differential, difference, differential-difference, etc. operators. They are contained in a certain non-commutative "universe" ring - a purely algebraic analogue of the ring of pseudodifferential operators on a manifold, and admit (under certain mild restrictions) a convenient algebraic-geometric description. An important algebraic part of this description is the Schur-Sato theory - a generalisation of the well known theory for ordinary differential operators. I'll talk about this theory in dimension n and about some of its unexpected applications related to the generalized Birkhoff decomposition and to the Abhyankar formula.

In this presentation, we discuss the geometry and topology of compact Kähler manifolds with RC-positive tangent bundle, and describe the relationship between RC-positivity and rational connectedness in algebraic geometry.