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An algebraic vector field is an algebraic variety equipped with a rational section to its tangent bundle, or equivalently a derivation on its function field. The goal of this talk will be to articulate several new results on the birational geometry of algebraic vector fields, obtained using the model theory of differentially closed fields.

A matrix cocycle is a non-commutative counterpart of random walk. The counterpart of the ergodic theorem, describing the almost sure asymptotic behaviour to leading order, is given by the theory of random matrix products originating in the works of Furstenberg—Kesten, Furstenberg, and Oseledec. On the other hand, the spectral theory of random one-dimensional second-order operators leads to the study of cocycles depending on an additional real number (the spectral parameter), and, a priori, the theory is applicable for almost all (rather than all) values of the parameter. The focus of the talk will be on the exceptional sets, where different asymptotic behaviour occurs: particularly, we shall discuss their rôle in spectral theory and their topologic and metric properties, including a result resembling the Jarnik theorem on Diophantine approximation. Based on joint work with Ilya Goldsheid.

This talk is devoted to the study of some spectral properties of the Almost Mathieu Operator – a one-dimensional discrete Schrödinger operator with potential generated by an irrational rotation with angle \alpha (called the frequency). The spectral properties of the Almost Mathieu operator depend sensitively on the arithmetic properties of the frequency. If the frequency is poorly approximated by rationals, the spectral properties are as nice as one would expect.

The focus of this talk will be on the complementary case of well-approximated frequencies, in which the state of affairs is completely different. We show that in this case several spectral characteristics of the Almost Mathieu Operator can be as poor as at all possible in the class of all discrete Schrödinger operators. For example, the modulus of continuity of the integrated density of states (that is, of the averaged spectral measure) may be no better than logarithmic (for comparison, for poorly approximated frequencies the integrated density of states satisfies a Hölder condition). Other characteristics to be discussed are the Hausdorff measure of the spectrum and the non-homogeneity of the spectrum (as a set).

Based on joint work with A. Avila, Y. Last, and Q. Zhou

Symplectic embeddings of balls into symplectic manifolds have been extensively studied since the famous non-squeezing theorem of Gromov (1985). However, even for basic closed symplectic manifolds, such as a complex projective space of real dimension 6 or higher, the classification of these embeddings up to a symplectomorphism of the target manifold is still unknown. I’ll discuss such a classification for a special kind of symplectic embeddings of balls - the so-called Kahler-type embeddings - that can be studied using complex geometry.

This is a joint work with M.Verbitsky.

The Teichmuller space of geometric structures of a given type is a quotient of the (generally, infinite-dimensional) space of geometric structures by the group of isotopies, that is, by the connected component of the diffeomorphism group. In several important qand smooth.uestions, such as for symplectic, hyperkahler, Calabi-Yau, G2 structures, this quotient is finite-dimenisional and even smooth. The mapping class group acts on the Teichmuller space by natural diffeomorphisms, and this action is in many important situations ergodic (in particular, it has dense orbits), bringing strong consequences for the geometry. I would describe the Teichmuller space for the best understood cases, such as symplectic and hyperkahler manifolds, and give a few geometric applications.

The group of diffeomorphisms acts on the space of symplectic structures on a given manifold. Taking a quotient by isotopies, we obtain the mapping class group action on the Teichmuller space of symplectic structures; the latter is a finite-dimensional manifold. The mapping class group action on the Teichmuller space is quite often ergodic, which leads to important consequences for symplectic invariants, such as symplectic packing problems. I would describe some of the problems which were solved using this approach. This is a joint work with Michael Entov.

I will discuss some of the interactions between number theory and the spectral theory of the Laplacian. Some have very classical background, such as the connection with lattice point problems. Others are newer, including connections with Random Matrix Theory, the zeros of the Riemann zeta function, and the work of Maryam Mirzakhani on the moduli space of hyperbolic surfaces.