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## Noncommutative geometry and Painleve equations

We outline recent developments relating Painlev ́e equations and 2D conformal field theory. Generic tau functions of Painlev ́e VI and Painlev ́e III_3 are written as linear combinations of c= 1 conformal blocks and their irregular limits. This provides explicit combinatorial series representations of the tau functions, and helps to establish connection formula for the tau function in the Painlev ́e VI case.

Gamayun, Iorgov and Lisovyy in 2012 proposed that tau function of the Painlevé equation is equal to the series of *𝑐*=1 Virasoro conformal blocks. We study similar series of *𝑐*=−2 conformal blocks and relate it to Painlevé theory. The arguments are based on Nakajima–Yoshioka blowup relations on Nekrasov partition functions. We also study series of *q*-deformed *𝑐*=−2 conformal blocks and relate it to *q*-Painlevé equation. As an application, we prove formula for the tau function of *q*-Painlevé *𝐴*(1)′7 equation.

We examine asymptotic expansions of the third Painlevé transcendents for *αδ ≠* 0 and *γ* = 0 in the neighborhood of infinity in a sector of aperture *<*2*π* by the method of dominant balance). We compare intermediate results with results obtained by methods of three-dimensional power geometry. We find possible asymptotics in terms of elliptic functions, construct a power series, which represents an asymptotic expansion of the solution to the third Painlevé equation in a certain sector, estimate the aperture of this sector, and obtain a recurrent relation for the coefficients of the series.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.