For many years researchers have tried to glean hints about quantum gravity from black hole thermodynamics. We try a different approach using a minimal statistical mechanical model for the horizon fluid based on Damour-Navier-Stokes (DNS) equation. For asymptotically flat black hole spacetimes in General Relativity, we show explicitly that at equilibrium the entropy of the horizon fluid is the Bekenstein-Hawking entropy. Further we show that, for the bulk viscosity of the fluctuations of the horizon fluid to be identical to Damour, a confinement scale exists for these fluctuations, implying quantization of the horizon area.

Thanks to recent progress in biotechnologies, the high-throughput data
in molecular biology are getting more
 and more interesting from the point of view of application of advanced
methods for data analysis, aimed
 at finding non-trivial topological characteristics in the data such as
branching points and holes. Existence
 of such non-trivial structures in the data can have direct biological
interpretations such as the process
 of cell fate decisions during cell differentiation or existence of
cyclic processes in a cell (e.g., cell cycle).

 I will present examples of application of such methods in studying
various biological systems and explain their general
 principles. I will focus on the universal method of elatic principal
graphs for topological data analysis developed by us.
 The method is based on application of the notion of harmonic graph
embedding into a multi-dimensional space,
 minimization of graph elastic energy and using graph grammars defining
a family of possible graph structures (such as trees).
 Simplest implementations of the approach already give very usefull
data approximators such as principal curves,
 principal closed curves, principal manifolds and principal trees.
Several ideas for making these data approximators
 robust to the noise and outliers in the biological data will be
presented.

I will discuss several questions and some results about algebraic flows, o-minimal flows and holomorphic flows on abelian varieties and Shimura varieties.

In this talk, we show that the elliptic deformation of W-algebra is naturally realized using quiver gauge theory in six dimensions compactified on a torus. This construction is based on the gauge theoretical realization of W-algebra proposed in our previous study [arXiv:1512.08533]. In particular, double quantization of Seiberg-Witten geometry for Γ-quiver gauge theory provides a generating current of W(Γ)-algebra in the free field realization. We also show that the partition function is given by a correlator of the corresponding W(Γ)-algebra, which is equivalent to the AGT relation under the gauge/quiver (base/fibre; spectral) duality. This talk is based on a collaboration with V. Pestun [arXiv:1608.04651]

Ten dimensional supergravities are well-known to arise as low-energy limits of string theory. I will show that in fact, these pure supergravities are complete string theories in their own right. The strings in question describe holomorphic maps to the space of null geodesics in complexified space-time, known as ambitwistor space, and have no massive states in their spectrum. Worldsheet correlators describe scattering amplitudes in pure supergravity, in a form found recently by Cachazo, He & Yuan, in which the integral over the moduli space of marked curves is localized to solutions of the Gross-Mende equations.

Mini-cours