We discuss thermodynamics of N M2- and M5-branes by using the method proposed by Smilga and Wiseman, which explains the black Dp-brane thermodynamics from the maximally supersymmetric U(N) Yang-Mills theories. As result we obtain the consistent results with the predictions from the eleven-dimensional supergravity by very simple calculations: The free energy of M2-branes is evaluated by using ABJM theory as F ~ N^{3/2}k^{1/2}T^3, and that of M5-branes is estimated by assuming some natural properties of 6d conformal field theory as F ~ N^3T^6.

We discuss tree level scattering of any number of massless open superstring states on a worldsheet of disk topology. The entire state dependence of the tree amplitude can be expressed in terms of gauge theory subamplitudes from the point particle limit. The string corrections entering through momentum dependent integrals over the disk boundary can be disentangled from the YM seeds and analyzed separately. Their power series expansion in the string length and momenta involves multiple zeta values (MZVs). We review some mathematical background on MZVs and the network of relations between them. The explicit form of any tree level string correction to YM theory is derived from the generating function of MZVs -- the Drinfeld associator. It interpolates between the worldsheet integrals in N-point and (N-1)-point scattering and leads to a recursive formula for the momentum expansion of any disk amplitude. Our results apply for any number of spacetime dimensions or supersymmetries and chosen helicity configurations.

Although at first sight it may seem an odd idea, I will argue that it is actually quite natural to investigate the properties of General Relativity and its black holes in the limit in which the number of spacetime dimensions grows to infinity. The theory simplifies dramatically: it reduces to a theory of non-interacting particles, of finite radius but vanishingly small cross sections, which do not emit nor absorb radiation of any finite frequency. This leads to efficient calculational approaches in an expansion around this limit, as well as to intriguing connections to low-dimensional string-theory black holes.

From the supersymmetric partition functions of gauge theories on S5 and CP2 x S1, we calculate the superconformal index of the 6d (2,0) SCFT on S5 x S1 and explore its physics.

I will review the basic techniques and ideas involved in the study of dualities in D=3,N=4 supersymmetric gauge theories using partition functions on a sphere and demonstrate how these partition functions can provide an extremely non-trivial check of mirror symmetry for a large class of quiver gauge theories. In particular, I will focus on theories whose Type IIB descriptions involve D3 branes ending on orbifold/orientifold 5-planes. In addition to providing a convincing check, this procedure allows one to extract important information about the duality like the "mirror map" rather trivially.

We explain how to define, in matrix (gauge) field theory, the notion of the effective action of k ``probe'' branes in the presence of N ``background'' branes on which the field theory lives. The analysis of the large N planar diagram expansion which computes the effective action yields a simple and generic mechanism explaining the emergence of holographic space dimensions. The resulting effective action matches the non-abelian D-brane action in the closed string background dual to the field theory. We shall discuss briefly a few explicit examples, and also provide a general introduction to the notion of emerging space.

We present a collection of results supporting holographic dualities between 3D higher spin gravity and 2D higher spin-symmetric minimal model CFTs at large central charge, focusing on those that are not fixed by higher spin symmetry alone. This includes the first bulk-boundary matching of correlators in 3D higher spin gravity whose functional form is not fixed by conformal invariance, namely, 4-point functions of certain scalar primary operators. In the bulk, this involves the study of propagating scalars in higher spin gravitational backgrounds, which will also help clarify what constitutes a higher spin black hole.

I will review the intermediate long wave equation which interpolates between the celebrated Korteweg-de Vries and Benjamin-Ono equations, describing the one-dimensional waves in a shallow and deep water, respectively. The quantum version of this equation is of some interest to the theory of quantum wires and other quasi-one dimensional condensed matter systems. I will explain the unexpected connection of this equation to the dynamics of gauge theories in two, four, and six dimensions. This connection is one of the applications of the BPS/CFT correspondence. Based on the work in progress with Andrei Okounkov

I will discuss recent developments in the duality between gauge-fields and Strings.

**Polygon Seminar** Over the past several years, it has become increasingly apparent that many interesting and tractable superconformal field theories (SCFTs) can be realized without resorting to a UV Lagrangian description. Although many of these theories have N=2 supersymmetry, these is an even larger class of N=1 theories that can be studied, certain aspects of which remain calculable despite the reduced amount of supersymmetry. In this talk, I will give an introduction to these N=1 theories, how they can be realized via M-theory and F-theory, and which tools we can use to explore them.

Computation of conformal dimensions in planar N=4 SYM using integrability techniques was a hot topic during the last decade, with more than thousand publications devoted to it. I will tell you about our new results in this domain: Instead of the Y-system used previously, we are now able to encode the conformal dimensions, at any value of the 't Hooft coupling, in much simpler way: through a Riemann-Hilbert problem. This appears to be not only a very beautiful mathematical setup, but also the most efficient approach to explicitly compute the dimensions. For instance, we've analytically computed the so called Konishi anomalous dimension up to 8 loops in perturbation theory. The talk will include a pedagogical overview of the subject, no special knowledge in this domain is required.

We study a holographic model dual to a CFT in 2+1 dimensions at finite temperature and chemical potential with a global U(2) symmetry. At large enough chemical potential spontaneous symmetry breaking occurs and breaks the symmetry to U(1). The non-abelian nature of the symmetry and the explicit Lorentz breaking by the chemical potential imply the presence of an ungapped mode with quadratic dispersion relation in the broken phase. Such modes are called Type II Goldstone bosons and have several distinguished features that we study within the framework of this holographic model.

In this talk I will explain a new method to numerically construct stationary black holes with non-Killing horizons. As an example, we will use AdS/CFT to describe a time-independent CFT plasma flowing through a static spacetime which asymptotes to Minkowski in the flow's past and future, with a varying spatial geometry in-between. When the boundary geometry varies slowly, the holographic stress tensor is well-described by viscous hydrodynamics. For fast variations it is not, and the solutions are stationary analogs of dynamical quenches, with the plasma being suddenly driven out of equilibrium. We find evidence that these flows become unstable for sufficiently strong quenches and speculate that the instability may be turbulent. The gravitational dual of these flows are the first examples of stationary black holes with non-Killing horizons.

I will present the quantized function algebras associated with various examples of generalized sine-Gordon models. These are quadratic algebras of the general Freidel-Maillet type, the classical limits of which reproduce the lattice Poisson algebra recently obtained for these models formulated as gauged Wess-Zumino-Witten models plus an integrable potential. More specifically, I will argue based on these examples that the natural framework for constructing quantum lattice integrable versions of generalized sine-Gordon models is that of affine quantum braided groups.

Quivers are directed graphs which encode information about the gauge groups and matter content of a large class of gauge theories, many of which have AdS/CFT duals. The counting of local gauge invariant operators and the computation of their correlators (in the free field limit) can be done by simple diagrammatic manipulations of the quiver, with the help of permutation group theory data. This data includes Young diagrams, Littlewood-Richardson numbers and branching coefficients of permutation groups. Riemann surfaces obtained by thickening the quivers are intimately related to these computations.

Tom Kibble was a pioneer of the symmetry breaking paradigm in fundamental physics. His view of physics is exceptionally broad, and Tom also led efforts to see how to test ideas of grand unification through exploring their consequences for the very early universe. Over time, this led to new paradigms for cosmology, like cosmic inflation, with a plethora of observational tests. In this talk, I shall review some of Tom's cosmological innovations and also look forward to new and even more fundamental paradigms capable of tackling the big bang singularity. [Symmetry and Fundamental Physics - Tom Kibble at 80]

In a seminal 1976 paper Tom Kibble pointed out that, in cosmological phase transitions, causality precludes coordination between local choices of broken symmetry, and, as a result, formation of topological defects is all but inevitable. I shall discuss consequences of this Kibble mechanism for condensed matter physics, where its experimentally testable extrapolation is being studied. [Symmetry and Fundamental Physics - Tom Kibble at 80]