Brane construction with certain boundary conditions are used to study knot invariants and Khovanov homology. We argue that seven-dimensional manifolds in M-theory give rise to the topological theories may appear from certain twisting of the G-flux action. We discuss explicit constructions of the seven-dimensional manifolds in M-theory, and show that both the localization equations and surface operators appear naturally from the Hamiltonian formalism of the theories. Knots and link invariants are then constructed using M2-brane states in both the models.

Exceptional geometry is an attempt to combine the geometric diffeomorphisms and matter gauge transformations in gravity-matter theories into a single geometric structure. I will review recent results associated with a 2+9 split of maximal supergravity where the affine symmetry group E9 plays a central role. The results also provide a general formula that is applicable to many other cases.

A powerful approach to compute multi-loop Feynman integrals is to reduce the integrals to a basis of integrals and set up a first-order linear system of partial differential equations for the basis integrals. In this talk I will discuss the differential equations that arise when the loop integrals are parametrized in Baikov representation. In particular, I give a proof that dimension shifts (which are undesirable) can always be avoided. I will moreover show that in a large class of two- and three-loop diagrams it is possible to avoid integrals with squared propagators in the intermediate stages of setting up the differential equations. This is interesting because it implies that the differential equations can be set up using a smaller set of reductions.

I will discuss the programme to understand conformal field theory via subfactors and twisted equivariant K-theory. This has also resulted in a better understanding of the double of the Haagerup subfactor, which was previously thought to be exotic and un-related to known models.

Squaring involves the tensoring between the state content of two super Yang-Mills (sYM) theories to obtain the state content of a supergravity theory. Understanding the YM origin of gravitational symmetries is a powerful tool towards classifying gravity theories which admit such a factorisation. In the first part of the talk I will show how the global symmetries of a pair of sYM theories combine to form those of the corresponding supergravity. In the second part I will discuss how these tools can be further extended to sYM coupled to matter such that squaring can give all ungauged N=2 supergravities with homogeneous scalar manifold.

The main purpose of this talk is to describe, by way of concrete examples, how the field of numerical relativity now contributes to our understanding of open questions in gravitational collapse, heavy-ion physics, and turbulence. I will begin by motivating these studies in terms of the physical systems they are intended to clarify,then provide specific examples of how to describe these systems with numerical simulations of asymptotically AdS spacetimes in the fully non-linear regime of general relativity.

The relation between black holes and thermodynamics leading to the holographic principle is well known. Formulating thermodynamics as the theory of transformations performing some work or task allows us to reinterpret recent developments in AdS/CFT, such as the holographic description of entanglement entropy, as a measure of the connectivity of space (resource). Whether spacetime in the interior of a black hole also allows an understanding as a resource is an interesting open question.

Type IIB superstring brane configurations can have a low energy dynamics described by an effective 3d N=4 gauge theory. The moduli space of the gauge theory is normally a Hyperkähler variety. Singular points in the variety correspond to brane configurations where some fields become massless, giving rise to the Higgs mechanism. I will explain the relevance of a set of theories whose moduli space is the closure of a nilpotent orbit of Lie(F), where F is the flavour symmetry group of the theory. I will show how the mathematical description of the "transverse slice" between two nilpotent orbits can be understood in terms of brane dynamics as a realisation of the Higgs mechanism.

These lectures will be focused on aspects of combinatorics relevant to gauge-string duality (holography). The physical theories we will discuss include two dimensional Yang Mills theory, four-dimensional N=4 super Yang Mills theory with U(N) gauge group, Matrix and tensor models. The key mathematical concepts include : Schur Weyl-duality, permutation equivalence classes and associated discrete Fourier transforms as an approach to counting problems and, branched covers and Hurwitz spaces. Schur-Weyl duality is a powerful relation between representations of U(N) and representations of symmetric groups. Representation theory of symmetric groups offers a method to define nice bases for functions on equivalence classes of permutations. These bases are useful in counting gauge invariant functions of matrices or tensors, as well as computing their correlators in physical theories. In AdS/CFT these bases have proved useful in identifying local operators in gauge-theory dual to giant gravitons in AdS. In the simplest cases of gauge-string duality, the known mathematics of branched covers and Hurwitz spaces provide the mechanism for the holographic correspondence between gauge invariants and stringy geometry. (Lecture 3: Hermitian matrix model. Tensor models and Permutation centralizer al- gebras. Using permutation equivalences to count matrix/tensor invariants and compute correlators. Relations to covering spaces.)

These lectures will be focused on aspects of combinatorics relevant to gauge-string duality (holography). The physical theories we will discuss include two dimensional Yang Mills theory, four-dimensional N=4 super Yang Mills theory with U(N) gauge group, Matrix and tensor models. The key mathematical concepts include : Schur Weyl-duality, permutation equivalence classes and associated discrete Fourier transforms as an approach to counting problems and, branched covers and Hurwitz spaces. Schur-Weyl duality is a powerful relation between representations of U(N) and representations of symmetric groups. Representation theory of symmetric groups offers a method to define nice bases for functions on equivalence classes of permutations. These bases are useful in counting gauge invariant functions of matrices or tensors, as well as computing their correlators in physical theories. In AdS/CFT these bases have proved useful in identifying local operators in gauge-theory dual to giant gravitons in AdS. In the simplest cases of gauge-string duality, the known mathematics of branched covers and Hurwitz spaces provide the mechanism for the holographic correspondence between gauge invariants and stringy geometry. (Lecture 2: Local gauge invariant operators and Hilbert space of CFTs. Young diagrams and Brane geometries. Half-BPS and quarter-BPS. Counting, construction and correlators in group theoretic combinatorics.)

The NJL model is a classic model of chiral symmetry breaking in QCD and the gauged NJL model underlies many BSM models. I investigate how to apply Witten's double trace prescription in holographic models of quarks to describe NJL interactions. A holographic realisation of NJL and gauged NJL is realised and can be applied to understanding QCD and extended technicolor models.

These lectures will be focused on aspects of combinatorics relevant to gauge-string duality (holography). The physical theories we will discuss include two dimensional Yang Mills theory, N=4 super Yang Mills theory with U(N) gauge group, Matrix and tensor models. The key mathematical concepts include : Schur Weyl-duality, permutation equivalence classes and associated discrete Fourier transforms as an approach to counting problems and, branched covers and Hurwitz spaces. Schur-Weyl duality is a powerful relation between representations of U(N) and representations of symmetric groups. Representation theory of symmetric groups offers a method to define nice bases for functions on equivalence classes of permutations. These bases are useful in counting gauge invariant functions of matrices or tensors, as well as computing their correlators in physical theories. In AdS/CFT these bases have proved useful in identifying local operators in gauge-theory dual to giant gravitons in AdS. In the simplest cases of gauge-string duality, the known mathematics of branched covers and Hurwitz spaces provide the mechanism for the holographic correspondence between gauge invariants and stringy geometry. (Lecture 1: Two dimensional Yang Mills theory. Exact solution. Large N expansion. Role of Schur-Weyl duality - relation between representation theory of symmetric groups and unitary groups. Hurwitz spaces and string interpretation of the large N expansion.)

In this talk, I will discuss the generalised and basic fuzzy 4-sphere in the context of the IKKT matrix model. These spaces arise as SO(5)-equivariant projections of quantised SO(6) coadjoint orbits and exhibit full SO(5) covariance. I will sketch how (basic and generalised) 4-sphere arise as solutions in a Yang-Mills matrix model, such that the fluctuations on the 4-sphere lead to a higher-spin gauge theory.

I will describe how to recast perturbative quantumgravity using non-perturbative techniques from conformal field theory,focussing on the case of N=4 super Yang-Mills theory. By resolving thedegeneracy among double trace operators at large N we are able to bootstrapone-loop supergravity corrections from the OPE of the CFT.

I'll give a basic introduction to particle-vortex duality in d=2+1 dimensions and its relation to 3d bosonization.

In this talk I shall give a review of two classes of integrable non-linear sigma models called \eta and \lambda deformations. Three reasons to be interested are 1) these are interesting examples of relatively rare integrable QFT displaying quantum group symmetries; 2) viewed as string theory sigma models they may have application to N=4 SYM via holography and 3) they provide concrete examples for generalised notions of T-duality. This talk will describe a variety of classical and quantum properties of these theories and their multi-parameter extensions drawing in part on arXiv:1711.00084; arXiv:1706.05322

Integrable lattice models in 1+1 dimensions have been a topic of much fascination in physics since the early days of quantum mechanics. Recently, a work by Costello has offered a new perspective on these models based on a 4D topological gauge theory. An interesting aspect of this proposal is the close connection with 3D Chern-Simons theory and knot invariants. I will explain how to derive the 4D topological theory using string dualities and localization techniques for a 5D twisted super Yang-Mills theory. Finally, I will explain how this construction might relate to other appearances of Yangian symmetry which played a prominent role in AdS/CFT physics. This is a joint work in progress with Joe Minahan.

In this talk we consider holographic duals of F-theory solutions to 2d SCFT's. We approach the problem by classifying a particular class of solutions of type IIB supergravity with AdS_3 factors and varying axio-dilaton. The class of solutions we discuss consist of D3 and 7-brane configurations and naturally fall into the realm of F-theory. We prove that for (0,4) supersymmetry in 2d the solutions are essentially unique and we match the holographic central charges to field theory results. We comment on future directions, including AdS_3 solutions of F-theory, preserving different amounts of supersymmetry.