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Factorization spaces (introduced by Beilinson and Drinfeld as "factorization monoids") are non-linear analogues of factorization algebras. They can be constructed using algebro-geometric methods, and can be linearised to produce examples of factorization algebras, whose properties can be studied using the geometry of the underlying spaces. In this talk, we will recall the definition of a factorization space, and introduce the notion of a module over a factorization space, which is a non-linear analogue of a module over a factorization algebra. As an example and an application, we will introduce a moduli space of principal G-bundles with parabolic structures, and discuss how it can be linearised to recover modules of the factorization algebra associated to the affine Lie algebra corresponding to a reductive algebraic group G.

This is based on my joint work with Yaping Yang. In this talk, we use the equivariant elliptic cohomology theory to study the elliptic quantum groups.  We define a sheafified elliptic quantum group for any symmetric Kac-Moody Lie algebra. This definition is naturally obtained using the cohomological Hall algebra associated to the equivariant elliptic cohomology. After taking suitable rational sections, the sheafified elliptic quantum group becomes a quantum algebra consisting of the elliptic Drinfeld currents. The Drinfeld currents satisfy the relations of the elliptic quantum group studied by Felder and Gautam-Toledano Laredo. We show the elliptic quantum group acts on the equivariant elliptic cohomology of Nakajima quiver varieties.

In particular, the sheafified elliptic quantum group is an algebra object in a certain monoidal category of sheaves on the colored Hilbert scheme of an elliptic curve. This monoidal structure is related to Mirkovic’s refinement of the factorization structure on semi-infinite affine Grassmannian over an elliptic curve. If time permits, I will also talk about a work in progress, joint with Mirkovic and Yang, towards a construction of a double loop Grassmannian and vertex representations of the toroidal algebra, which in turn is related to representations of the elliptic quantum groups

Although entanglement harvesting was first posited over 25 years ago, it is only in recent years that this phenomenon has been the subject of active study. The basic idea of entanglement harvesting is to transfer correlations from the vacuum of some quantum field to a pair of detectors. The result provides a new probe of the structure of spacetime via quantum correlations. I shall describe recent work on some of the first results in harvesting entanglement in curved spacetime, in particular anti de Sitter spacetime and black holes.

The observations of gravitational waves from coalescing compact binary systems allow us to test gravity in its strong field regime. In order to better constrain alternative theories of gravity, one has to build template waveforms for these theories. In this talk, I will present a post-Newtonian Lagrangian approach adapted to the specificities of scalar-tensor theories. I will derive the equations of motion of a compact binary system at 3PN order in harmonic coordinates. This result is primordial in order to compute the scalar and gravitational waveforms at 2PN order. I will conclude by a short discussion on tidal effects.

Adiabatic quantum computation (AQC) is a method for performing universal quantum computation in the ground state of a slowly evolving local Hamiltonian, and in an ideal setting AQC is known to capture all of the computational power of the  quantum circuit model.  However, despite having an inherent robustness to noise as a result of the adiabatic theorem and the spectral gap of the Hamiltonian, it has been a longstanding theoretical challenge to show that fault-tolerant AQC can in principle be performed below some fixed noise threshold.  There are many aspects to this challenge, including the difficulty of adapting known ideas from circuit model fault-tolerance as well as the need to develop an error model that is appropriately tailored for open system AQC.  In this talk I will introduce a scheme for combining Feynman-Kitaev history state Hamiltonians with topological quantum error correction, in order to show that universal quantum computation can be encoded not only in the ground state but also in the finite temperature Gibbs state of a local Hamiltonian.  Using only local interactions with bounded strength and a polynomial overhead in the number of qubits, the scheme is intended to serve as a proof of principle that universal AQC can be performed at non-zero temperature, and also to further our understanding of the complexity of highly entangled quantum systems in thermal equilibrium.