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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.

Topological phases of matter entail a prominent research theme, featuring
distinct characteristics that include protected metallic edge states and
the possibility of fractionalized excitations. With the advent of symmetry
protected topological (SPT) phases, many of these phenomena have
effectively become accessible in the form of readily available band
structures. Whereas the role of (anti-)unitary symmetries in such SPT
states has been thoroughly understood, the inclusion of lattice symmetries
provides for an active area of research.

In this talk, I will present a short overview of results on defects in SPT
states that directly motivate the existence of additional physics beyond
the characterization based on (anti-)unitary symmetries. More importantly,
I will then connect these ideas to recent work in which we were able to map
out all different gapped phases of free fermion systems in the presence of
solely lattice symmetries. This revolves around a very simple algorithm
that matches a rather involved mathematical perspective in terms of a
framework called K-theory. I will then discuss the implications of these
combinatorial arguments, such as their impact on the description of Weyl
phases, and sketch related ideas and future perspectives.

The entanglement properties of random quantum states or dynamics are important to the study of a broad spectrum of disciplines of physics, ranging from quantum information to high energy and many-body physics. This work investigates the interplay between the degrees of entanglement and randomness in pure states and unitary channels, by employing tools from random matrix theory, representation theory, combinatorics, Weingarten calculus etc. We reveal strong connections between designs (distributions of states or unitaries that match certain moments of the uniform Haar measure) and generalized entropies (entropic functions that depend on certain powers of the density operator), by showing that Renyi entanglement entropies averaged over designs of the same order are almost maximal. This strengthens the celebrated Page's theorem. Moreover, we find that designs of an order that is logarithmic in the dimension maximize all Renyi entanglement entropies, and so are completely random in terms of the entanglement spectrum. Our results relate the behaviors of Renyi entanglement entropies to the complexity beyond scrambling/thermalization/chaos in terms of the degree of randomness, and suggest a generalization of the fast scrambling conjecture.  (Refs: 1703.08104, 1709.04313)