Talks

Let G be a complex reductive group, and X be any smooth projective G-variety. In this talk, we will construct an algebra homomorphism from the G-equivariant homology of the affine Grassmannian Gr_G to the G-equivariant quantum cohomology of X. The construction uses shift operators in quantum cohomologies. We will also discuss the possible extension to the loop rotation equivariant setting and the relation with the Peterson isomorphism when X is the flag variety associated with G. This is based on joint work with Alexander Braverman.

There has long been interest in Beilinson-Bernstein localization for the affine Grassmannian (or affine flag variety). First, Kashiwara-Tanisaki treated the so-called negative level case in the 90's. Some ten years later, Frenkel-Gaitsgory (following work of Beilinson-Drinfeld and Feigin-Frenkel) formulated a conjecture at the critical level and made some progress on it. Their conjecture is more subtle than its negative level counterpart, but also more satisfying. We will review the necessary background from representation theory of Kac-Moody algebras at critical level, formulate the Frenkel-Gaitsgory conjecture, and outline a proof for GL_2. Time permitting, we will discuss how our result provides a test of the Frenkel-Gaitsgory proposal for local geometric Langlands.

You’ve likely heard of quantum computing. Maybe you’re even familiar with the basic principles of how this emerging form of technology harnesses counter-intuitive properties of the subatomic realm to perform tasks that would overwhelm even today’s most powerful “classical” computers. But do you know what that will mean for the ways you work, communicate, play, and live?

Does anyone truly know? Well, no, because some very big challenges remain before quantum computers are commonplace. But for expert perspectives on what the future may hold and how to prepare for the quantum future, you’d be hard-pressed to find a better source than Shohini Ghose.

In her March 4 public lecture at Perimeter Institute, Ghose will guide the audience through the latest advances in the quantum world and share her own journey in quantum science. Ghose, a professor of physics and computer science at Wilfrid Laurier University, studies how the laws of quantum physics can be exploited to transform computing and communications, and to achieve feats such as teleportation.

Ghose is a leading expert, and sought-after speaker, in quantum information science, as well as how to make the global science community more vibrant and inclusive. She is the founding director of the Laurier Centre for Women in Science and President of the Canadian Association of Physicists, and she served as Perimeter Institute’s first Diversity, Equity, and Inclusion Specialist.

Among her many honours, Ghose is a TED Senior Fellow and member of the Royal Society of Canada’s College of New Scholars, Artists, and Scientists. In 2019, she was among 25 leading women scientists from around the world featured in a UNESCO exhibit in Paris.

The average entanglement entropy of subsystems of random pure states is (nearly) maximal. In this talk, we discuss the average entanglement entropy of subsystems of highly excited eigenstates of integrable and nonintegrable many-body lattice Hamiltonians. For translationally invariant quadratic models (or spin models mappable to them) we prove that, when the subsystem size is not a vanishing fraction of the entire system, the average eigenstate entanglement entropy exhibits a leading volume-law term that is different from that of random pure states. We argue that such a leading term is likely universal for translationally invariant (noninteracting and interacting) integrable models. For random pure states with a fixed particle number (random canonical states) away from half filling and normally distributed real coefficients, we prove that the deviation from the maximal value grows with the square root of the system's volume when the size of the subsystem is one half of that of the system. We then show that the average entanglement entropy of highly excited eigenstates of a particle number conserving quantum chaotic model is the same as that of random canonical states.

Defining a generic quantum system requires, together with a Hilbert space and a Hamiltonian, the introduction of an algebra of observables, or equivalently a tensor product structure. Assuming a background time variable, Cotler, Penington and Ranard showed that the Hamiltonian selects an almost-unique tensor product structure. This result has been advocated by Carrol and collaborators as supporting the Everettian interpretation of quantum mechanics and providing a pivotal tool for quantum gravity. In this talk I argue against this, on the basis of the fact that the Cotler-Penington-Ranard result does not hold in the generic background-independent case where the Hamiltonian is replaced by a Hamiltonian constrain. This reinforces the understanding that entropy and entanglement, that in the quantum theory depend on the tensor product structure, are quantities that are observable dependent. To conclude, I would like to pose the question of whether clocks can be thought as a resource, and how thinking of time in terms of physical clocks can inform our interpretation of quantum mechanics