GAUGE THEORY AND STRING GEOMETRY (Mini Courses + Conference) Monday, 24 Nov 2025, 9am - Friday, 5 Dec 2025, 5pm

The ANU Futures Workshop: Recent Progress of Harmonic Analysis in General Relativity and Quantum Information will be followed by the Gauge Theory and String Geometry (Mini Courses + Conference). More information.

About

Recently, there have been some remarkable breakthroughs of harmonic analysis techniques in general relativity and quantum information.As some featured examples, we refer to a result in 2021 of Hintz and Vasy and now in 2025 of Hintz, Petersen, and Vasy, in proving global nonlinear stability of Kerr-de Sitter spacetimes, extending Fourier-harmonic analytic methods into nonlinear geometric PDES, and the solution of the L2 curvature conjecture for the Einstein Equations by Klainerman, Rodnianski, and Szeftel.

Some main themes of this workshop will be to explore:

• How harmonic analysis can be used in General Relativity settings, which often involve obstacles like a lack of global Fourier transforms, noncompactness of Lorentzian manifolds and the associated scattering theory, and nonlinear and geometric structure of PDEs.
• The analysis of Schrodinger operators that arise in quantum computing (including themes such as Lieb-Robinson bounds).

One of the main goals of the workshop will be to introduce researchers with some background in harmonic analysis and dispersive PDE to some open and accessible problems in general relativity and quantum information.  The structure of the workshop will accommodate time for plenary talks, problem sessions, and discussion and collaboration time.  Thus the workshop is ideal for researchers in harmonic analysis or dispersive PDE seeking an introduction to how these techniques may be applied in a general relativity or quantum information setting, and to be acquainted with some accessible open problems in those fields, or conversely researchers in General Relativity or Quantum Information seeking to incorporate more harmonic analysis techniques into their research.

Participation is in person only. There will be talks that are delivered by video link.

Speakers

Di Fang (Duke University) -  online talk

Jesse Gell-Redman (University of Melbourne)

Zihua Guo (Monash University)

Andrew Hassell (Australian National University)

Xiao Ma (University of Michigan) -  online talk

Todd Oliynyk (Monash University)

David Ou Yang (Ludwig-Maximilians-Universität München)

Volker Schlue (University of Melbourne)

Andras Vasy (Stanford University) -  online talk

Jingxuan Zhang (Tsinghua University)

Scientific and Organising Committee

Alexandria Rose (Australian National University)

Pierre Portal (Australian National University)

Xiaoxu Wu (Australian National University)

Tony Martin (Australian National University)

Registration as a participant

Registrations for participants is open. Please note:

• there is no support available from ANU for participants
• we are unable to issue letters of invitation to participants

Register

Program

Important info:

• All in person talks and group discussions will be in the Hanna Neumann building, room 1.33.
• All online talks and office hours will happen early in the morning (exact times to be confirmed). Participants are expected to use their own devices and follow the talks from wherever they like.
• There will be in person follow up group discussions for each online talk.
• The program is light on purpose, so that participants can have lots of small group discussions. This editable shared spreadsheet allows participants to organise and join these discussions (first tab).
• This editable shared spreadsheet also allows participants to organise and join dinner plans (second tab).

Monday 17: 2-3pm Talk. Zihua Guo. Well-posedness for the compressible Navier-Stokes equations in critical Besov space.5-6pm Welcome reception.

Tuesday 18:8-9am: Online talk. Di Fang. Time-dependent Hamiltonian Simulation: Quantum Algorithm and Superconvergence.10-11am: Group discussion on the talk by Prof. Di Fang.1-2pm: Talk. Todd Oliynyk. Big bang singularities in General Relativity.

Wednesday 19:8-9am: Online Office Hour with Prof. Di Fang.9-10am: Online talk. Andras Vasy. Kerr-de Sitter spacetimes: Stability of the black hole exterior and of the expanding region of Kerr-de Sitter spacetimes.11am-12pm: Group discussion on the talk by Prof. Andras Vasy.1-2pm: Talk. David Ou Yang. Gagliardo-Nirenberg inequality for Hartree-Fock model for neutron star and blow-up of limiting profile of Hartree-Fock ground states.2-3pm: Talk. Jingxuan Zhang. Macroscopic Suppression of Supersonic Quantum Transport.

Thursday 20:8-9am: Online talk. Xiao Ma. Recent Advances on Hilbert’s Sixth Problem.9-10am: Online Office Hour with Prof. Andras Vasy.11am-12pm: Group discussion on the talk by Prof. Xiao Ma.1-2pm: Talk. Volker Schlue. Expanding black hole cosmologies: On the non-linear stability of Kerr de Sitter spacetimes.

Friday 21:9-10am: Online Office Hour with Prof. Xiao Ma.11-12am: Talk. Jesse Gell-Redman. The Feynman propagator for the Klein-Gordon equation.2-3pm: Talk. Andrew Hassell. A microlocal approach to the Nonlinear Schrödinger equation.

Saturday 22:9am-11am: Bushwalk 11am: Lunch, then walk or bus back.

 

Abstracts

Zihua Guo. Well-posedness for the compressible Navier-Stokes equations in critical Besov space.The Cauchy problem to the barotropic compressible Navier-Stokes equation in critical Besov spaces is considered. I will talk about two recent results obtained in (I) arXiv:2409.01031 and (II) arXiv:2509.17005.I. Local well-posednessWe prove the continuity of the solution map by combining the frequency envelope method and the Lagrangian approach. This result bridges the Eulerian and Lagrangian methods in the study of compressible Navier-Stokes equation.II. Global well-posednessWe prove global well-posedness with small data in the optimal critical Besov space assuming some low frequency condition on the initial density and mo mentum. The main ingredients of the proof consist of: a novel nonlinear transformthat uses momentum formulation for low-frequency and effective velocity method for high frequency, and estimate of parabolic-dispersive semigroup that enables a Lq-framework for low frequency.

Di Fang. Time-dependent Hamiltonian Simulation: Quantum Algorithm and Superconvergence.

Hamiltonian simulation becomes more challenging as the underlying unitary becomes more oscillatory. In such cases, an algorithm with commutator scaling and a weak dependence, such as logarithmic, on the derivatives of the Hamiltonian is desired. We introduce a family of new time-dependent Hamiltonian simulation algorithm based on the Magnus series expansion that exhibits both features. Importantly, when applied to unbounded Hamiltonian simulation in the interaction picture, we prove that the commutator in the p-order algorithm leads to a surprising 2p-order superconvergence, with an error preconstant independent of the number of spatial grids. The proof of superconvergence is based on semiclassical analysis that is of independent interest.

Todd Oliynyk. Big bang singularities in General Relativity.Since the 1920s, it has been known that spatially homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes generically develop curvature singularities in the contracting time direction along spacelike hypersurfaces, known as big bang singularities, both in vacuum and for a wide range of matter models. For many years, it remained unclear whether these big bang singularities were physically relevant. A partial resolution came in 1967 when Hawking established his singularity theorem, guaranteeing that a cosmological spacetime is geodesically incomplete for a large class of matter models and initial data sets, including highly anisotropic ones. While Hawking’s theorem ensures geodesic incompleteness for many initial data sets, it is silent on the cause of that incompleteness. It has long been anticipated that the incompleteness arises from the formation of curvature singularities, and it remains an outstanding problem in mathematical cosmology to rigorously determine the conditions under which this expectation holds and to understand the dynamical behaviour of cosmological solutions near singularities.In this talk, I will begin by introducing the Kasner family of solutions to the Einstein-scalar field equations, which are exact, spatially homogeneous solutions, including the FLRW family, determined by a set of parameters called the Kasner exponents. These solutions play a distinguished role in the analysis of big bang singularities, and I will define what it means for the big bang singularities appearing in the Kasner family to be nonlinearly stable. With this notion in hand, I will discuss recent results establishing the nonlinear stability of Kasner big bang singularities over the so-called subcritical range. A particular aspect of these stability proofs to which I will draw attention is the need to consider initial data whose regularity depends on the choice of Kasner exponents and diverges as the exponents approach the boundary of the subcritical range. This phenomenon now appears to be genuine, rather than a deficiency of the proofs. Indeed, in a simplified linear setting, sharp microlocal results show that the regularity of solutions must depend on the Kasner exponents. What remains unclear is how these results extend to the nonlinear regime, where the situation is far more delicate because as the big bang singularity is approached, the limiting Kasner solution is expected to depend on the spatial point of the final singular manifold in a possibly discontinuous manner.

Andras Vasy. Kerr-de Sitter spacetimes: Stability of the black hole exterior and of the expanding region of Kerr-de Sitter spacetimes.Based on joint work with Peter Hintz and partly with Oliver Petersen, I will discuss the nonlinear stability of Kerr–de Sitter spacetimes as solutions of the Einstein vacuum equations with positive cosmological constant. The first part concerns the conditional stability of a neighborhood of the black hole exterior (more precisely, the domain of outer communication) in the subextremal range of black hole parameters, namely stability under the assumption of mode stability for the linearized Einstein equation. The second part discusses the stability of the expanding (cosmological) region in a modification of a generalized harmonic gauge introduced by Ringström; in a different type of gauge such a result was obtained recently by Fournodavlos and Schlue. Due to the hyperbolic character of the gauge, the stability result is local near points on the conformal boundary. I will also discuss the smoothness of the metric up to the future conformal boundary, with a Fefferman–Graham type asymptotic expansion.

David Ou Yang. Gagliardo-Nirenberg inequality for Hartree-Fock model for neutron star and blow-up of limiting profile of Hartree-Fock ground states.

In this talk, I will introduce the relativistic Hartree-Fock (HF) model for neutron star, where particles interact through Newtonian gravity, and present a Gagliardo-Nirenberg (GN) type inequality for such systems.  The relativistic HF model for neutron stars has a finite range of total number of particles for which the system is stable; beyond this range, the system is unstable and undergoes gravitational collapse.  In a dual manner, there is a critical coupling parameter for each fixed total number of particle.  I will show how the GN inequality is used to describe the limiting behavior (blow-up/mass concentration) of ground states as coupling parameter is approaching its critical value.  This talk is based on a joint work with B. Chen, Y. Guo and P. T. Nam.

Jingxuan Zhang. Macroscopic Suppression of Supersonic Quantum Transport.In 1972, Lieb and Robinson proved that for a nonequilibrium quantum spin system, there exists an upper bound on the speed at which physical effects can propagate. Constrained by particle interactions, quantum transport with speed beyond this bound is exponentially suppressed. This upper bound is far below the speed of light and is therefore often referred to as the “speed of sound.” In a recent joint work with Faupin, Lemm, and Sigal, we identified a macroscopic suppression mechanism for “supersonic” quantum transport, valid for a broad class of quantum many-body systems. The suppression strength grows exponentially with the total particle number, rendering supersonic transport effectively impossible at the macroscopic level. To fix ideas, in line with realistic optical lattice experiments, consider a one-dimensional Bose–Hubbard model with 18 particles, lattice spacing of 500 nm, and hopping amplitude of 500 s⁻¹. Our estimates show that within an observation time of 6.67 × 10⁻⁴ s, the probability of supersonic transport across more than six sites is bounded by 1.52 × 10⁻¹⁸—17 orders of magnitude smaller than conventional estimates (≈ 0.37). The goal of this talk is to explain the main result and the key idea behind its proof. 

Xiao Ma. Recent Advances on Hilbert’s Sixth Problem.

In this talk, we present our recent progress on Hilbert’s sixth problem—deriving fluid equations from microscopic dynamics. We will provide the necessary physical background and introduce the key techniques developed in our work, including the Feynman diagram representation and the cutting algorithm.

Volker Schlue. Expanding black hole cosmologies: On the non-linear stability of Kerr de Sitter spacetimes.The Kerr de Sitter geometry models a rotating black hole in an expanding universe. I will review its stability properties in the context of the Einstein vacuum equations with positive cosmological constant, and present a recent resolution of the non-linear stability problem for the cosmological region. The talk is based on joint work with G Fournodavlos, and describes among others contributions by H Friedrich, P Hintz and A Vasy. 

Jesse Gell-Redman. The Feynman propagator for the Klein-Gordon equation.We construct the Feynman propagator for Klein-Gordon (KG) equation on Minkowski space perturbed by a decaying spatial potential. In particular, we construct global in time solutions to the inhomogeneous KG equation, each of whose wavefront sets is contained in the flowout of the wavefront set of the source in the direction of the Hamilton flow. Such solutions were shown to exist locally by Duistermaat-Hormander. To accomplish this, we prove a global Fredholm estimate. The persistence of the potential in time means that estimates for KG can be obtained using positive commutator estimates with operators in the three-body calculus of Vasy. This is joint work with Dean Baskin and Moritz Doll.

Andrew Hassell. A microlocal approach to the Nonlinear Schrödinger equation.I will discuss a new approach to the Nonlinear Schrödinger equation based on Fredholm analysis and microlocal propagation estimates, and show how it provides a direct approach to showing existence of global solutions and scattering for NLS.

Accommodation options

ANU Apartments (on campus)

Novotel Canberra - Google Maps

Quest Canberra - Google Maps

QT Canberra - Google Maps

Peppers Gallery Hotel Canberra - Google Maps

Ovolo Nishi - Google Maps

Canberra Accommodation Centre. They are a short tram ride from ANU.

Forrest Hotel and Apartments. They are about a 10 min drive from ANU.

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