An old question of Poincaré concerns creating periodic orbits via perturbations of a flow/diffeomorphism. While pseudoholomorphic methods have successfully addressed this question in dimensions 2-3, the higher-dimensional case remains less understood. I will describe a connection between this question and Gromov-Witten invariants, which goes through a new class of invariants of symplectic cobordisms.
Stochastic Analysis is concerned with solving differential equations in the presence of highly irregular random noise terms. The field has evolved from the foundational works by Itô in the 1940s and its method are used today in numerous modelling contexts. In the first half of this talk I will present my personal take on some of this history and some of the key ideas used. In the second half, I will discuss exciting developments of the last 10 years that show how methods developed for stochastic differential equations allow to give a new perspective on the classical problem to rigorously construct quantum fields.
Extremal (maximally rotating or maximally charged) and near-extremal black holes are of intense interest both for real astrophysics and in the context of fashionable speculations in high energy physics. They remain perhaps the most misunderstood objects in classical general relativity. In this talk, I will first introduce extremal black holes to a general mathematical audience. I will then discuss the stability problem for extremal (and near-extremal) black holes and describe a new conjectural picture of the moduli space of solutions of the Einstein equations describing gravitational collapse.
Wigner’s surmise states that the spectrum of the Hamiltonian of heavy nuclei is distributed like that of a large random matrix. Since it was proposed by Wigner in 1956, the eigenvalue distribution of large random matrices has been used as a toy model to study the distribution of more complex mathematical objects such as random tiles or the longest increasing subsequence of a random perturbation. However, this universality phenomenon generally concerns distributions derived from Gaussian matrices, known as the Gaussian ensembles. In this talk, we will discuss more general universality classes that appear in the theory of random matrices.
Dissipative processes can drive different magnetic orders in quantum spin chains. Using a non-perturbative analytic mapping framework, we systematically show how to structure different magnetic orders in spin systems by controlling the locality of the attached baths. Our mapping approach reveals analytically the impact of spin-bath couplings, leading to the suppression of spin splittings, bath-dressing and mixing of spin-spin interactions, and emergence of non-local ferromagnetic interactions between spins coupled to the same bath, which become long-ranged for a global bath. Our general mapping method can be readily applied to a variety of spin models: We demonstrate (i) a bath-induced transition from antiferromagnetic (AFM) to ferromagnetic ordering in a Heisenberg spin chain, (ii) AFM to extended Neel phase ordering within a transverse-field Ising chain with pairwise couplings to baths, and (iii) a quantum phase transition in the fully-connected Ising model. We also demonstrate how the mapping approach can be applied to higher dimensions, larger spin systems, and fermionic systems.
Describing strongly interacting electrons is one of the crucial challenges of modern quantum physics. A comprehensive solution to this electron correlation problem would simultaneously exploit both the pairwise interaction and its spatial decay. By taking a quantum information perspective, we explain how this structure of realistic Hamiltonians gives rise to two conceptually different notions of correlation and entanglement. The first one describes correlations between orbitals while the second one refers more to the particle picture. We illustrate those two concepts of orbital and particle correlation and present measures thereof. Our results for different molecular systems reveal that the total correlation between molecular orbitals is mainly classical, raising questions about the general significance of entanglement in chemical bonding. Finally, we also speculate on a promising relation between orbital and particle correlation and explain why this may replace the obscure but widely used concept of static and dynamic correlation.
An exciting promise of quantum simulators is to enable a first-principles look into the real-time dynamics of matter after high-energy collisions of hadrons and nuclei, which mimic conditions in the early universe. To realize such a promise, first the gauge theories of the Standard Model should be mapped to quantum simulators. Then complex initial states, in the form of moving wave packets of composite (bound) states of elementary constituents, need to be prepared. While much progress has happened in the former in recent years, developments in the latter are just starting to gain momentum. In this talk, I will provide three examples from our recent work to demonstrate concrete proposals and algorithms for hadronic wave-packet preparations in confining models, from Ising spin systems to the low-dimensional abelian lattice gauge theories. These examples involve a range of platforms, from (solid-state and atomic) analogue quantum simulators to digital quantum computers. I will further present results for numerical studies of expected scattering outcomes, and conditions for observing inelastic channels, along with a demonstration of a high-fidelity meson wave packet generated on a trapped-ion quantum computer.
Presentation of work on (asymptotic) analysis of non-linear time-dependent PDEs modelling fast, self-interacting charged fermions in relativistic quantum mechanics, from the Dirac-Maxwell to Vlasov/Euler-Poisson equations. I focus on intermediate first and second order in 1/c models, such as the Pauli-Poisswell and Euler-Darwin equations, which are useful e.g. in plasma physics.
We investigate a micro-scale model of superfluidity derived by Pitaevskii in 1959 to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional non-linear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of solutions in 𝕋d (d=2,3) for a power-type non-linearity, beginning from small initial data. Depending upon the strength of the nonlinear self-interactions, we obtain solutions that are global or almost-global in time.
The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while obtaining time-independent a priori estimates. We present two different approaches (purely energy based, versus a combination of energy estimates and maximal regularity) based on the dimension.
In the last few years, much progress has been made in connecting the field of QFT amplitudes calculations to that of classical physical observables, such as gravitational waveforms and power emitted of merging black holes. These observables typically arise from highly energetic mergers, where point-particle descriptions and flat space approximations start to break down. On the side of classical relativity, this has naturally led to alternative approximation schemes, such as the self-force expansion (valid for extreme mass ratios of the two bodies). However, on the side of amplitudes, flat space QFT is not well-adapted to capture the full non-linearities of this problem. In this talk, I will present recent developments in addressing this gap via amplitudes on strong backgrounds.
I consider the scattering of charged particles on particular electromagnetic fields which have properties analogous to gravitational horizons. Classically, particles become causally excluded from regions of spacetime beyond a null surface which I identify as an 'electromagnetic horizon'. In the quantum theory there is pair production at the horizon via the Schwinger effect, but only one particle from the pair escapes the field. Furthermore, unitarity appears to be violated when crossing the horizon, and there is no well-defined S-matrix. Despite this, the perturbiner method can be used to construct 'amplitudes' which contain all the dynamical information required to construct observables related to pair creation, and to radiation from particles scattering on the background.
The double copy programme aims to construct gravitational quantities from suitably defined "products" of analogous quantities in gauge theory. It was initially developed in the context of scattering amplitudes and hence appears biased towards perturbation theory in flat backgrounds. I will give an overview of recent efforts to extend the double copy beyond flat spacetimes, and comment on possible connections to cosmology and holography.
In this talk, I will introduce a method to bound higher order operators of Effective Field Theories (EFTs) by assuming causal propagation in the IR. I will give two examples of these bounds; one is in scalar EFTs and the other one is in photon EFTs. In these examples, we will see how causality bounds can be similar or complementary to positivity bounds, which are derived using UV assumptions, in different regions of the Wilson coefficient space.
We develop a formalism to understand the full gravitational two-body dynamics of classical scattering and bound states and their matrix elements using the Schwinger-Dyson equations, with the aim of computing bound observables from amplitudes. Starting with the familiar case of on-shell scattering and bound wavefunctions defined on a Schwarzschild background, we show that they can be analytically continued into each other in the partial wave basis by using a definite branch cut prescription for the incoming energy. The map involves also taking the residue on the bound state pole, which can be avoided by resuming superclassical iterations: this prompts us to study the classical Bethe-Salpeter recursion in the conservative and the radiative case, which can be solved in impact parameter space in terms of an exponential structure connected to two-massive particle irreducible (2MPI) kernels. The relation of these kernels with the Hamilton-Jacobi action and with the waveform is then established. We find that the scattering waveform admits then a natural analytic continuation to the bound waveform at tree-level order, which we explicitly checked by studying the Post-Newtonian expansion of the time-domain multipoles at large angular momentum. Our boundary to bound map agrees also with the Damour-Deruelle prescription for the orbital elements in the quasi-Keplerian parametrization, which enters into the direct evaluation of the time-domain multipoles. Finally, we discuss how our map is consistent with the analytic continuation of the fluxes (i.e., radiated energy and angular momentum) at 3PM order entirely in terms in the binding energy of the system.
In space dimension larger or equal to two, the non-linear Klein-Gordon equation with small, smooth, decaying initial data has global in time solutions. This no longer holds true in one space dimension, where examples of blowing up solutions are known. On the other hand, it has been proved that if the non-linearity satisfies a convenient compatibility condition, the 'null condition', one recovers global existence and that the solutions satisfy the same dispersive bounds as linear solutions. The goal of this talk is to show that, in the case of cubic semi-linear nonlinearities, this null condition is optimal, in the sense that, when it does not hold, one may construct small, smooth, decaying initial data giving rise to solutions that display inflation of their L∞ and L2 norms in finite time.
Many interesting gravitational-wave sources can be effectively described in a large-mass ratio expansion. The leading effect of the lighter secondary in such sources can be read off from the dynamics of a spinning test particle in the space-time of the heavy primary. I will discuss the integrability and exact solution of motion of spinning test particles in black hole space-times, focusing on bound motion but also mentioning some results on scattering.
Classical observables for Kerr black hole dynamics can be constructed from scattering amplitudes. Elegant three-point spin-s amplitudes exist for Kerr black holes, however constructing the corresponding four-point Compton amplitudes is an open problem. In this talk, I will discuss the origin of the Kerr three-point amplitudes from a higher-spin theory perspective. Guided by higher-spin constraints and classical-limit analysis, I will propose quantum and classical tree-level Compton amplitudes relevant for Kerr to all orders in spin. I will also comment on upcoming results for scattering observables that require the classical Compton amplitude as input.
In 2007, Wolfgang Rump introduced algebraic objects called braces, these generalize Jacobson radical rings and are related to involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation (YBE). These objects were subsequently generalized to skew braces by Leandro Guarnieri and Leandro Vendramin in 2017, and a similar relation was shown to hold for non-degenerate set-theoretic solutions of the YBE which are not necessarily involutive. In this talk, we will describe this interplay between skew braces and the YBE. We will also discuss their relation to Hopf-Galois structures and see how this extends the classical Galois theory in an elegant way.
The Higgs mechanism is a part of the Standard Model of quantum mechanics that allows certain kinds of particles to have non-zero mass. In spite of its great importance, there is no rigorous proof that the Higgs mechanism can indeed generate mass in situations that are relevant for the Standard Model. In technical terms, this corresponds to the "coupling parameter" of the model being small, and the "gauge group" being non-abelian (the most important cases are SU(2) or SU(3)). I will present the first rigorous proof in this direction, showing that SU(2) lattice Yang-Mills theory coupled to a Higgs field transforming in the fundamental representation of SU(2) has a mass gap at any value of the coupling parameter, provided that the interaction of the Higgs field with the gauge field is strong enough. No background is needed.
