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.
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