Virtual

I will present a theorem on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.  The proof employs a double null gauge, is expressed entirely in physical space, and utilises the analysis of Dafermos--Holzegel--Rodnianski on the linear stability of the Schwarzschild family.  This is joint work with M. Dafermos, G. Holzegel and I. Rodnianski.

In this talk I will explain how a subfactor (ie an inclusion of type II_1 factors) give rise to a diagrammatic algebra called the Temperley-Lieb-Jones algebra. We will observe the connection between the index of the subfactor, and the TLJ algebra. In the TLJ algebra setting, we will observe that indices below four are discrete, while any number above four can be an index.

I will discuss a precise connection between renormalization group (RG) and quantum chaos. Every RG flow between two conformal fixed points can be described in terms of the dynamics of Nambu-Goldstone bosons of broken symmetries. The theory of Nambu-Goldstone bosons can be viewed as a theory in anti-de Sitter space with the flat space limit. This enables an equivalent formulation of these 4d RG flows in terms of spectral deformations of a generalized free CFT in 3d. This approach provides a precise relation between C-functions associated with 4d RG flows and certain out-of-time-order correlators that diagnose chaos in 3d. As an application, I will show that the 3d chaos bound imposes constraints on the low energy effective action associated with unitary RG flows in 4d with a broken continuous global symmetry in the UV. These bounds, among other things, imply that the proof of the 4d a-theorem remains valid even when additional global symmetries are broken.

Many systems exhibit complex temporal dynamics due to the presence of different processes taking place simultaneously. Temporal networks provide a framework to describe the time-resolve interactions between components of a system. An important task when investigating such systems is to extract a simplified view of the temporal network, which can be done via dynamic community detection or clustering. Several works have generalized existing community detection methods for static networks to temporal networks, but they usually rely on temporal aggregation over time windows, the assumption of an underlying stationary process, or sequences of different stationary epochs. Here, we derive a method based on a dynamical process evolving on the temporal network and restricted by its activation pattern that allows to consider the full temporal information of the system. Our method allows dynamics that do not necessarily reach a steady state, or follow a sequence of stationary states. Our framework encompasses several well-known heuristics as special cases. We show that our method provides a natural way to disentangle the different natural dynamical scales present in a system. We demonstrate our method abilities on synthetic and real-world examples.

arXiv link: https://arxiv.org/abs/2101.06131

Online social networks such asTwitter, Facebook, Instagram and TikTokserve as mediafor the spread of information between their users.We areinterested in developing models forthis information diffusion to gain a greater understanding of its drivers. Some models forthe spread ofonlinebehaviour and informationassume that the information behaves similarly to a virus, where infection is equally likely after each exposure, these dynamics are known as a simple contagion. In a simple contagion, the exposures are independent of each other.However,online adoption of some behaviour and content has been empirically observed to be more likely after multiple exposures from their network neighbours, the exposures are not independent of each other, we refer to this as a complex contagion.Analytically tractable descriptions of complex contagions havebeendeveloped for continuous-time dynamics. These extend mean-field and pair approximation methods to account for clustering in the network topologies; however, no such analogous treatments for discrete-time cascade processes exist using branching processes. We describe a novel definition of complex contagion adoption dynamics and show how to construct multi-type branching processeswhichaccount for clustering on networks. We achieve this by tracking the evolution of a cascade via different classes of clique motifs whichaccount for the different numbers of active, inactive and removed nodes. This description allows for extensive MonteCarlo simulations (which are faster than network-based simulations), accurate analytical calculation of cascade sizes, determination of critical behaviour and other quantities of interest

We use machine learning to approximate Calabi-Yau and SU(3)-structure metrics, including for the first time complex structure moduli dependence. Our new methods furthermore improve existing numerical approximations in terms of accuracy and speed. Knowing these metrics has numerous applications, ranging from computations of crucial aspects of the effective field theory of string compactifications such as the canonical normalizations for Yukawa couplings, and the massive string spectrum. In the case of SU(3) structure, our machine learning approach allows us to engineer metrics with certain torsion properties. Our methods are demonstrated for Calabi-Yau and SU(3)-structure manifolds based on a one-parameter family of quintic hypersurfaces in ℙ4.

I briefly give an overview on the key ML frameworks involved in this analysis (neural networks, auto-differentiation). This talk is mainly based on 2012.04656.

Plebanski generating functions give a compact encoding of the geometry of self-dual Ricci-flat space-times or hyper-Kahler spaces.  They have applications as generating functions for BPS/DT/Gromov-Witten invariants.  We first show that Plebanski's first fundamental form also provides a generating function for the gravitational MHV amplitude.  We then obtain these Plebanski generating functions from the corresponding twistor spaces as the value of the action of new sigma models for holomorphic curves in twistor space.   
In four-dimensions, perturbations of the hyperk¨ahler structure corresponding to positive helicity gravitons. The sigma model’s perturbation theory gives rise to a sum of tree diagrams for the gravity MHV amplitude observed previously in the literature, and their summation via a matrix tree theorem gives a first-principles derivation of Hodges’ determinant formula directly from general relativity. We generalise the twistor sigma model to higher-degree (defined in the first instance with a cosmological constant), giving a new generating principle for the full tree-level graviton S-matrix in general with or without  cosmological constant.  This is joint work with Tim Adamo and Atul Sharma in https://arxiv.org/abs/2103.16984.  

The 4D 4-point scattering amplitude of massless scalars via a massive exchange can be expressed in a basis of conformal primary particle wavefunctions. In this talk I will show that the resulting celestial amplitude admits a decomposition as a sum over 2D conformal blocks. This decomposition is obtained by contour deformation upon expanding the celestial amplitude in a basis of conformal partial waves. The conformal blocks include intermediate exchanges of spinning light-ray states, as well as scalar states with positive integer conformal weights. The conformal block prefactors are found as expected to be quadratic in the celestial OPE coefficients. Finally, I will comment on implications of this result for celestial holography and discuss some open questions.

A topology optimization problem for Stokes flow finds the optimal material distribution of a Stokes fluid that minimizes the fluid’s power dissipation under a volume constraint. In 2003, T. Borrvall and J. Petersson [1] formulated a nonconvex optimization problem for this objective. They proved the existence of minimizers in the infinite-dimensional setting and showed that a suitably chosen finite element method will converge in a weak(-*) sense to an unspecified solution. In this talk, we will extend and refine their numerical analysis. We will show that there exist finite element functions, satisfying the necessary first-order conditions of optimality, that converge strongly to each isolated local minimizer of the problem.

[1] T. Borrvall, J. Petersson, Topology optimization of fluids in Stokes flow, International Journal for Numerical Methods in Fluids 41 (1) (2003) 77–107. doi:10.1002/fld.426.

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We study the model of randomly perturbed dense graphs, which is the union of any $n$-vertex graph $G_\alpha$ with minimum degree at least $\alpha n$ and the binomial random graph $G(n,p)$. In this talk, we shall examine the following central question in this area: to determine when $G_\alpha \cup G(n,p)$ contains $H$-factors, i.e. spanning subgraphs consisting of vertex disjoint copies of the graph $H$. We offer several new sharp and stability results.
This is joint work with Julia Böttcher, Olaf Parczyk, and Jozef Skokan.