In this article we prove that iterated renormalisations of circle diffeomorphisms with breaks, , with given size of breaks, converge to an invariant family of piecewise Moebius maps, of dimension . We prove that this invariant family identifies with a \textit{relative character variety} $χ(…
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48 results for “Renormalisation Group”
Defines renormalised energies for singular harmonic maps into compact manifolds.
problem Analyzing harmonic maps with singularities in planar domains.
method Introduces renormalised energies and synharmony to study singularities and minimising configurations.
result Renormalised energies are coercive and Lipschitz-continuous, and associated with minimising singular harmonic maps.
The thesis explores IMP, a process that identifies winning tickets in DNNs, and its universality.
problem Understanding how winning subnetworks (tickets) perform across different problems.
method Iterative Magnitude Pruning (IMP) and comparison with Renormalisation Group (RG) theory.
result IMP identifies winning subnetworks that can perform similarly across various problems.
New formula connects Loewner energy to moving frames' renormalised energy.
problem Calculating Loewner energy of Jordan curves.
method Using renormalised energy of moving frames.
result Loewner energy as Kähler potential for Weil-Petersson space.
New approach to abstract neural network representations using renormalization group.
problem Developing truly abstract representations in neural networks.
method Renormalization group approach to expand representations to encompass broader data sets.
result Representations in neural networks become more abstract as data breadth increases and depth increases.
The paper studies harmonic maps to the circle with complex singular sets.
problem Finding harmonic maps with prescribed singular sets in higher-dimensional spaces.
method Considered variational relaxations of the problem, showing energy convergence to a renormalised volume plus lower-order interaction energy.
result The energy of minimisers converges, after renormalisation, to the volume of the singular set plus a lower-order interaction energy.
HR in 8D encodes unique conformal gravity with negative curvature.
problem Holographic Renormalisation in 8D Einstein Gravity.
method Relating HR to Topological Regularisation and adding the Euler term.
result The unique conformal gravity theory reproduces the polynomial and cancels divergent terms.
New metrics found in hyperbolic manifolds as volume-minimizers.
problem Finding critical points of volume-renormalized mass.
method Critical points of the volume-renormalized mass over asymptotically hyperbolic manifolds.
result V-static metrics are critical points of volume-renormalized mass.
Hairy Cantor setsmath.GN
We introduce a topological object, called hairy Cantor set, which in many ways enjoys the universal features of objects like Jordan curve, Cantor set, Cantor bouquet, hairy Jordan curve, etc. We give an axiomatic characterisation of hairy Cantor sets, and prove that any two such objects in the plane are ambiently homeo…
We provide local expressions for Chern-Weil type forms built from superconnections associated with families of Dirac operators previously investigated in work by S. Scott and later work by S. Scott and the second author. When the underlying fibration of manifolds is trivial, the even degree forms can be interpreted as …
Symmetry in neural networks affects generalization, as shown by CLT and RG transformations.
problem Improving generalization in neural networks by incorporating physical symmetries.
method Evaluation of symmetry constraints and expressivity in MLPs and GNNs using the CLT as a test case.
result Overly complex or overconstrained models generalize poorly, revealing a competition between symmetry constraints and expressivity.
The susceptibility of deep learning to adversarial attack can be understood in the framework of the Renormalisation Group (RG) and the vulnerability of a specific network may be diagnosed provided the weights in each layer are known. An adversary with access to the inputs and outputs could train a second network to clo…
The Renormalisation Group (RG) provides a framework in which it is possible to assess whether a deep-learning network is sensitive to small changes in the input data and hence prone to error, or susceptible to adversarial attack. Distinct classification outputs are associated with different RG fixed points and sensitiv…
The perpendicular Neumann problem for mean curvature flow with a timelike cone boundary conditionmath.DG
This paper demonstrates existence for all time of mean curvature flow in Minkowski space with a perpendicular Neumann boundary condition, where the boundary manifold is a convex cone and the flowing manifold is initially spacelike. Using a blowdown argument, we show that under renormalisation this flow converges toward…
The paper proves weighted monotonicity theorems in various spaces and applies them to minimal surfaces.
problem Proving weighted monotonicity theorems in different spaces.
method Proving weighted monotonicity theorems for functions proportional to the metric tensor in Riemannian manifolds.
result Weighted monotonicity theorems in hyperbolic space imply unweighted theorems, leading to bounds on minimal surface areas.
Researchers investigate extremal eigenvalues of GJMS operators in fixed conformal classes.
problem Investigating extremal eigenvalues of GJMS operators in fixed conformal classes.
method Developed a general framework for existence theory of extremals, defined and investigated generalised eigenvalues, and established semi-continuity results and Euler-Lagrange equations.
result Proved several new (non)-existence results for extremals of renormalised eigenvalues over the conformal class.
Generalizes Landau-Ginzburg mirrors for Frobenius manifolds in Dynkin type A.
problem Classifying Frobenius manifold structures in Dynkin type A.
method Generalizing the method from previous works, developing a pole-collision framework.
result Structural result at the level of prepotential for arbitrary rank and dimension.
New framework for gravitational perturbations of Kerr spacetimes, focusing on stability.
problem Stability of Kerr spacetimes to gravitational perturbations.
method New geometric framework with tailored null frames and gauge, reformulating Einstein equations.
result Derivation of linearised vacuum Einstein equations in the new framework.
On the scattering operators for ACHE metrics of Bergman type on strictly pseudoconvex domainsmath.DG
The scattering operators associated to an ACHE metric of Bergman type on a strictly pseudovonvex domain are a one-parameter family of CR-conformally invariant pseudodifferntial operators of Heisenberg class with respect to the induced CR structure on the boundary. In this paper, we mainly show that if the boundary Webs…
Constructs explicit solutions to Spin(7)-structures gradient flow.
problem Finding explicit solutions to Spin(7)-structures gradient flow.
method Expressed Spin(7)-torsion tensor and gradient flow in terms of torsion forms; used these formulae to find solutions.
result Found explicit solutions including a shrinking soliton on SU(3) and another on a -bundle over .
The main goal of the present paper is to construct new invariants of knots with additional structure by adding new gradings to the Khovanov complex. The ideas given below work in the case of virtual knots, closed braids and some other cases of knots with additional structure. The source of our additional grading may be…
Circular variables arise in a multitude of data-modelling contexts ranging from robotics to the social sciences, but they have been largely overlooked by the machine learning community. This paper partially redresses this imbalance by extending some standard probabilistic modelling tools to the circular domain. First w…
We analyse the behaviour of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalisation of the strike variable with the property that the implied volatility converges to a non-constant limiting shape, which is a function of …
A new framework predicts hidden Markov model regimes online.
problem Efficiently identify hidden Markov model regimes in streaming data.
method Develops a predictive-first optimisation framework for streaming HMMs, approximating the full posterior predictive distribution.
result The method provides competitive prequential performance compared to Online EM and Sequential Monte Carlo.
General Relativity in 4 dimensions can be equivalently described as a dynamical theory of SO(3)-connections rather than metrics. We introduce the notion of asymptotically hyperbolic connections, and work out an analog of the Fefferman-Graham expansion in the language of connections. As in the metric setup, one can solv…
The paper shows instability in Minkowski spacetime for a quantum system.
problem Linear instability of the semiclassical Einstein-Klein-Gordon system in Minkowski spacetime.
method Formulated a forcing problem for metric and state perturbations, used tensor decomposition and quantum Møller operator.
result Metric perturbations grow exponentially, bounded by a universal scale H, indicating quantum backreaction.
Conformal geodesics are distinguished curves on a conformal manifold, loosely analogous to geodesics of Riemannian geometry. One definition of them is as solutions to a third order differential equation determined by the conformal structure. There is an alternative description via the tractor calculus. In this article …
New link groups are derived from torus necklaces, connecting braid groups to reflection groups.
problem Understanding the relationship between braid groups and reflection groups.
method Constructing torus necklaces and linking them to braid groups of -reflection groups.
result Link groups of torus necklaces are precisely braid groups of -reflection groups, with meridians as braid reflections.
The study proves super-rigidity of Gromov's random monster group for various types of groups.
problem Super-rigidity of Gromov's random monster group in various group types.
method Proof of morphisms having finite image and introduction of hereditary super-rigidity.
result Gromov's random monster group has super-rigidity and hereditary super-rigidity with respect to certain groups.
We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi--direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi--direct product of free groups. From these results we obtain a normal form of words…
Virtual twin groups map to symmetric groups, revealing automorphism structure.
problem Understanding homomorphisms between virtual twin groups and symmetric groups.
method Using irreducible right-angled Coxeter groups and right-angled Artin groups.
result A complete description of homomorphisms between virtual twin groups and symmetric groups, including the structure of the automorphism group of .
Characterizes group connections on group bundles.
problem Understanding connections on group bundles.
method Characterizes connections as affine spaces and uses the Ambrose-Singer theorem.
result Group connections form an affine space over cocycles.
Totally Symmetric Setsmath.GR
Study on totally symmetric sets with group applications.
problem Understanding totally symmetric sets and their group applications.
method Survey of existing theory and applications to various groups.
result Exploration of totally symmetric sets in multiple group contexts.
Affine cactus groups are CAT(0) and hyperbolic.
problem Characterizing geometric properties of affine cactus groups.
method Analyzing CAT(0) and hyperbolic properties through group theory.
result Affine cactus groups of degree three are hyperbolic.
The study restricts groups in graph of groups structures.
problem Realizing groups as fundamental groups of graph of groups with restricted vertex groups.
method Analyzes restrictions on groups that can be realized and applies to manifold construction.
result Places constraints on groups that can be realized in graph of groups structures.
New Garside structures found for torus knot groups and related braid groups.
problem Finding Garside structures for torus knot groups and related braid groups.
method Introducing a new Garside monoid for -torus knot groups and other braid groups.
result New Garside structures for -torus knot groups and related braid groups are constructed.
Logarithmic separation profile in hyperbolic groups shows hierarchical structure.
problem Understanding hierarchical structure in hyperbolic groups with logarithmic separation.
method Proving groups with logarithmic separation split over cyclic groups and providing counterexamples.
result Not all groups with hierarchical structure have logarithmic separation profile.
New Garside structures derived from groups, leading to new group properties.
problem Creating Garside structures from groups and Artin groups.
method Method for turning direct product of a group G by Z into a Garside group.
result Proved new cases of K(π,1)-conjecture for some hyperbolic type Artin groups.
We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pur…
The group of 2-by-2 matrices with integer entries and determinant can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional torus. Thus this group is the beginning of three natural sequences of groups, name…
The study restricts normal subgroups of Kähler groups, proving specific cases and general restrictions.
problem Characterizing normal subgroups of Kähler groups.
method Analyzing embeddings and conjugation actions of surface groups and one-ended hyperbolic groups.
result Restrictions on normal subgroups of Kähler groups, including virtual direct products and surface group properties.
Toric reflection groupsmath.GR
New reflection groups derived from torus knots with finite meridians.
problem Understanding reflection groups derived from torus knot groups with finite meridians.
method Using the theory of J-groups and Coxeter groups, study quotients of torus knot groups.
result Classification of toric reflection groups and their properties.
Graphically discrete groups have strong rigidity properties.
problem Understanding the rigidity of group actions on graphs.
method Introducing graphical discreteness and proving rigidity properties.
result Free products of graphically discrete groups are action rigid.
Paper proves vanishing homology groups for certain hyperbolic groups.
problem Understanding homology groups of specific hyperbolic groups.
method Using twisted Wirtinger presentations to prove homology group vanishing.
result Second homology groups vanish for certain Gromov hyperbolic groups.
Study fundamental groups of geometric transformation groups using loop spaces.
problem Understanding fundamental groups of geometric transformation groups.
method Use differential forms on loop spaces to prove infinite fundamental groups.
result Proves infinite fundamental groups for specific geometric transformation groups.
Simple construction of Lie 2-groups from loop group extensions.
problem Constructing Lie 2-groups from loop group extensions.
method Using conjugation action of loop group on its central extension.
result Simple construction of string 2-group as a strict Fréchet Lie 2-group.
Study knot invariants using automorphism groups of free nilpotent groups.
problem Developing knot invariants using automorphism groups.
method Nilpotently -localization of knot groups and automorphism groups of free nilpotent groups.
result Maps from outer automorphism groups yield knot invariants.
We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dens…