Local index theorem for manifolds with Lie structure at infinity using rescaling and renormalized supertrace.
problem Proving an Atiyah-Singer type index theorem for manifolds with Lie structure at infinity.
method Rescaling technique similar to Getzler's, integrating Lie groupoid, functional calculus, heat kernel expansion, Lichnerowicz theorem.
result Proof of a local index theorem for Dirac operators on Lie manifolds.
The Gauss-Bonnet Theorem is studied for edge metrics as a renormalized index theorem. These metrics include the Poincaré-Einstein metrics of the AdS/CFT correspondence. Renormalization is used to make sense of the curvature integral and the dimensions of the L2-cohomology spaces as well as to carry out the heat equa…
Derives an index formula for families of end-periodic Dirac operators.
problem Calculating the index of families of end-periodic Dirac operators.
method Using the renormalized Chern character and Fourier-Laplace transform of the Bismut superconnection.
result Establishes an index formula involving a new end-periodic eta form.
Study of harmonic oscillators on singular geometries using supersymmetry.
problem Global invariants on singular spaces.
method Supersymmetric localization, equivariant localization, Morse theory, geometric quantization.
result Renormalized Lefschetz numbers and Morse polynomials for singular spaces.
The paper addresses the expansion of Berezinian and super exterior powers, revealing new insights into supertraces.
problem The breakdown of classical volume elements in supergeometric settings and the need for generalized forms.
method Introduces and analyzes r∣s-forms, demonstrates the expansion of Ber(E+zA), and identifies supertraces. result Intermediate expansions in annular regions encode supertraces of representations on vector spaces.
We use invariance theory to compute the divergence term am+2,md+δ in the super trace for the twisted de Rham complex for a closed Riemannian manifold.
Develops analytic methods for Lefschetz and Morse theories on stratified pseudomanifolds.
problem Analytic framework for Lefschetz and Morse theories on stratified pseudomanifolds.
method Heat kernel and Witten deformation based techniques for global and local Lefschetz numbers and Morse polynomials.
result Formulas for Lefschetz numbers and Morse polynomials as supertraces over cohomology groups of Hilbert complexes.
A notion of n-Lie algebra introduced by V.T. Filippov can be viewed as a generalization of a concept of binary Lie algebra to the algebras with n-ary multiplication law. A notion of Lie algebra can be extended to Z_2-graded structures giving a notion of Lie superalgebra. Analogously a notion of n-Lie algebra can be ext…
Novel threefold partitioning of L2-spinor space on cone links.
problem Understanding equivariant Riemann-Roch defects in complex spaces with conic singularities.
method Investigates supertraces over local cohomology groups and spectral asymmetry.
result Novel complex equivariant ξT and ηT invariants defined. Defines a new geometric quantity for hyperbolic manifolds, showing it's well-defined and invariant.
problem Defining a mass for asymptotically hyperbolic manifolds under weaker conditions.
method Volume-renormalized mass defined as a linear combination of ADM mass and renormalized volume.
result Volume-renormalized mass is well-defined and diffeomorphism invariant under weaker conditions.
After analyzing renormalization schemes on a Poincaré-Einstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is well-known, and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms and their behavior und…
Paper constructs super integrable systems on color Lie algebra.
problem Super integrable systems on color Lie algebra.
method Using non-isospectral problems with matrices from color Lie algebra sp1(6), constructing (1+1)- and (2+1)-dimensional systems. result Super integrable systems and their Hamiltonian structures constructed on color Lie algebra sp1(6). Minimal surfaces in hyperbolic space have a renormalized area criterion.
problem Minimal surfaces in hyperbolic space
method Renormalized area criterion
result Y must be a totally geodesic disk
Study shows continuity of renormalized volume in geometric limits.
problem Continuity of renormalized volume in geometric limits of hyperbolic manifolds.
method Analyzing sequences of convex-cocompact hyperbolic manifolds and their renormalized volumes.
result Renormalized volume is continuous under geometric limits.
Study uses renormalized area to determine metric expansion from minimal surfaces.
problem Recovering the expansion of asymptotically hyperbolic metrics from minimal surfaces.
method Uses renormalized area functional on minimal submanifolds to recover metric expansion.
result Proves rigidity for log-analytic metrics and determines obstruction tensor.
Defines renormalized volume for bounded regions in asymptotically hyperbolic Einstein spaces.
problem Calculating the volume of bounded regions in complex geometries.
method Defines renormalized volume, proves Gauss-Bonnet theorem, computes derivative under variations.
result Derives a Gauss-Bonnet theorem for the renormalized volume.
Study calculates the renormalized area of catenoids in hyperbolic spaces.
problem Calculating the renormalized area of catenoids in hyperbolic spaces.
method Variational characterization and Chern--Gauss--Bonnet formulas for locally conformally flat manifolds.
result Renormalized area of catenoids varies continuously from negative infinity to twice the area of totally geodesic hypersurfaces.
Ricci flow connects Poisson-Lie T-duality to renormalization.
problem Compatibility of Poisson-Lie T-duality with renormalization.
method Generalized Ricci tensor in Courant algebroids.
result Poisson-Lie T-duality is compatible with 1-loop renormalization.
New Bayesian method uses momentum-space renormalization for image modeling.
problem Bayesian image modeling challenges in Gaussian graphical models.
method Combines marginal likelihood maximization with momentum-space renormalization.
result Scheme for computing hyperparameters and mean square errors.
Defines and proves properties of weighted renormalized volume coefficients.
problem None explicitly stated; focuses on mathematical definitions and proofs.
method Defines weighted renormalized volume coefficients and proves their variational nature and polynomial representation.
result Weighted renormalized volume coefficients are variational and can be expressed as polynomials of specific tensors.
We study the critical points of the renormalized volume for acylindrical geometrically finite hyperbolic 3-manifolds that include rank-1 cusps, and show that the renormalized volume is locally convex around these critical points. We give a modified definition of the renormalized volume that is additive under gluing, an…
For a strictly pseudoconvex domain in a complex manifold we define a renormalized volume with respect to the approximately Einstein complete Kähler metric of Fefferman. We compute the conformal anomaly in complex dimension two and apply the result to derive a renormalized Chern--Gauss--Bonnet formula. Relations between…
We study the renormalized volume of a conformally compact Einstein manifold. In even dimensions, we derive the analogue of the Chern-Gauss-Bonnet formula incorporating the renormalized volume. When the dimension is odd, we relate the renormalized volume to the conformal primitive of the Q-curvature. We show how all t…
Researchers define and analyze the renormalized volume of 4D Ricci-flat ALE spaces.
problem Understanding the properties of 4D Ricci-flat ALE spaces.
method Introducing a new definition of renormalized volume and proving its properties.
result The renormalized volume is always less than or equal to zero, with equality for specific spaces.
We define and study the renormalized volume for geometrically finite hyperbolic 3-manifolds, including with rank-1 cusps. We prove a variation formula, and show that for certain families of convex co-compact hyperbolic metrics $g_\eps$ degenerating to a geometrically finite hyperbolic metric g0 with rank-1 cus…
We interpret the physical B-field renormalization group flow in the language of Courant algebroids, clarifying the sense in which this flow is the natural "Ricci flow" for generalized geometry. Next we show that the B-field renormalization group flow preserves T-duality in a natural sense. As corollaries we obtain …
New adapted renormalized volume for hyperbolic 3-manifolds with compressible boundary.
problem Analyzing convex co-compact hyperbolic 3-manifolds with compressible boundaries.
method Defining and analyzing a new version of the renormalized volume.
result The adapted renormalized volume is bounded and has properties analogous to the classical renormalized volume.
Study shows continuity of renormalized volume for geometrically convergent hyperbolic structures.
problem Continuity of renormalized volume under geometric limits.
method Extended renormalized volume concept for geometrically finite hyperbolic 3-manifolds and showed continuity for geometrically convergent sequences.
result Renormalized volume attains its minimum at the geodesic class.
Paper derives a formula for renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
problem Calculating the renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
method Derives a Gauss-Bonnet formula for renormalized area in terms of scalar invariants and characterizes minimal hypersurfaces in terms of conformal geometry.
result Derives a formula expressing renormalized area in terms of integrals of scalar Riemannian invariants.
Proves energy expression on Poincaré-Einstein spaces.
problem Computing renormalized Yang-Mills energy on Poincaré-Einstein manifolds.
method Generalizes Chang-Qing-Yang method for renormalized volumes and uses scattering theory for Schrödinger operators.
result Agrees with anomaly boundary integrand in seven dimensions.
New formula for volume in 4D hyperbolic manifolds.
problem Volume calculation for specific geometric manifolds.
method Derive new renormalized volume formula.
result Generalizes existing formulas for Poincare-Einstein manifolds.
Upper bounds on renormalized volume for Schottky groups derived from extremal lengths.
problem Comparing renormalized volumes of Schottky and Fuchsian manifolds with the same boundary.
method Bounding renormalized volume in terms of genus and extremal lengths of curves on the boundary Riemann surface.
result Upper bounds on renormalized volume for Schottky groups established.
New properties are derived of renormalized volume functionals, which arise as coefficients in the asymptotic expansion of the volume of an asymptotically hyperbolic Einstein (AHE) manifold. A formula is given for the renormalized volume of an even-dimensional AHE manifold in terms of an arbitrary totally geodesic compa…
Renormalization in neural networks linked to quantum field theory.
problem Implementing renormalization in neural networks.
method Mapping neural networks to quantum field theory, applying renormalization techniques.
result Changing weight standard deviation corresponds to a renormalization flow.
Formula derived for 4D manifolds with boundary involving renormalized volume and boundary integral.
problem Calculating the Euler characteristic of 4D manifolds with boundary.
method Derives a Chern-Gauss-Bonnet formula for a specific type of metric.
result Sum of renormalized volume and boundary integral is a conformal invariant when boundary is umbilic.
We study the evolution of the renormalized volume functional for asymptotically Poincare-Einstein metrics (M,g) which are evolving by normalized Ricci flow. In particular, we prove that the time derivative of the renormalized volume along the flow is the negative integral of scal(g(t)) + n(n-1) over the manifold. This …
Paper renormalizes volume of singular Yamabe metrics.
problem Renormalizing volume of singular Yamabe metrics.
method Volume renormalization for singular Yamabe metrics.
result Existence of a conformally invariant energy.
New compactification of Teichmüller space via renormalized volume.
problem Compactify Teichmüller space with new distance.
method Horocompactification with renormalized volume.
result Translation length of pseudo-Anosov mapping classes equals hyperbolic volume of mapping tori.
Renormalized pruning improves neural network accuracy.
problem Over-parameterized neural networks waste many parameters.
method Propose renormalizing sparse neural networks.
result Renormalized pruning converges to zero error.
Formula for tube volume on special geometric manifolds.
problem Calculating volumes of geometric structures.
method Using Einstein constant and polarization volume.
result Formula for renormalized tube volume.
Formula for renormalized area of hypersurfaces in hyperbolic spaces.
problem Calculating the renormalized area of asymptotically minimal hypersurfaces in hyperbolic spaces.
method Combining Chen's conformal invariant quantity and Chern-Gauss-Bonnet formulas.
result Extension of renormalized area formulas to higher dimensions and non-minimal cases.
We compute renormalized curvature integrals on Poincaré-Einstein manifolds.
problem Computing renormalized curvature integrals on Poincaré-Einstein manifolds.
method General procedure that connects Gauss-Bonnet-type formulas and identifies scalar conformal invariants.
result Explicit conformally invariant Gauss--Bonnet-type formulas for compact Einstein manifolds.
New method calculates volume-renormalized mass from Hamiltonian perspective.
problem Calculating volume-renormalized mass for asymptotically hyperbolic manifolds.
method Using Michel's mass invariants and a reduced Hamiltonian perspective, the volume-renormalized mass is deduced.
result The reduced Hamiltonian recovers the volume-renormalized mass and its variations.
The paper studies circle packings using renormalization and subdivision rules.
problem Characterizing and proving properties of circle packings with specific subdivision rules.
method Iterations of skinning maps on Teichmüller spaces, renormalization theory, subdivision rules.
result Uniformly contracting renormalization operator and geometric inflexibility of circle packings.
Minimal surfaces in hyperbolic space have a sharp area bound.
problem Bounding the renormalized area of minimal surfaces.
method Proving an inequality using conformal length of ideal boundary.
result Sharp isoperimetric property of renormalized area.
The renormalized volume is reinterpreted using isoperimetric profiles.
problem Understanding the renormalized volume of convex co-compact hyperbolic 3-manifolds.
method Using isoperimetric profiles and Minkowski inequalities.
result A sharp Minkowski inequality for horospherically convex sets in H3. The paper generalizes CR invariants using renormalized characteristic forms.
problem Defining new CR invariants via renormalized characteristic forms.
method Introducing new curvatures for each renormalized characteristic form.
result The new curvatures' integrals match CR invariants constructed by Marugame.
Ricci flow emerges from renormalizing nonlinear Sigma models.
problem Understanding the renormalization group flow in nonlinear Sigma models.
method Using Euclidean algebraic quantum field theory and Wick ordering.
result The first-order renormalization group flow of nonlinear Sigma models equals the Ricci flow.