The paper surveys the Guillemin-Uribe trace formula for magnetic Laplacians.
problem Analyzing dynamics of magnetic geodesic flows through eigenvalues.
method Semiclassical version of the Selberg trace formula for magnetic Laplacians.
result Concrete examples computed for specific magnetic fields.
We give a complete proof of the fact that the trace of the curvature of the connection associated to a planar d-web (d>3) is the sum of the Blaschke curvatures of its sub 3-webs.
Research examines properties of special metrics on complex manifolds.
problem Properties of Hermitian metrics with proportional curvature traces.
method Study of basic properties and examples of Chern-Einstein metrics.
result Examples of Chern-Einstein metrics are provided.
Researchers compute trace formula for magnetic Laplacian on hyperbolic surfaces.
problem Analyzing the magnetic Laplacian on compact hyperbolic surfaces.
method Computed the trace formula for magnetic Laplacian energies above the Mane critical level.
result Asymptotic behavior of trace formula coefficients near the Mane critical level.
New heat trace coefficients reveal curvature effects in polygonal domains.
problem Understanding heat trace behavior in polygonal domains with curved corners.
method Local heat trace expansion through order t1/2, analyzing both Dirichlet and Neumann boundary conditions. result Sharp sign law for the Dirichlet angular factor of the first corner-curvature heat invariant.
A new method for optimizing deep neural networks using TKFAC.
problem Optimizing deep neural networks with second-order methods.
method Proposes Trace-restricted Kronecker-factored Approximate Curvature (TKFAC) for Fisher information matrix approximation.
result TKFAC improves performance on deep network architectures compared to state-of-the-art algorithms.
We prove a dynamical wave trace formula for asymptotically hyperbolic (n+1) dimensional manifolds with negative (but not necessarily constant) sectional curvatures which equates the renormalized wave trace to the lengths of closed geodesics. A corollary of this dynamical trace formula is a dynamical resonance-wave trac…
Based on ideas of L. Alías, D. Impera and M. Rigoli developed in "Hypersurfaces of constant higher order mean curvature in warped products", we develope a fairly general weak/Omori-Yau maximum principle for trace operators. We apply this version of maximum principle to generalize several higher order mean curvature est…
Ancient Ricci flows with nonnegative curvature operator have bounded entropy.
problem Conditions for bounded entropy in ancient Ricci flows.
method Used Perelman's entropy and Hamilton's trace Harnack inequality.
result Curvature operator nonnegativity is not necessary for bounded entropy.
Wave trace singularity formula for fibre bundles generalizes Poisson summation.
problem Wave trace singularity formula for fibre bundles.
method Proved a wave trace singularity formula for a family of generalised Laplacians defined by a Riemannian fibre bundle.
result Generalizes Poisson summation formulae for families.
The main objective of the paper is to prove a geometric version of sharp trace and product estimates on null hypersurfaces with finite curvature flux. These estimates play a crucial role to control the geometry of such null hypersurfaces. The paper is based on an invariant version of the classical Littlewood -Paley the…
The paper proves a Connes trace theorem for curved noncommutative tori.
problem Recovering scalar curvature in curved noncommutative tori.
method Proving a version of Connes' trace theorem for noncommutative tori of any dimension.
result Establishes a curved version of Connes' integration formula for scalar curvature.
Study on heat trace expansion for thermoelastic Dirichlet-to-Neumann map.
problem Asymptotic expansion of heat trace for thermoelastic Dirichlet-to-Neumann map.
method Provided a method to obtain all coefficients of the asymptotic expansion.
result Explicitly gave the first two coefficients involving volume and total mean curvature of the boundary.
Study spectral invariants for polygons and orbisurfaces.
problem Compute spectral invariants for polygons and orbisurfaces.
method Study the heat kernel and asymptotic expansion of the heat trace.
result Explicit formulas for heat invariants of polygons and orbisurfaces.
An important and natural question in the analysis of Ricci flow singularity formation in dimensions four and above is as follows: What are the weakest conditions that provide control of the norm of the Riemann curvature tensor? In this short note, we show that on a compact manifold, the trace-free Ricci tensor is contr…
Study of curvature tensor algebra with nonstandard multiplications.
problem Understanding curvature tensor algebra structures.
method Investigates orthogonally invariant commutative nonassociative multiplications on curvature tensors.
result Characterizes curvature tensor algebras in low dimensions and proves their simplicity in higher dimensions.
Study of spectral geometry on noncommutative tori using functional metrics.
problem Understanding the spectral properties of noncommutative tori.
method Introduction of functional metrics and analysis of their Laplace type operators and spectral invariants.
result Explicit computation of scalar curvature and total scalar curvature for certain functional metrics.
The paper connects curvature data to polynomial coefficients in gluing formulas.
problem Understanding polynomial coefficients in gluing formulas for zeta-determinants.
method Expressing coefficients of a polynomial in terms of scalar and principal curvatures of a 2D hypersurface.
result Coefficients of the polynomial are expressed in terms of curvature data.
A general approach to proving that the length spectrum of a compact Riemannian manifold is an invariant of the Laplace spectrum comes from considering the wave trace, a spectrally determined tempered distribution. The Poisson relation states that the singularities of the wave trace can only occur at lengths of closed g…
Early training phase affects deep neural network optimization and generalization.
problem The choice of learning rate influences generalization in deep learning models.
method Showed that SGD implicitly penalizes the trace of the Fisher Information Matrix (FIM) from the start of training, and explicitly penalizing the trace of FIM improves generalization.
result Catastrophic Fisher explosion (large trace of FIM early in training) is linked to poor generalization.
Equations link metrics with tensors, revealing curvature constraints.
problem Understanding curvature properties of geometric structures.
method Formal analogies to Einstein-Maxwell equations, studying Codazzi and conformal Killing equations.
result Constraints on scalar curvature of metrics in solutions.
The wave trace of certain convex domains can be smooth near some points in the length spectrum.
problem Understanding the relationship between the wave trace and the length spectrum of convex domains.
method Constructing silent periodic billiard orbits with the same length but different Maslov indices, using a microlocal parametrix for wave invariants.
result The wave trace can be smooth near some points in the length spectrum, showing potential limitations for inverse spectral problems.
A vector field on a Riemannian manifold is called conformal Killing if it generates one-parameter group of conformal transformations. The class of conformal Killing symmetric tensor fields of an arbitrary rank is a natural generalization of the class of conformal Killing vector fields, and appears in different geometri…
We examine questions of geometric realizability for algebraic structures which arise naturally in affine and Riemannian geometry. Suppose given an algebraic curvature operator R at a point P of a manifold M and suppose given a real analytic (resp. C-k for finite k at least 2) pseudo-Riemannian metric on M defined near …
In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form H and extend the result to n-dimensions. We also obtain an estimate for the gradient of the smaller principal curvature in 2 dimension…
In this note we consider some properties of GLn(R) with the Semi-Riemannian structure induced by the trace metric g. In particular we study geodesics and curvature tensors. Moreover we prove that GLn has a suitable foliation, whose leaves are isometric to (SLn(R),g), while its component of…
The paper examines rigidity of Einstein metrics using curvature functionals.
problem Characterizing rigidity of Einstein metrics.
method Critical points of quadratic curvature functionals and integral inequalities involving Weyl curvature, trace-less Ricci curvature, and Sobolev constant.
result Rigidity results for Einstein metrics on complete manifolds.
New gauge preserves Einstein metrics' interactions, proving rigidity on negatively curved manifolds.
problem Stability and deformation theory of Einstein metrics.
method Introduces Chen-Nagano gauge condition, linking Lichnerowicz Laplacian to shifted scalar operator.
result Chen-Nagano gauge collapses to classical transverse-traceless gauge under spectral pinching assumptions.
In this paper we characterize compact and complete hypersurfaces with some constant higher order mean curvature into warped product spaces. Our approach is based on the use of a new trace operator version of the Omori-Yau maximum principle which seems to be interesting in its own.
Let Γ be a nonelementary discrete subgroup of SU(n,1) or Sp(n,1). We show that if the trace field of Γ is contained in R, Γ preserves a totally geodesic submanifold of constant negative sectional curvature. Furthermore if Γ is irreducible, Γ is a Zariski dense irreducible discrete subgroup of SO(n,1…
The paper calculates spectral invariants of magnetic Steklov eigenvalues on Riemannian manifolds.
problem Investigating spectral invariants of magnetic Steklov eigenvalues on Riemannian manifolds.
method Established an effective procedure to calculate all coefficients of the heat trace asymptotic expansion.
result Explicitly provided expressions for the first four coefficients of the heat trace asymptotic expansion.
New equations for rigid body motion on infinite-dimensional spaces of operators.
problem Integrating rigid body dynamics on infinite-dimensional spaces of operators.
method Introducing pseudo-Riemannian metrics and adapting classical integrability theory.
result Existence of geodesics and integrals of motion for the rigid body equations.
Study the spectral geometry of surfaces with curved conic singularities.
problem Understanding the spectral properties of surfaces with conic singularities.
method Using the heat trace expansion, express spectral geometry terms through the geometry and curvature of the singularities.
result The first few terms in the heat trace expansion are expressed through the geometry and curvature of the singularities.
The paper proves rigidity of Einstein metrics as critical points of quadratic curvature functionals.
problem Characterizing Einstein metrics as critical points of quadratic functionals.
method Analyzing Einstein metrics on closed manifolds using quadratic curvature functionals and point-wise inequalities.
result Rigidity results for Einstein metrics involving Weyl curvature, trace-less Ricci curvature, and Yamabe invariant.
The study quantifies geometric differences between axonal branches using splines.
problem Neuromorphology's focus on macroscopic features neglects neuron internal geometry.
method Fitting splines to neuron traces, using Frenet-Serret formulas to compute curvature and torsion.
result Parameters of curvature and torsion are distributed differently between axonal branches.
In this note we investigate the behaviour at finite-time singularities of the mean curvature flow of compact Riemannian submanifolds M^m_t\hookrightarrow (N^{m+n}, h). We show that they are characterized by the blow-up of a trace A = H \cdot II of the square of the second fundamental form.
Compactness proven for specific dimensions of manifolds with boundary conditions.
problem Compactness of scalar-flat conformal metrics on low-dimensional manifolds with boundary.
method Quantitative analysis of a linear equation associated with the boundary Yamabe problem, positive mass theorem, and properties of the trace-free second fundamental form.
result Proven C2-compactness for 4- and 5-manifolds, and a sum of second-order derivatives of the trace-free second fundamental form is non-negative at possible blow-up points for 5-manifolds. New scalar curvature defined from Ollivier-Ricci curvature for graphs.
problem Defining scalar curvature for graphs and point clouds.
method Defining a new scalar version of Ollivier-Ricci curvature and proving its convergence.
result The new scalar curvature converges to scalar curvature for sampled manifolds.
We prove a linear trace Li-Yau-Hamilton inequality for the Kaehler-Ricci flow. We then use this sharp differential inequality to study the Liouville properties of the plurisubharmonic functions on complete Kaehler manifolds with nonnegative bisectional curvature.
Deform moment map on symplectic connections using star product algebras.
problem Understanding symplectic connections and their deformations.
method Study vector bundle of Fedosov star product algebras, formal connection, curvature, and star product trace.
result Showed star product trace as a formal symplectic form and moment map.
Let (Mn,g)(n≥3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and Rm˚ the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that Rm˚ goes to ze…
The paper proves new curvature estimates in quaternionic contact geometry.
problem Estimating the curvature of compact qc manifolds.
method Establishing quaternionic contact versions of the Almost Schur Lemma.
result Curvature is a constant in terms of specific norms and components.
The study proves Liouville theorems on curved manifolds with convex boundaries.
problem Proving Liouville theorems on manifolds with nonnegative curvature and strictly convex boundary.
method Analyzing smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary.
result Derives Liouville theorems and verifies a conjecture about eigenvalues and inequalities.
The Riemannian Bures metric on the space of (normalized) complex positive matrices is used for parameter estimation of mixed quantum states based on repeated measurements just as the Fisher information in classical statistics. It appears also in the concept of purifications of mixed states in quantum physics. Here we d…
The paper shows how to uniquely determine a connection up to gauge.
problem Determining a connection up to gauge from its holonomies.
method Using hyperbolic dynamical systems and transport operators.
result The primitive trace map is locally injective near generic points.
We obtain an estimate from below for the remainder in Weyl's law on negatively curved surfaces. In the constant curvature case, such a bound was proved independently by Hejhal and Randol in 1976 using the Selberg zeta function techniques. Our approach works in arbitrary negative curvature, and is based on wave trace as…
Study geometric properties of loss functions to understand neural network performance.
problem Understanding the geometric properties of high-dimensional loss functions to improve neural network performance.
method Combine concepts from high-dimensional probability and differential geometry to study curvature properties in lower-dimensional loss representations.
result Mean curvature in the original loss space determines if saddle points appear as minima, maxima, or flat regions.
In a previous paper, the authors showed that metrics which are asymptotic to Anti-de Sitter-Schwarzschild metrics with positive mass admit a unique foliation by stable spheres with constant mean curvature. In this paper we extend that result to all asymptotically hyperbolic metrics for which the trace of the mass term …