Schwartz functions smoothly extend to real projective spaces.
problem Identifying functions that extend smoothly to compactifications.
method Showed a similar result for real projective spaces.
result Schwartz functions extend smoothly to real projective spaces.
Defines Schwartz and tempered functions on o-minimal manifolds.
problem Defining Schwartz and tempered functions on non-polynomially bounded o-minimal manifolds.
method Defining Schwartz and tempered functions on manifolds definable in polynomially bounded o-minimal structures, and showing classical properties hold.
result The theory of Schwartz and tempered functions can be constructed on manifolds definable in polynomially bounded o-minimal structures but not on non-polynomially bounded ones.
Compactifies semi-simple Lie groups to manifolds with corners.
problem Compactification of semi-simple Lie groups.
method hd-compactification to a manifold with corners, 1-1 correspondence with conjugacy classes of parabolic subgroups.
result Harish-Chandra's Schwartz space identified with conormal functions of rapid-logarithmic decay.
The infinite matrix `Schwartz' group G−∞ is a classifying group for odd K-theory and carries Chern classes in each odd dimension, generating the cohomology. These classes are closely related to the Fredholm determinant on G−∞. We show that while the higher (even, Schwartz) loop groups of $G^{-\infty…
Study on knot invariants from Teichmüller TQFT, proving equivalence and new calculations.
problem Calculating knot invariants from Teichmüller TQFT.
method Formal stationary phase analysis, explicit isomorphism, and equivalence proofs.
result Two formulations of Teichmüller TQFT give equivalent knot invariants.
Compactifications of SL(2) groups are studied for complex and real numbers.
problem Compactification of SL(2) groups for complex and real numbers.
method Analysis of Schwartz and Harish-Chandra Schwartz spaces as relatively standard spaces of conormal functions on compact manifolds with boundary.
result Closure under convolution and other module properties follow from the structure of generalized product spaces and functorial properties of conormal functions.
Defines a new algebra for singular foliations, extending Schwartz kernels.
problem Extending Schwartz kernel operators to singular foliations.
method Defines convolution algebra of transverse distributions, proves representation as operators on spaces of functions.
result Generalizes Schwartz kernel operators to singular foliations.
The paper explores global index formulas for one-dimensional holomorphic foliations.
problem Global index formulas for one-dimensional holomorphic foliations.
method Microlocal point of view and short proofs for existing index formulas.
result Generalizations of existing index formulas.
The Schwartz-Smith model parameters are estimated using Kalman Filter with additional constraints.
problem Estimating parameters of the Schwartz-Smith model for risk-neutral pricing of futures contracts.
method Kalman Filter method with additional constraints to address parameter identification problem.
result The obtained parameter estimates are the conditional Maximum Likelihood Estimators (MLEs) evaluated within the Kalman Filter.
A beta function for double layers is defined and analyzed.
problem Defining and analyzing a beta function for double layers.
method Holomorphic function definition and analytic continuation.
result Residues of the beta function are integrals of invariants.
Describes representations of modular group into SL(3,R)/SO(3).
problem Understanding representations of modular group into isometry group of symmetric space.
method Analyzes connected component of conjugacy classes of representations.
result Certain representations in the component are Anosov.
The paper constructs Ricci-flat Kähler manifolds with specific decay properties.
problem Finding Ricci-flat Kähler manifolds with controlled decay rates.
method Geometric existence proof and construction of ansatz.
result Existence of 39 distinct Ricci-flat Kähler 3-folds with specific asymptotic angles.
The paper develops a method for stochastic differential equations on manifolds using Schwartz morphisms and diffusion generators.
problem Representing stochastic differential equations on smooth manifolds.
method Using Schwartz morphisms and diffusion generators to construct SDEs on manifolds.
result An extended Ito formula for SDEs on manifolds.
Researchers developed a rigorous mathematical formulation of functional integration for fields on paracompact manifolds.
problem Overcoming the problematic aspect of measures in the space of fields.
method Using Schwartz-Sobolev spaces, open coverings with subordinate partition-of-unity test functions, and convolution operations.
result Validated the basic assumption of differential geometry that fields live on differentiable manifolds.
We construct an algebra of smooth functions over the tangent groupoid associated to any Lie groupoid. This algebra is a field of algebras over the closed interval [0, 1] which fiber at zero is the algebra of Schwartz functions over the Lie algebroid, whereas any fiber out of zero is the convolution algebra of the initi…
Two sweeps of the Brennan-Schwartz algorithm solve American options under negative rates.
problem Inability of the Brennan-Schwartz algorithm to solve American options under negative interest rates.
method Two sweeps of the Brennan-Schwartz algorithm in two directions.
result Recovery of the exact solution for American options under negative rates.
Characterizes the range of a tensor field transform in Schwartz space.
problem Range characterization of a tensor field transform in Schwartz space.
method Differential and integral equations for characterizing the range in different dimensions.
result Range characterization for the operator on Schwartz space of rank m tensor fields.
PDSim simulates and estimates commodity futures prices using polynomial diffusion models.
problem Simulating and estimating commodity futures prices using polynomial diffusion models.
method Developed an R package with a Shiny app for simulation and estimation of commodity futures prices using polynomial diffusion models.
result PDSim is the only package specifically designed for the simulation and estimation of the polynomial diffusion model.
The paper proves estimates for Hodge Laplacians on Lie groups.
problem Estimating Hodge Laplacians on semisimple Lie groups.
method Proves Schwartz estimates for Hodge Laplacian and Dirac operators.
result Generalizes results on symmetric spaces to Lie groups.
The paper explores Schwartz representations and their connection to Anosov representations.
problem Understanding the relationship between Schwartz representations and Anosov representations of the modular group.
method Constructing families of Anosov representations and analyzing their limits.
result Schwartz representations are limits of Anosov representations of the modular group.
Study characteristic classes of a specific type of determinantal varieties.
problem Understanding the geometric properties of a special class of determinantal varieties.
method Used Schubert calculus to derive explicit formulas for Chern-Schwartz-MacPherson and Chern-Mather classes.
result Explicit formulas for sectional Euler characteristics, characteristic cycles, and polar classes were obtained.
A new two-step LSMC method improves game option pricing accuracy.
problem Improving game option pricing accuracy using Monte Carlo methods.
method Proposed a two-step Longstaff Schwartz Monte Carlo approach with two regression models fitted at each time step.
result Our method produces more reliable results compared to the original LSMC.
Classifies complex hyperbolic triangle groups by types.
problem Classifying complex hyperbolic triangle groups.
method By types defined by the ellipticity of two short words.
result Improves Schwartz conjecture.
The paper studies asymptotic expansions of operators related to Bochner-Schrödinger on Riemannian manifolds.
problem Asymptotic expansions of operators related to Bochner-Schrödinger on Riemannian manifolds.
method Analyzes the Bochner-Schrödinger operator Hp and its function φ(Hp) in L2(X,Lp⊗E), providing an asymptotic expansion of its smooth Schwartz kernel. result The trace of the operator φ(Hp) admits a complete asymptotic expansion in powers of p−1/2 as po∞. We analyze the possibility of defining infinite-dimensional manifolds as ringed spaces. More precisely, we consider three definitions of manifolds modeled on locally convex spaces: in terms of charts and atlases, in terms of ringed spaces, and in terms of functored spaces, as introduced by Douady in his thesis. It is s…
Study fundamental solutions for a class of ultra-hyperbolic operators on pseudo H-type groups.
problem Investigating fundamental solutions for a specific class of ultra-hyperbolic operators on pseudo H-type groups. method Analyzing the algebraic structure of pseudo H-type Lie groups and their associated operators, using families of fundamental solutions and properties of classical special functions. result Found fundamental solutions for the operator in the case r=0, s>0 and proved that no fundamental solution exists in the space of tempered distributions for the case r>0. Survey recent constructions of cyclic cocycles for Lie groups.
problem Higher index theory for proper cocompact G-actions. method Constructions of cyclic cocycles on Harish-Chandra Schwartz algebra.
result Applications to higher index theory.
Study compares synthetic and distributional Ricci curvature bounds.
problem Comparing synthetic and distributional approaches to lower Ricci curvature bounds.
method Analyzes synthetic via weak displacement convexity and distributional via non-negativity of Ricci-tensor.
result Distributional bounds imply entropy bounds for C1 metrics and vice versa for C1,1 under convergence condition. Resolving Schwartz's quadratic meander number conjecture
problem Meander number of cyclic permutations
method Constructing families of cyclic permutations
result Meander number is bounded above and below quadratically in n
We provide a characterization of quotients of three-dimensional complex tori by finite groups that act freely in codimension one via a vanishing condition on the first and second orbifold Chern class. We also treat the case of actions free in codimension two, using instead the "birational" second Chern class, as we cal…
R. Schwartz's inequality provides an upper bound for the Schwarzian derivative of a parameterization of a circle in the complex plane and on the potential of Hill's equation with coexisting periodic solutions. We prove a discrete version of this inequality and obtain a version of the planar Blaschke-Santalo inequality …
New findings show stabilization isn't enough for certain knotted surfaces.
problem The necessity of stabilization for exotic knotted surfaces.
method Examined smooth 4-manifolds with nonempty boundaries containing exotically knotted 2-spheres.
result The 'one is enough' theorem does not apply to closed surfaces with characteristic homology classes.
Study asymptotics of extension and orthogonal Bergman kernels for high tensor powers of positive line bundles.
problem Asymptotic behavior of Bergman kernels for high tensor powers of positive line bundles.
method Analyzing the Schwartz kernel of the Ohsawa-Takegoshi extension operator and orthogonal Bergman projector, proving exponential estimates and asymptotic expansions.
result Explicit asymptotic expansions for the Ohsawa-Takegoshi extension operator and orthogonal Bergman projector.
We prove a conjecture of R. Schwartz about the type of some complex hyperbolic triangle groups.
Study extends surface embedding theorems to non-orientable cases.
problem Embedding non-orientable surfaces in 4-manifolds.
method Conditions for embedding non-orientable surfaces as sums of surfaces and projective planes.
result Extends Gabai and Auckly-Kim-Melvin-Ruberman-Schwartz theorems to non-orientable surfaces.
Paper optimizes neural networks for Bermudan option pricing with faster convergence and risk management tools.
problem Efficiently pricing Bermudan options with static hedging and risk management.
method Monte-Carlo-based artificial neural network framework with novel optimisation algorithm.
result The proposed neural network accelerates convergence and provides improved risk management tools.
Extends calculus of variations to generalized functions.
problem Extending calculus of variations to generalized functions.
method Category of generalized smooth functions, proving connections and conditions.
result Full connections between extremals and Euler-Lagrange equations, necessary conditions for minimizers.
Characterizes polyhomogeneous symbols and applies to Heisenberg calculus.
problem Understanding polyhomogeneous symbols and their applications.
method Simple characterisation and generalization of A.~Connes' tangent groupoid.
result Heisenberg calculus on contact manifolds coincides with groupoid calculus.
New theory of distributions on spaces with singular submanifolds.
problem Defining distributions on spaces with singular submanifolds.
method Construction of thick distributions, operations, and special distributions.
result Clarified connection between thick and classical distributions.
We analyze the probabilistic variance of a solution of Liouville's equation for curvature, given suitable bounds on the Gaussian curvature. The related systolic geometry was recently studied by Horowitz, Katz, and Katz, where we obtained a strengthening of Loewner's torus inequality containing a "defect term", similar …
Wave equation map reveals manifold's structure.
problem Reconstructing Lorentzian manifold from wave equation map.
method Analyzing Schwartz kernel and boundary light observation set.
result Full Lorentzian structure can be recovered under geometric assumptions.
We deduce a recent theorem by R. Schwartz on the structure of the so-called Poncelet grid from complete integrability of the billiard in an ellipse
Paper applies subdiffusive dynamics to American and barrier options pricing.
problem Valuation of American and barrier options in subdiffusive financial models.
method Proposes weighted finite difference and Longstaff-Schwartz methods for valuation.
result Numerical valuation of American and barrier options demonstrated.
We deal with the interest rate model proposed by Schaefer and Schwartz, which models the long rate and the spread, defined as the difference between the short and the long rates. The approximate analytical formula for the bond prices suggested by the authors requires a computation of a certain constant, defined via a n…
We construct analogues of FI-modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings and prove basic structural properties such as Noetherianity. Applications include a proof of the Lannes--Schwartz Artinian conjecture in the generic representatio…
This paper consists of two parts. In the first part we show that in odd dimension, as well as in even dimension below the critical weight (i.e. half the dimension), the logarithmic singularities of Schwartz kernels and Green kernels of conformal invariant pseudodifferential operators are linear combinations of Weyl con…
Consider Least Squares Monte Carlo (LSM) algorithm, which is proposed by Longstaff and Schwartz (2001) for pricing American style securities. This algorithm is based on the projection of the value of continuation onto a certain set of basis functions via the least squares problem. We analyze the stability of the algori…
We study random Morse functions on a Riemann manifold (Mm,g) defined as a random Gaussian weighted superpositions of eigenfunctions of the Laplacian of the metric g. The randomness is determined by a fixed Schwartz function w and a small parameter ε>0. We first prove that as ε→0 the ex…