We explore completely exceptional 2nd order scalar PDEs and their connection to Monge-Ampère equations.
problem Identifying a specific class of nonlinear PDEs that are not genuinely nonlinear.
method Unified geometric background review and definition of completely exceptional PDEs and Monge-Ampère equations.
result The class of completely exceptional 2nd order scalar PDEs reduces to Monge-Ampère equations.
Paper studies shock wave transitions and completely exceptional PDEs via conformal geometry.
problem Understanding shock wave transitions and completely exceptional PDEs.
method Recasting conditions in terms of characteristics, embedding in Lagrangian Grassmannian, using BGG resolution.
result Completely exceptional PDEs, including Monge-Ampere, can be described via conformal geometry and BGG resolution.
Two geometric realizations of F(4) Lie superalgebra found.
problem Geometric realization of F(4) Lie superalgebra. method Explicit geometric realizations as supersymmetries of super-PDE systems.
result Two orders of super-PDE systems found for F(4). Unified PDE models from Lie algebras.
problem Developing PDE models from Lie algebra symmetries.
method Local descriptions of parabolic contact structures, flat models, and explicit PDEs.
result Explicit PDEs with symmetry algebras isomorphic to all complex simple Lie algebras except sl2. Complete exceptional surgeries identified for two-bridge links.
problem Identifying complete exceptional surgeries on two-bridge links.
method Enumerated all hyperbolic two-bridge links with complete exceptional surgeries.
result Listed all candidate slopes for complete exceptional surgeries.
We give a complete classification of exceptional surgeries on hyperbolic alternating knots in the 3-sphere. As an appendix, we also show that the Montesinos knots M (-1/2, 2/5, 1/(2q + 1)) with q at least 5 have no non-trivial exceptional surgeries. This gives the final step in a complete classification of exceptional …
Study the rank of Nijenhuis tensor on parallelizable almost complex manifolds.
problem Understanding the rank of Nijenhuis tensor on parallelizable almost complex manifolds.
method Reduction of computations to solving PDEs, explicit solutions on specific manifolds, analysis of curve of almost complex structures, classification of Lie algebras.
result Classification of Lie algebras admitting almost complex structures with specific Nijenhuis tensor ranks.
Study on Gauss maps of minimal surfaces in 4D space.
problem Understanding the Gauss map of minimal surfaces in Euclidean 4-space.
method Systematic study of Gauss map properties for complete minimal surfaces.
result Optimal results for the maximal number of exceptional values of the Gauss map.
Classifies exceptional Legendrian realizations of Hopf link connected sums.
problem Classifying exceptional Legendrian realizations of Hopf link connected sums.
method Complete coarse classification using Legendrian knot theory.
result First classification result about exceptional Legendrian representatives for Hopf link connected sums.
Study maximal antipodal sets in exceptional symmetric spaces.
problem Classify maximal antipodal sets in exceptional symmetric spaces.
method Combining existing literature and new results, classify maximal antipodal sets.
result Complete classification of maximal antipodal sets in all exceptional compact symmetric spaces.
Study decay and compact support of solutions to certain nonlinear PDEs.
problem Decay and compact support properties of positive solutions to Δpu≥Λ(u) on manifolds. method Nonlinear PDE analysis, Feller property, integral Ricci curvature conditions.
result Characterization of stochastic completeness for the p-Laplacian. Character scheme of small Seifert 3-manifolds is reduced if and only if no exceptional abelian character exists.
problem Character scheme of small Seifert 3-manifolds and its reduction condition.
method Complete description of the SL2(C)-character scheme of small Seifert 3-manifolds.
result Character scheme is reduced if and only if no exceptional abelian character exists, and exceptional abelian character has multiplicity 2.
We construct a complete bounded immersed null holomorphic curve in C^3, which is a recovery of the previous version of the paper Calc. Var. and PDE's vol 36 (2009); Erratum: to appear in Calc. Var. and PDE's, doi:10.1007/s00526-009-0226-5 by the last three authors on this subject.
A Liouville theorem on a specific PDE yields quadratic solutions.
problem Characterizing solutions to a particular PDE on plurisubharmonic functions.
method Analyzing the growth and completeness properties of the metric associated with the solution.
result A Liouville theorem proves that solutions are quadratic.
In this paper, we give a complete classification of exceptional Dehn surgeries on a component of a hyperbolic two-bridge link in the 3-sphere.
The study lists all exceptional Dehn fillings on specific 3-manifolds.
problem Understanding exceptional Dehn fillings on hyperbolic 3-manifolds.
method Examined 1-cusped hyperbolic 3-manifolds with up to 9 ideal tetrahedra.
result Consistent with standard conjectures, suggests new ones.
Study proves existence of Kähler-Einstein metrics and Ricci flat Kähler metrics in 4-manifolds.
problem Existence of Kähler-Einstein metrics and Ricci flat Kähler metrics in 4-manifolds.
method Proves existence through cohomogeneity one triaxial Kähler-Einstein metrics and generalized PDEs for Ricci flat Kähler metrics.
result Proves existence of complete cohomogeneity one triaxial Kähler-Einstein metrics and local existence of Ricci flat Kähler metrics.
It has been observed that most manifolds in the Callahan-Hildebrand-Weeks census of cusped hyperbolic 3-manifolds are obtained by surgery on the minimally twisted 5-chain link. A full classification of the exceptional surgeries on the 5-chain link has recently been completed. In this article, we provide a complete cl…
The thesis explores integrable systems and rigidity in PDEs with symmetry.
problem Understanding the deformation theory and rigidity of PDEs with symmetry.
method The approach involves studying completely integrable systems, their equivalence relations, and the deformation theory of PDEs with pseudogroups of symmetries.
result A solution is rigid if its deformation cohomology vanishes and certain estimates hold.
Study complete space-like stationary surfaces with graphical Gauss image, estimating exceptional values and classifying degenerate surfaces.
problem Estimating exceptional values and classifying degenerate surfaces in Minkowski spacetime.
method Generalizing Fujimoto's theorem, estimating upper bounds, introducing conjugate similarity, and establishing structure theorems.
result Sharp contrast to Bernstein type results for minimal surfaces, estimating upper bounds of exceptional values.
The main goal of this paper is to reveal the geometric meaning of the maximal number of exceptional values of Gauss maps for several classes of immersed surfaces in space forms, for example, complete minimal surfaces in the Euclidean three-space, weakly complete improper affine spheres in the affine three-space and wea…
The paper explores symmetry in solutions of semilinear PDEs on Riemannian domains.
problem Symmetry phenomena in solutions of semilinear PDEs on Riemannian domains.
method General framework for formulating the symmetry problem; evidence from stable solutions; consideration of manifolds with density.
result Evidence that the framework is natural, with results for stable solutions.
We study the general model of self-financing trading strategies in illiquid markets introduced by Schoenbucher and Wilmott, 2000. A hedging strategy in the framework of this model satisfies a nonlinear partial differential equation (PDE) which contains some function g(alpha). This function is deep connected to an utili…
A new method infers parameters from PDEs using Gaussian processes.
problem Estimating unknown parameters in PDEs from noisy data.
method PDE-Informed Gaussian Process (PIGP) method.
result The method bypasses numerical solvers for PDEs and provides uncertainty quantification.
Proving NP-completeness of (d,k)-collapsibility for d≥k+2 except (2,0).
problem Determining if a d-dimensional simplicial complex can be collapsed to a k-dimensional subcomplex. method Extending previous works to show NP-completeness for d≥k+2. result NP-completeness of (d,k)-collapsibility for d≥k+2 except (2,0). Combining M-algebra and hyperbolic involutory algebra extends exceptional tangent spaces to 11 dimensions.
problem Combining symmetries in M-theory to extend exceptional tangent spaces.
method Combining known results to show hyperbolic involutory algebra acts on M-algebra through brane-rotating symmetry.
result Extends the hierarchy of exceptional tangent spaces from n ≤ 7 to n = 11.
Classifies fibered ribbon pretzels, except for a few cases.
problem Classifying fibered ribbon pretzel knots up to mutation.
method Combining lattice embedding techniques with Gabai's classification of fibered pretzel knots, and exhibiting ribbon disks.
result Complete classification except for a few cases.
We give different proofs and prove new results on the non complete solvability of some systems of complex first order p.d.e.'s, especially related to the analysis on CR manifolds.
Maximal representations in exceptional Hermitian Lie groups classified for complex hyperbolic lattices.
problem Classifying maximal representations of complex hyperbolic lattices in exceptional Hermitian Lie groups.
method Unified approach using cominuscule representation of complexified groups, focusing on exceptional cases.
result Complete description of maximal representations for n=2 and G=mE6. New systems of linear PDEs discovered in 3D contact manifolds.
problem Investigating linear PDEs of sl3-type. method Complete local classification using extrinsic geometry.
result 7 new systems of second-order linear PDEs with 8-dimensional solution spaces.
Physics-informed DeepONets solve PDEs without paired data, predicting solutions quickly.
problem Lack of paired input-output data for solving PDEs.
method Physics-informed DeepONets use automatic differentiation to enforce physical laws as soft penalty constraints.
result Physics-informed DeepONets can solve PDEs without paired data, predicting solutions up to 3 orders of magnitude faster.
Deep autoencoder finds linear PDE coordinates for nonlinear equations.
problem Discovering linear coordinates for nonlinear PDEs.
method Residual network architecture for finding intrinsic coordinates.
result Deep learning autoencoder transforms nonlinear PDEs into linear ones.
Darboux Wronskian formulas allow to construct Darboux transformations, but Laplace transformations, which are Darboux transformations of order one cannot be represented this way. It has been a long standing problem on what are other exceptions. In our previous work we proved that among transformations of total order on…
Deep learning approximates PDE evolution operators from solution data.
problem Recovering unknown time-dependent PDEs from solution data.
method Approximate evolution operator in modal space, train deep neural network.
result Deep learning method accurately approximates PDE solutions.
New PDEs for k-harmonic maps link to calibrated fibrations.
problem Understanding k-harmonic maps and their relation to calibrated fibrations. method Analyzing two special classes of k-harmonic maps between Riemannian manifolds. result Explicit noncompact examples of the second class of maps.
No standard compact Clifford-Klein forms found for exceptional Lie groups.
problem Proving the non-existence of standard compact Clifford-Klein forms for exceptional Lie groups.
method Computer-aided approach, algorithmic methods for classifying semisimple subalgebras, and invariant calculations.
result Proves the non-existence of standard compact Clifford-Klein forms for homogeneous spaces of exceptional Lie groups.
The paper shows how profinite completions can reveal 4D geometries except for specific cases.
problem Detecting 4D geometries via profinite completions of fundamental groups.
method Analyzing profinite completions of fundamental groups of 4-manifolds.
result Not all 4D manifolds are geometric, but some Seifert fibred ones are.
Diffeologies unify infinite-dimensional geometry and PDEs, enhancing classical function spaces.
problem Combining infinite-dimensional geometry and PDEs for optimization problems.
method Review and extension of classical function spaces and mapping spaces.
result Diffeologies provide a unified framework for evolution equations and optimization problems.
We refine Osserman's argument on the exceptional values of the Gauss map of algebraic minimal surfaces. This gives an effective estimate for the number of exceptional values and the totally ramified value number for a wider class of complete minimal surfaces that includes algebraic minimal surfaces. It also provides a …
Study on Dirichlet problem for mean curvature type PDEs in Riemannian manifolds.
problem Existence and asymptotic behavior of solutions to a specific PDE in Riemannian manifolds.
method Three parts: general existence proof, asymptotic analysis on hyperbolic space, and non-existence of singularities.
result Existence of solutions in general domains and non-existence of isolated singularities in asymptotic cases.
New connection between complex analysis and PDE for spherical conical metrics.
problem Understanding spectral properties of spherical conical metrics.
method Analyzing monodromy, eigenfunctions, and developing maps.
result Monodromy reducibility implies real-valued eigenfunction with eigenvalue 2.
New equivalences found linking parabolicity, comparison principle, and capacity on Riemannian manifolds.
problem Understanding parabolicity and related concepts on Riemannian manifolds.
method Establishing new equivalences between parabolicity, comparison principle, and capacity.
result Equivalence between p-parabolicity and the comparison principle for the p-Laplace equation. We present three large classes of examples of conformal structures for which the equations for the Fefferman-Graham ambient metric to be Ricci-flat are linear PDEs, which we solve explicitly. These explicit solutions enable us to discuss the holonomy of the corresponding ambient metrics. Our examples include conformal …
We study the de Rham complex on a smooth manifold with a periodic end modeled on an infinite cyclic cover X' \to X. The completion of this complex in exponentially weighted L^2-norms is Fredholm for all but finitely many exceptional weights determined by the eigenvalues of the covering translation map H_*(X') \to H_*(X…
Exceptional Dehn surgeries on arborescent knots have been classified except for Seifert fibered surgeries on Montesinos knots of length 3. There are infinitely many of them as it is known that 4n+6 and 4n+7 surgeries on a (-2, 3, 2n+1) pretzel knot are Seifert fibered. It will be shown that there are only finitely many…
Bi-invariant Einstein metric on G2 unstable under Ricci flow.
problem Stability of bi-invariant Einstein metrics on Lie groups under Ricci flow.
method Analysis of the bi-invariant Einstein metric on G2 using Ricci flow. result The bi-invariant Einstein metric on G2 is dynamically unstable. We investigate the case of the Kahler-Ricci flow blowing down disjoint exceptional divisors with normal bundle O(-k) to orbifold points. We prove smooth convergence outside the exceptional divisors and global Gromov-Hausdorff convergence. In addition, we establish the result that the Gromov-Hausdorff limit coincides wi…
Modeling market impacts leads to perfect hedging strategies.
problem Trading with permanent market impacts and nonlinearity.
method Modeling market impacts using g-expectation and nonlinear stochastic integrals; introducing completeness condition for perfect replication.
result Under certain conditions, derivatives can be perfectly hedged dynamically.