We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as , to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.
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New representation of field equations for QMAG, presenting new solutions.
problem Finding new explicit solutions for quadratic metric-affine gravity.
method Developed a new explicit representation of field equations without assumptions on torsion properties.
result Presented two conjectures on new types of solutions of QMAG.
New ancient solutions found for curvature flow in 2D.
problem Ancient solutions for curvature flow in 2D.
method Constructing and classifying convex ancient solutions.
result All convex ancient solutions classified for .
New proof shows solutions to Lichnerowicz equation exist.
problem Existence of solutions to the Lichnerowicz equation in general relativity.
method A new proof approach.
result Existence and uniqueness of solutions proven.
Exact solutions found for a new SV model with stationary volatility.
problem Finding exact solutions for a new SV model.
method Analytical solutions for transition probability density, option values, and martingale defect.
result First example of an SV model with exact solutions, GBM volatility, and stationary volatility.
New periodic solutions found for a specific equation.
problem Finding solutions to a specific partial differential equation.
method Using Bäcklund transformation for the sine-Gordon equation to generate solutions.
result New periodic exact solutions of the constant astigmatism equation.
Two new proofs show Ricci flow breathers are special solutions.
problem Characterize solutions to Ricci flow on closed manifolds.
method Use singularity models and ancient solutions to show they are gradient Ricci solitons.
result Ricci flow breathers are gradient Ricci solitons.
New solutions found for a complex boundary problem.
problem One-phase free boundary problem in geometry.
method New connection with minimal surfaces.
result Disproved a conjecture and constructed homogeneous solutions.
New varifold solutions for mean curvature flow converge and are unique.
problem Mean curvature flow and Allen-Cahn equation convergence and uniqueness.
method Evolving varifolds coupled to phase volumes, weak-strong uniqueness principle.
result Limits of Allen-Cahn solutions are varifold solutions, and classical flows are unique.
New cylindrical solutions found for Grushin-type problem.
problem Critical Grushin-type problem on CR sphere.
method Local Pohozaev identities for non-degeneracy, Lyapunov-Schmidt reduction for solutions.
result New type of multi-bubbling cylindrical solutions constructed.
Study on Ricci flow of invariant metrics on spheres, finding new ancient solutions.
problem Analyzing the Ricci flow of Sp(n+1)-invariant metrics on spheres.
method Determine forward and ancient solutions, classify them, and classify their behavior under flow.
result Exhibit a new one-parameter family of ancient solutions on spheres with larger isometry groups.
New smooth solutions of the Strominger system with non vanishing flux, non-trivial instanton and non-constant dilaton based on the quaternionic Heisenberg group are constructed. We show that through appropriate contractions the solutions found in the -heterotic case converge to the heterotic solutions on 6-dimensi…
New solutions found for G2 system on squashed manifolds.
problem Finding solutions to the heterotic G2 system on squashed manifolds.
method Constructing families of solutions using squashed metrics on 7-sphere or Aloff-Wallach space.
result Obtained AdS3 solutions for most squashing parameters, excluding a specific case.
New flow solves LYZ equation on Kähler manifolds.
problem Solving the LYZ equation on compact Kähler manifolds.
method Introduced a new flow and showed its longtime solution converges to the LYZ equation solution under certain conditions.
result The flow converges to a singular solution on compact Kähler surfaces under specific conditions.
We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as , to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.
Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniquenessmath.AP
Novel weak solutions for volume-preserving mean curvature flow established.
problem Existence and uniqueness of solutions to volume-preserving mean curvature flow.
method Introducing varifold solutions coupled with phase volumes and new calibrations.
result Uniqueness of classical solutions among varifold solutions.
New solutions found using rational exponents in complex-valued geometry.
problem Difficult non-linear problems in differential geometry.
method Generalizing from polynomial solutions to rational exponents.
result Many new solutions found.
New tools identify potential counter-examples to string theory conjectures.
problem Classical de Sitter solutions with specific conditions.
method Developed new tools and constraints to explore parameter space.
result Identified corners of parameter space where counter-examples could be found.
New solutions found for complex-valued harmonic morphisms with rational exponents.
problem Finding new solutions to complex-valued harmonic morphisms.
method Generalizing from polynomial solutions to rational exponents.
result Many new solutions to the problem.
New solutions found for complex structures on specific manifolds.
problem Constructing smooth solutions to the Hull-Strominger system.
method Using fibrations over K3 orbisurfaces.
result Proved existence of solutions for certain manifolds.
A new presentation of the -string braid group is studied. Using it, a new solution to the word problem in is obtained which retains most of the desirable features of the Garside-Thurston solution, and at the same time makes possible certain computational improvements. We also give a related solution to t…
New solutions found for Yamabe problem on spheres with foliations.
problem Yamabe problem on spheres with singular Riemannian foliations.
method Variational methods, symmetries from foliations, Sobolev embedding theorem, Principle of Symmetric Criticality.
result Existence of sign-changing and positive solutions with specific symmetries.
Exact solution found for k-price auctions, introducing a new fair auction type.
problem Finding fair solutions for auction mechanisms.
method Exact analytical solution for k-price auctions and introduction of a new auction type.
result New auction type provides fair solutions.
New solutions found for elliptic sinh-Gordon and sine-Gordon equations.
problem Elliptic sinh-Gordon and sine-Gordon equations on the real plane.
method Backlund transformation connecting the equations.
result New families of solutions introduced.
Improved option pricing formula using relativistic mechanics.
problem Improving Black-Scholes formula for better option pricing accuracy.
method Developed a relativistic version of the Black-Scholes formula and derived a closed-form solution.
result New formula offers significant improvements over existing solutions and better fits empirical data.
New proof for convex solutions of Monge-Ampère equation.
problem Interior regularity of strictly convex solutions
method Doubling inequality for Hessian in extrinsic distance function
result Interior regularity established
We classify generalised supersymmetric fluxbranes in type II string theory obtained as Kaluza-Klein reductions of the Minkowski space vacuum of eleven-dimensional supergravity. We obtain two families of smooth solutions which contains all the known solutions, new solutions called nullbranes, and solutions interpolating…
Smooth compactly supported solutions of some underdetermined elliptic PDE, with gluing applicationsmath.FA
We give sufficient conditions for some underdetermined elliptic PDE of any order to construct smooth compactly supported solutions. In particular we show that two smooth elements in the kernel of certain underdetermined linear elliptic operators can be glued in a chosen region in order to obtain a new smooth soluti…
New periodic solutions found in 2n-body problem, braids of pseudo-Anosov type with stretch factors as metallic ratios.
problem Periodic solutions of the 2n-body problem and their braid types.
method Analyzing braid types and stretch factors associated with pseudo-Anosov braids.
result Braids from new periodic solutions are of pseudo-Anosov type with stretch factors as metallic ratios.
Study of 4D flows with nilpotent symmetry, showing immortal solutions and blowdown limits.
problem Understanding 4D generalized Ricci flows with nilpotent symmetry.
method Immortal solutions, type III curvature and diameter estimates, new energy monotonicity.
result Blowdown limits lie in a finite-dimensional family of solutions.
We construct new type II ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as , to a tower of two spheres. Their curvature operator changes sign. We allow two time-dependent parameters in our ansatz. We use perturbation theory, via fixed point arguments,…
New findings on -solutions with round cylinder as asymptotic shrinker.
problem Characterizing -solutions with specific asymptotic behavior.
method Analysis of Ricci flow in dimensions .
result Uniformly Positive Isoperimetric Constant (PIC) for -solutions.
New method proves regularity for small energy solutions of Yang-Mills-Higgs equations.
problem Proving regularity for small energy solutions of Yang-Mills-Higgs equations.
method Improved Kato inequality and Weitzenböck formulae.
result Obtains bounded curvature without Coulomb gauges.
New proof shows origin-centred balls are unique solutions to curvature problems.
problem Proving uniqueness of solutions to curvature problems.
method Local Brunn-Minkowski inequality and Alexandrov-Fenchel inequality.
result Origin-centred balls are the only solutions to curvature and related problems.
Bifurcation of periodic orbits for the -body problem, from a non geometrical family of solutionsmath.DS
New periodic solution found in 4-body problem, not part of expected geometrical family.
problem Finding new periodic solutions in the 4-body problem not fitting the expected geometrical family.
method Analytic continuation of a numerical solution to discover a new family of solutions.
result Existence of a non-planar periodic solution for any pair of masses and integer n.
Study finds solutions for degenerate affine curve shortening flow.
problem Analyzing degenerate affine curve shortening flow.
method Solved equations for affine self-similar solutions.
result New special solutions discovered for affine curve shortening flow.
New solutions found for Hull-Strominger system on torus bundles.
problem Finding solutions to the Hull-Strominger system with torus symmetry.
method Constructing solutions on torus bundles over K3 orbifolds.
result Smooth manifolds with complex structures have solutions to the Hull-Strominger system.
New solutions found for complex geometry problem.
problem Solving the Hull-Strominger system on non-Kähler manifolds.
method Constructing solutions with a specific connection ansatz and using moduli spaces of sheaves.
result First T-dual solutions on compact non-Kähler manifolds with different topology.
Solves a PDE for Landsberg surfaces using new Finsler surface insights.
problem Solving the Landsberg's PDE for Finsler surfaces.
method Reduces the system of non-linear PDEs to a single PDE, the Landsberg's PDE, and solves it.
result Obtains a class of solutions for the Landsberg's PDE.
New proof shows 3D κ-solutions are always rotationally symmetric.
problem Uniqueness and symmetry of 3D κ-solutions.
method Alternative proof using Strong Stability Theorem for singular Ricci flows.
result Compact 3D κ-solutions are rotationally symmetric.
New technique for multiple-source adaptation without density estimation.
problem Multiple-source adaptation problem.
method Discriminative technique that uses conditional probabilities from unlabeled data.
result Our technique outperforms previous generative solutions and other domain adaptation baselines.
Paper shows non-uniqueness of solutions to a specific Minkowski problem.
problem Investigating non-uniqueness in the dual -Minkowski problem for .
method Proved existence of solutions and constructed an example.
result Non-uniqueness of solutions to the dual -Minkowski problem.
We provide new exact Taylor's series with fixed coefficients and without the remainder. We demonstrate the usefulness of this contribution by using it to obtain very simple solutions to (non-linear) PDEs. We also apply the method to the portfolio model.
New solutions found for a free boundary problem using refined methods.
problem Free boundary problem with two component boundaries.
method Refined Lyapunov-Schmidt reduction method and variational arguments.
result Global minimizers of energy functional found in dimension 8.
New methods solve complex PDEs with mixed boundary conditions.
problem Solving inhomogeneous Robin type boundary value problems for linear PDEs.
method Odd and even Hilbert transforms.
result Non-standard solutions to various PDEs in finance, stochastic analysis, etc.
Kähler-Ricci Shrinkers and Ancient Solutions with Nonnegative Orthogonal Bisectional Curvaturemath.DG
Study classifies Ricci solitons with specific curvature conditions.
problem Classifying gradient shrinking Ricci solitons with nonnegative orthogonal bisectional curvature.
method Proves classification results under invariant conditions without curvature bounds.
result Obtains new results on ancient solutions for Ricci and Kähler-Ricci flows.
New solutions found for a complex equation, diverging from a cone.
problem Finding new solutions to a complex equation with specific symmetry properties.
method Careful study of a Jacobi-Toda system on an invariant manifold, asymptotic to a cone.
result Constructs new families of two-ended solutions diverging logarithmically from a cone.
New steady Euler flows found on 3-sphere and Sasakian manifolds.
problem Finding new steady Euler solutions on specific manifolds.
method Bifurcating from an existing ansatz and extending it to Sasakian 3-manifolds.
result Previously known solutions are not isolated, and new solutions found on 3-sphere and other Sasakian manifolds.