Study of Poisson homeomorphisms and rigidity of coisotropic submanifolds.
problem Rigidity and non-rigidity phenomena in Poisson geometry.
method Study of Poisson homeomorphisms, use of clean intersection points, and analysis of characteristic partitions.
result Poisson homeomorphisms preserve symplectic foliations and coisotropic submanifolds are flexible.
Develops a method to compute Morse homology for clean but not necessarily transverse intersections.
problem Computing Morse homology for clean but not necessarily transversely intersecting manifolds.
method Constructs minimal semi-global Kuranishi structures for moduli spaces of Morse trajectories, generalizing obstruction bundle gluing.
result Obtains iterated gluing equals simultaneous gluing, maintaining computability.
Clean intersections of Lagrangian knots in 3D are impossible.
problem Prohibiting clean intersections of certain knots in 3D symplectic geometry.
method Symplectic field theory and algebraic constraints on augmentation varieties.
result No Hamiltonian diffeomorphism can cleanly intersect a specific type of knot's conormal bundle.
Paper computes Atiyah class for DG manifolds of amplitude +1.
problem Computing the Atiyah class for DG manifolds of specific amplitude.
method Computed the Atiyah class by encoding the derived intersection of sections and zero sections of vector bundles.
result Atiyah class vanishes if and only if the intersection is clean.
The study of quotient structures in multi-graded bundles, including double vector bundles.
problem Understanding quotients of multi-graded bundles, especially double vector bundles.
method Analyzing quotients as towers of affine bundles and constructing normal bundles.
result Any quotient of multi-graded bundles fits into a tower of affine bundles.
We prove: If a complete connected smooth surface M in euclidean 3-space has general position, intersects some plane along a clean figure-8 (a loop with total curvature zero) and all compact intersections with planes have central symmetry, then M is a (geometric) cylinder over some central figure-8. On the way, we estab…
This paper uses sheaf theory to constrain knot types in clean intersections.
problem Understanding constraints on knot types in clean intersections.
method Microlocal sheaf theory and 3-manifold theory.
result Existence of a surjective homomorphism preserving longitude and meridian.
We show that the cardinality of the transverse intersection of two compact exact Lagrangian submanifolds in a cotangent bundle is bounded from below by the dimension of the Hom space of sheaf quantizations of the Lagrangians in Tamarkin's category. Our sheaf-theoretic method can also deal with clean and degenerate Lagr…
A geometric method optimizes over the intersection of two manifolds.
problem Optimizing over the intersection of two manifolds with coupled geometry.
method Geometric method using retraction on one manifold and orthogonal updates.
result Convergence to first-order stationarity under intrinsic transversality.
The paper studies knot types of clean intersections in a 3D space.
problem Identifying knot types in clean intersections.
method Using compactly supported Hamiltonian isotopy and DGA maps.
result Constraints on knot types of intersections.
This is primarily an expository note showing that earlier work of Lai on CR geometry provides a clean interpretation, in terms of a Gauss map, for an adjunction formula for embedded surfaces in an almost complex four manifold. We will see that if F is a surface with genus g in an almost complex four-manifold M, then 2 …
Floer homotopy theory applies to Lagrangians, overcoming curvature issues.
problem Curvature phenomena in high dimensions for monotone Lagrangians.
method Introduces N-truncated, R-oriented flow categories and module prospectrum. result Well-defined invariants for closed embedded monotone Lagrangians.
New concept of coisotropic structures for differentiable stacks defined.
problem Defining coisotropic structures for differentiable stacks.
method Using twisted Dirac structures and Morita equivalences.
result 1-shifted coisotropic structures transfer through Morita equivalences.
For a real valued periodic smooth function u on R, n≥0, one defines the osculating polynomial φs (of order 2n+1) at a point s∈R to be the unique trigonometric polynomial of degree n, whose value and first 2n derivatives at s coincide with those of u at s. We will say that a point s is a clean maximal flex …
Paper shows how to evenly distribute intersections in hyperbolic spaces.
problem Equidistribution of intersections in hyperbolic manifolds.
method Properly immersed totally geodesic submanifolds, hyperbolic volume measure.
result Effective equidistribution of intersection points as submanifolds grow.
In this paper we present the algorithms for calculating the differential geometric properties {t,n,b1,b2,b3,k1,k2,k3,k4} along-with geodesic curvature and geodesic torsion of the transversal intersection curve of four hypersurfaces (given by parametric representation) in Euclidean space R^5. In transversal intersection…
Trimming helps in conformal prediction when it separates anomaly scores.
problem Effectiveness of trimming in conformal prediction under contamination.
method Analyse fixed-threshold trimming as a replacement of the contaminated calibration law with a retained law.
result Trimming helps when it separates anomaly scores, reducing clean-target coverage to a one-dimensional score-CDF transfer problem.
Study properties of self-similar continua with finite intersection property.
problem Characterize self-similar continua with finite intersection property.
method Prove intersection graph criterion, finite order theorem, and parameter matching theorem.
result All Jordan arcs starting from a intersection point in such continuum on a plane should have the same slope parameter at that point.
The problem on the minimal number (with respect to deformation) of intersection points of two closed curves on a surface is solved. Following the Nielsen approach, we define classes of intersection points and essential classes of intersection points, which "are preserved under deformation" and whose total number is cal…
The paper proves the exact number of singular points in the intersection of convex shapes.
problem Determining the exact number of singular points in the intersection of convex shapes.
method Analyzing the intersections of n translates of a strictly convex, smooth, convex body in the Euclidean plane.
result The intersection of n translates of a convex body has exactly n points of singularity along its boundary.
The paper provides an algorithm to create curves touching a smooth cubic at specific intersection points.
problem Creating curves that touch a smooth cubic at specific intersection points.
method Algorithm based on divisions and Zariski tuples to produce n-contact curves. result An algorithm to generate n-contact curves to a smooth cubic. Computes expected number of real intersection points of essential variety with random linear spaces.
problem Computing the expected number of real intersection points of the essential variety with random linear spaces.
method Two probability distributions for linear spaces: invariant under orthogonal group action and one motivated from computer vision. Used Monte Carlo simulation for the latter.
result Expected number of real intersection points lies in the interval (3.95 - 0.05, 3.95 + 0.05) with high probability.
Suppose a smooth planar curve γ is 2π-periodic in the x direction and the length of one period is ℓ. It is shown that if γ self-intersects, then it has a segment of length ℓ−2π on which it self-intersects and somewhere its curvature is at least 2π/(ℓ−2π). The proof involves the projection Γ …
SSMs can be poisoned with clean labels, leading to generalization failure.
problem The implicit bias of SSMs can be manipulated by including special training examples with clean labels.
method Formal proof and empirical demonstration of the phenomenon.
result SSMs can fail to generalize even with clean labels, due to the inclusion of special training examples.
Based on Nielsen fixed point theory and Gröbner-Shirshov basis, we obtain a simple method to compute geometric intersection numbers and self-intersection geometric numbers of loops on surfaces.
Adversarial training leads to clean data generalization with significant robust overfitting gap.
problem Significant robust generalization gap in adversarial training.
method Two theoretical views: representation complexity and training dynamics.
result ReLU nets with O(ND) extra parameters can achieve CGRO. Improved robust regression with clean covariates achieves better rates than Huber's model.
problem Robust regression under adaptive contamination of responses with clean covariates.
method Exploiting clean covariates to construct an estimator achieving better rates than Huber's model.
result Improved estimation rate even with constant contamination, achieving consistency.
Method detects intersections between ellipses for Borromean linking.
problem Detecting Borromean linking between ellipses.
method Transforming one ellipse to a unit circle, examining intersections.
result Efficiently determines Borromean linking between ellipses.
A new method clusters intersecting lines using hypergraphs.
problem Clustering intersecting lines in subspace clustering.
method Constructing a geometric hypergraph and using spectral algorithm.
result Achieves information-theoretic bounds for line clustering.
New method finds optimal training stop point with noisy labeled data.
problem Finding optimal training stop point with noisy labeled data.
method Analyzed training accuracy rate changes for different noise ratios to identify a training stop region. Developed a heuristic algorithm based on a small-learning assumption.
result Identified optimal training stop point at or close to maximum obtainable test accuracy.
The paper studies the number of normals to ellipsoids and their intersections with caustics.
problem The number of normals to an ellipsoid passing through a given point.
method Intersection points of the ellipsoid and its caustics are used to study the problem in 3D space.
result The number of normals is dependent on the position of the given point with respect to the caustics of the ellipsoid.
Method cleans covariance matrices for better statistical inference.
problem Reducing estimation noise in covariance matrices for better statistical inference.
method Robust yet flexible hierarchical ansatz with bootstrap procedure.
result Lower realized risk in global minimum variance portfolios.
A self-transverse immersion of a smooth manifold M^{k+2} in R^{2k+2} has a double point self-intersection set which is the image of an immersion of a smooth surface, the double point self-intersection surface. We prove that this surface may have odd Euler characteristic if and only if k is congruent to 1 modulo 4 or k+…
Growth rate of Dehn twist lattice points in Teichmüller space is slower than mapping class group lattice points.
problem Analyzing the growth rate of Dehn twist lattice points in Teichmüller space.
method Comparing growth rates of Dehn twist, mapping class group, and multi-twist lattice points.
result The growth rate of Dehn twist lattice points is coarsely asymptotic to $e^{ rac{h}{2}R}$, slower than the mapping class group.
We describe a family of hyperbolic knots whose character variety contain exactly two distinct components of characters of irreducible representations. The intersection points between the components carry rich topological information. In particular, these points are non-integral and detect the Seifert surface.
Using an elementary argument, we prove new fixed point theorems for classical elliptic complexes. We obtain new results for conformal relations and coisotropic intersections. We obtain theorems for the average intersections of families of special lagrangian and lagrangian varieties in certain homogeneous spaces.
We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from the plane curve as states to take the sum of the warping crossing polynomials for…
In this article, we give proofs on the Arnold Lagrangian intersection conjecture on the cotangent bundles, Arnold-Givental Lagrangian intersection conjecture and the Arnold fixed point conjecture.
A new proof shows almost every normal to a smooth convex body intersects at least 6 normals from different points.
problem The conjecture about normals to convex bodies in high dimensions.
method Short proof of Y. Martinez-Maure's result for n≥3. result Almost every normal through a boundary point intersects at least 6 normals from different points.
Monotone adversarial corruptions degrade optimal learning algorithms.
problem Optimal learning algorithms' reliance on exchangeability and independence is challenged.
method Introduces a monotone adversarial corruption model where an adversary adds monotone corruptions to a clean dataset.
result Optimal learning algorithms achieve suboptimal expected error on new test points.
Geodesics on hyperbolic surfaces become evenly spread over time.
problem Distribution of geodesics on hyperbolic surfaces.
method Equidistribution analysis of geodesics with length less than T as T approaches infinity.
result Closed geodesics on hyperbolic surfaces of finite area become evenly spread over time.
Paper proves curves can be smoothed to reduce self-intersection by exactly 1.
problem Prove that the shortest closed geodesic self-intersects exactly k times.
method Carefully smoothing intersection points reduces self-intersection by exactly 1.
result The shortest closed geodesic self-intersects exactly k times for hyperbolic and Riemannian metrics.
The paper examines how closed curves on surfaces intersect and how this intersection determines the curves.
problem Determining closed curves on surfaces based on their intersections.
method Constructing and studying k-equivalent curves, analyzing intersections with other curves. result Curves are determined by their intersections with all other curves, but non-simple curves require infinitely many intersections to distinguish.
Neural networks learn clean data patterns first, then noisy data, leading to improved performance initially but deteriorating later.
problem Improvement in prediction error on clean data during early training of neural networks with noisy labels.
method Theoretical analysis and experiments to explore the dynamics of gradient descent and the impact of clean and noisy data.
result Neural networks prioritize learning clean data patterns first, leading to improved performance initially but deteriorating later due to diminishing gradient dominance of clean samples over noisy ones.
Novel approach for large genus intersection number asymptotics.
problem Computing intersection numbers in large genus.
method Resurgent analysis of n-point functions with quantum curve.
result Extension of Aggarwal's results and new r-spin and Theta-class intersection numbers. In this paper we treat the intersection of fixed point subgroups by the involutive automorphisms of exceptional Lie group G=F4,E6,E7. We shall find involutive automorphisms of G such that the connected component of the intersection of those fixed point subgroups coincides with the maximal torus of G.
The paper proves that any smooth curve can have two similar inscribed rectangles.
problem Finding two similar inscribed rectangles in a smooth Jordan curve.
method Lagrangian Floer homology and differential topological computation.
result Generic doubling of inscribed rectangles in smooth Jordan curves.
In the eighties Goldman discovered a Lie algebra structure on the vector space generated by the free homotopy classes of oriented curves on an oriented surface. The Lie bracket [a,b] is defined as the signed sum over the intersection points of a and b of the loop product of at the intersection points. If one of the cla…