Paper predicts pedestrian trajectories at intersections with varying geometries.
problem Accurately predicting pedestrian trajectories at intersections with different geometries.
method Utilizes contravariant components of trajectories in curbside coordinate system and ASNSC formulation.
result Improves prediction accuracy by 7.2% in same train and test intersections.
Recovering manifold geometry from geodesic intersections.
problem Recovering the geometry of a Riemannian manifold from geodesic intersection lengths.
method Applying stitching data to solve the delayed collision data problem.
result Geometry of the manifold can be recovered from geodesic intersection lengths.
Research characterizes intersection cohomology groups of gauge theories and cotangent bundles.
problem Characterizing intersection cohomology groups of Coulomb branch gauge theories.
method Uses geometric Satake correspondence for Kac-Moody settings.
result Sketches proof of conjecture in affine type A.
Proves intersection properties of minimal hypersurfaces in various spaces.
problem Intersection properties of minimal hypersurfaces in different geometric settings.
method Two approaches: classifications of stable minimal hypersurfaces and conformal change with comparison geometry.
result Intersection properties for minimal hypersurfaces in specific geometric settings, including free boundary minimal hypersurfaces.
Improved bounds on shortest geodesics with self-intersections on hyperbolic surfaces.
problem Quantifying the complexity of non-simple closed geodesics on hyperbolic surfaces.
method Analyzing the geometry of shortest figure eight curves and constructing geodesic representatives.
result Explicit upper bounds for the length of shortest geodesics with k self-intersections improved from 512 to 128. Defines quantum intersection number on pants decompositions and relates it to hyperbolic geometry.
problem Quantum and geometric intersection numbers on surfaces and 3-manifolds.
method Using asymptotic expansions of curve operators in skein theory, we define quantum intersection numbers and relate them to geometric intersection numbers and Teichmüller geometry.
result The pants graph equipped with a metric derived from quantum intersection numbers is quasi-isometric to the Teichmüller space with the Weil-Petersson metric.
In this paper we study relations between intersection numbers on moduli spaces of curves and Hurwitz numbers. First, we prove two formulas expressing Hurwitz numbers of (generalized) polynomials via intersections on moduli spaces of curves. Then we show, how intersection numbers can be expressed via Hurwitz numbers. An…
Study the intersection form on Kähler manifolds of dimension 4 and above.
problem Understand the intersection form on higher-dimensional Kähler manifolds.
method Investigate fundamental properties and applications to birational geometry.
result Present open problems in the relationship between birational invariants and topological invariants.
We investigate the geometry and topology of a standard moduli space of stable bundles on a Riemann surface, and use a generalization of the Verlinde formula to derive results on intersection pairings.
Klainerman, Luk and Rodnianski derived an anisotropic criterion for formation of trapped surfaces in vacuum, extending the original trapped surface formation theorem of Christodoulou. The effort to understand their result led us to study the intersection of a hyperplane with a lightcone in the Minkowski spacetime. For …
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.
Researchers found explicit solutions to a complex equation in advanced geometry.
problem Critical Yamabe type equation in sub-Finsler geometry.
method Computed a two-parameter family of explicit positive solutions.
result Explicit solutions to a critical equation in sub-Finsler geometry.
Graph dynamics link combinatorics to geometry, revealing manifold intersections and stability.
problem Understanding the geometry of graph dynamical systems with odd interactions.
method Proved geometry and stability of manifolds governed by graph homology and coverings.
result Derived upper and lower bounds on the dimension of the equilibrium set.
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.
Paper certifies intersection of minimum-volume confidence sets for multinomial outcomes.
problem Certifying intersection of minimum-volume confidence sets for multinomial outcomes.
method Exploits likelihood ordering to induce halfspace constraints, enabling adaptive geometric partitioning and computable bounds on p-values.
result Efficient and provably sound algorithm for certifying intersection, disjointness, or indeterminate result.
For a fully irreducible automorphism φof the free group F_k we compute the asymptotics of the intersection number n \mapsto i(T,T'φ^n) for trees T,T' in Outer space. We also obtain qualitative information about the geometry of the Guirardel core for the trees T and T'φ^n for n large.
We present a survey of the calibrated geometries arising in the study of the local singularity structure of supersymmetric fivebranes in M-theory. We pay particular attention to the geometries of 4-planes in eight dimensions, for which we present some new results as well as many details of the computations. We also ana…
The investigation of strings and M-theory involves the understanding of various BPS solitons which in a certain approximation can be thought of as solutions of ten- and eleven-dimensional supergravity theories. These solitons have a brane or a intersecting brane interpretation, saturate a bound and are associated with …
Paper connects volumes of moduli spaces of super Riemann surfaces to integrals over stable Riemann surfaces.
problem Relating volumes of moduli spaces of super Riemann surfaces to integrals over stable Riemann surfaces.
method Relates volumes of moduli spaces of super Riemann surfaces to integrals over the moduli space of stable Riemann surfaces Mg,n. result Proves recursion between volumes of moduli spaces of super hyperbolic surfaces using algebraic geometry.
The paper classifies a specific type of quadratic variety with a small codimension.
problem Classifying nondegenerate smooth projective varieties of dimension n=2c−1 defined by quadratic equations. method Using the Hartshorne conjecture on complete intersections and classification techniques.
result The paper classifies varieties with n=2c−1 and proves they are complete intersections. Combines topological and geometric approaches to data analysis.
problem Understanding when and how geometric objects intersect.
method Connects topological and geometric concepts of curvature.
result Reconceptualizes curvature and links it to hyperconvexity.
The study of smoothing arcs and curves on surfaces, proving tautness and arc length spectrum properties.
problem Analyzing the geometric and combinatorial effects of smoothing intersections in arcs or curves.
method Geometric and combinatorial analysis, proving tautness and arc length spectrum properties.
result Shortest arcs with self-intersections have exactly or at most one more self-intersection than the self-intersection number.
The study extends inscription problems to non-Euclidean geometries.
problem Generalizing inscription problems to non-Euclidean geometries.
method Symplectic and Riemannian geometry techniques.
result Proved generalized inscription theorems for hyperbolic and spherical surfaces.
This is an expanded and updated version of a lecture series I gave at Seoul National University in September 1997. It is in some sense an update of the 1979 Griffiths and Harris paper with a similar title. I discuss: Homogeneous varieties, Topology and consequences Projective differential invariants, Varieties with deg…
By Markowitz geometry we mean the intersection theory of ellipsoids and affine subspaces in a real finite-dimensional linear space. In the paper we give a meticulous and self-contained treatment of this arch-classical subject, which lays a solid mathematical groundwork of Markowitz mean-variance theory of efficient por…
Smooth complex surfaces with triple intersections using differential geometry.
problem Smooth complex surfaces with trivial canonical bundle and triple intersections.
method Explicit construction of local smoothings and solutions to nonlinear elliptic PDEs.
result Existence of smoothings for d-semistable SNC complex surfaces with trivial canonical bundle. In this paper, we give a framework for the study of the extremal length geometry of Teichmüller space after S. Kerckhoff, F. Gardiner and H. Masur. There is a natural compactification using extremal length geometry introduced by Gardiner and Masur. The compactification is realized in a certain projective space. We deve…
Study connects flow dynamics to 3D geometry via surface intersections.
problem Relating flow dynamics to geometric properties of 3-manifolds.
method Relates pseudo-Anosov flow dynamics to hyperbolic geometry via curve graphs.
result Established a link between flow invariants and geometric features of 3-manifolds.
HADES detects data singularities quickly and accurately.
problem Detecting singularities in data efficiently.
method Kernel goodness-of-fit test based on differential geometry and optimal transport theory.
result Correctly detects singularities with high probability.
In the mid eighties Goldman proved an embedded curve could be isotoped to not intersect a closed geodesic if and only if their Lie bracket (as defined in that work) vanished. Goldman asked for a topological proof and about extensions of the conclusion to curves with self-intersection. Turaev, in the late eighties, aske…
We show that there are minimal graphs in R^{n+1} whose intersection with the portion of the horizontal hyperplane contained in the unit ball has any prescribed geometry, up to a small deformation. The proof hinges on the construction of minimal graphs that are almost flat but have small oscillations whose geometry we c…
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.
The study proves spherical polygon analogs of curve theorems, finding bounds on intersections and inflections.
problem Finding bounds on intersections and inflections for spherical polygons.
method Adapting smooth curve theorems to spherical polygons using discrete tools.
result Proves discrete analogs of four-vertex theorems for spherical polygons.
This paper constructs symplectic surfaces in 4-manifolds with transversal intersections.
problem Constructing symplectic surfaces in symplectic 4-manifolds with transversal intersections.
method Revision and generalization of existing constructions.
result Produces the ramification locus of Seifert bundles over symplectic 4-orbifolds.
This paper extends the C*-signature to non-Witt spaces using noncommutative geometric methods.
problem Extending the signature to non-Witt spaces with noncommutative geometric methods.
method Noncommutative geometric methods, combinatorial framework, and comparison with analytical signature.
result Constructing the C*-signature on non-Witt spaces.
Study on ruled surfaces over elliptic curves with unique foliations and parallelizable 4-webs.
problem Characterizing the geometry of ruled surfaces over elliptic curves.
method Analysis of foliations, minimal self-intersection sections, and 2-webs.
result The 4-web defined by fibration, foliation, and minimal self-intersection sections is locally parallelizable.
We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperkahler manifold. We establish a characterization, a Chern number inequality, topological and geometrical properties of Lagrangian submanifolds. We discuss a category of Lagrangian subvarieties and its relationship with the…
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.
This paper introduces an intersection theory problem for maps into a smooth manifold equipped with a stratification. We investigate the problem in the special case when the target is the unitary group and the domain is a circle. The first main result is an index theorem that equates a global intersection index with a f…
We show the intersection of a compact almost complex subvariety of dimension 4 and a compact almost complex submanifold of codimension 2 is a J-holomorphic curve. This is a generalization of positivity of intersections for J-holomorphic curves in almost complex 4-manifolds to higher dimensions. As an applicat…
In a recent paper, Ohta and Townsend studied the conditions which must be satisfied for a configuration of two intersecting M5-branes at angles to be supersymmetric. In this paper we extend this result to any number of M5-branes or any number of M2-branes. This is accomplished by interpreting their results in terms of …
PINNs solve differential geometry problems in complex shapes.
problem Solving differential geometry problems in complex shapes.
method Training neural networks with loss functions inspired by differential conditions.
result PINNs are effective for differential geometry problems.
Overview of algebraic geometry for almost complex manifolds.
problem Developing algebraic geometry for almost complex manifolds without genericity.
method Reviewing results based on pseudoholomorphic maps and intersection theory.
result Introduction of birational morphism between almost complex manifolds.
Decomposes J-energy into simpler intersection numbers for stability analysis.
problem Analyzing J-stability in algebraic geometry.
method Proves a decomposition formula for J-energy and shows equivalence of stability conditions.
result Equivalence of J-stability and K-stability for surfaces under pseudoeffective conditions.
Gradient of harmonic functions tied to level hypersurface geometry.
problem Understanding how the gradient of harmonic functions changes.
method Analyzes how the gradient of harmonic functions changes along gradient flows.
result The gradient's change is determined by the mean curvature of level hypersurfaces.
A quadrisecant line is one which intersects a curve in at least four points, while an essential secant captures something about the knottedness of a knot. This survey article gives a brief history of these ideas, and shows how they may be applied to questions about the geometry of a knot via the total curvature, ropele…
Average teaching complexity for locating target regions among halfspace intersections is Θ(d).
problem Teaching the location of a target region among intersections of halfspaces.
method Novel insights from computational geometry to count convex polytopes and faces.
result Average-case teaching complexity is Θ(d), contrasting with Θ(n) worst-case complexity.
We extend the results given by Colbois, Dryden and El Soufi on the relationships between the eigenvalues of the Laplacian and an extrinsic invariant called intersection index, in two directions. First, we replace this intersection index by invariants of the same nature which are stable under small perturbations. Second…