Engel structures and flows on 4-manifolds linked via constant rank intersections.
problem Pairs of Engel structures on 4-manifolds with constant rank intersections.
method Established a correspondence with weakly hyperbolic flows.
result Engel structures and flows on 4-manifolds are linked via constant rank intersections.
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…
Study calculates volumes and constants from intersection theory on abelian differential strata.
problem Calculating volumes and constants from intersection theory on abelian differential strata.
method Intersection numbers on strata with prescribed zeros orders.
result Evaluation of large genus limits and saddle connection Siegel-Veech constants for all strata.
3-manifold triangulation can be reconstructed from its intersection matrix.
problem Reconstructing the triangulation of 3-manifolds from their intersection matrix.
method Using the intersection matrix of a simplicial complex to determine the triangulation of a 3-manifold up to isomorphism.
result The intersection matrix is sufficient to determine the triangulation of a 3-manifold up to isomorphism.
We prove formulas (found by Witten in 1992 using physical methods) for intersection pairings in the cohomology of the moduli space M(n,d) of stable holomorphic vector bundles of rank n and degree d (assumed coprime) on a Riemann surface of genus g greater than or equal to 2. We also use these formulas for intersection …
Proposes causal modeling for intersectional fairness in rankings.
problem Fairness in rankings, especially intersectional fairness.
method Causal modeling approach for intersectional fairness, flexible ranking computation.
result Experimental evaluation shows the approach's effectiveness under different assumptions.
Study refines Siegel-Veech constants for abelian differentials.
problem Computing Siegel-Veech constants for abelian differentials.
method Intersection theory and quasimodular forms.
result New identity for Siegel-Veech constants of cylinders.
Let H and K be subgroups of a free group of ranks h and k \geq h. We prove the following strong form of Burns' inequality: rank(H \cap K) - 1 \leq 2(h-1)(k-1) - (h-1)(rank(H \vee K) -1). A corollary of this, also obtained by L. Louder and D. B. McReynolds, has been used by M. Culler and P. Shalen to obtain information …
A classical result attributed to Joachimsthal in 1846 states that if two surfaces intersect with constant angle along a line of curvature of one surface, then the curve of intersection is also a line of curvature of the other surface. In this note we prove a global analogue of this result, as follows. Suppose that two …
We give the diffeomorphism classification of complete intersections with S^1-symmetry in dimension less than or equal to 6. In particular, we show that a 6-dimensional complete intersection admits a smooth non-trivial S^1-action if and only if it is diffeomorphic to the complex projective space or the quadric. We also …
We derive various inequalities involving the intersection number of the curves contained in geodesics and tight geodesics in the curve graph. While there already exist such inequalities on tight geodesics, our method applies in the setting of geodesics. Furthermore, the method gives inequalities with a uniform constant…
Study curves' intersections and distances, with applications in graph and group studies.
problem Understanding intersections and distances of curves.
method Using a relationship between intersection numbers and subsurface projection distances, applications in curve graphs and mapping class groups.
result Explicit quasi-constants for the relationship between intersection numbers and subsurface projection distances.
Compact spacelike hypersurface with constant curvature and boundary angles must be part of a hyperboloid.
problem Characterizing compact spacelike hypersurfaces in Minkowski space with constant curvature and boundary conditions.
method Using an auxiliary function and an associated integral equality, the authors prove the rigidity of the hypersurface.
result Compact spacelike hypersurfaces with constant curvature and boundary angles are rigid, being parts of hyperboloids unless entirely in the boundary hyperplane.
SL(3,Z) contains subgroups whose intersection is not finitely generated.
problem Identifying subgroups of SL(3,Z) whose intersection is not finitely generated.
method Explicit construction of subgroups H and K, using Schreier graph of an affine action of a free group on Z^2.
result Intersection of two 2-generated subgroups H and K in SL(3,Z) is not finitely generated.
The paper constructs homology spheres with large correction terms.
problem Finding homology spheres with large correction terms.
method Constructing families of homology spheres that bound 4-manifolds with intersection forms isomorphic to -E8.
result Homology spheres have arbitrarily large correction terms.
Composite fluxbrane and S-brane solutions for a wide class of intersection rules are considered. These solutions are defined on a product manifold R_{*} x M_1 x ... x M_n which contains n Ricci-flat spaces M_1, ..., M_n with 1-dimensional factor spaces R_{*} and M_1. They are determined up to a set of functions obeying…
The paper proves embedding conditions for complexes in manifolds using matrix rank criteria.
problem Embedding k-dimensional simplicial complexes into (k−1)-connected PL manifolds. method Proves embedding conditions using a skew-symmetric matrix with low rank over Q. result Embedding conditions for k-complexes in 2k-manifolds are equivalent to low-rank matrix conditions. In this paper we consider the topological side of a problem which is the analogue of Sen's S-duality testing conjecture for Hitchin's moduli space of rank 2 stable Higgs bundles of fixed determinant of odd degree over a Riemann surface. We prove that all intersection numbers in the compactly supported cohomology vanish…
The paper calculates area Siegel--Veech constants for specific submanifolds of REL zero.
problem Calculating area Siegel--Veech constants for affine invariant submanifolds of REL zero.
method Using volumes of the principal boundary strata and intersection theory.
result Proves a conjectural formula for the area Siegel--Veech constant in the case of REL zero.
For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting, we prove the existence of a canonical and functorial "intersection" connection on the Deligne pairing. A relationship is…
A homology stratification is a filtered space with local homology groups constant on strata. Despite being used by Goresky and MacPherson [Intersection homology theory: II, Inventiones Mathematicae, 71 (1983) 77-129] in their proof of topological invariance of intersection homology, homology stratifications do not appe…
The paper calculates large genus limits for quadratic differential volumes and constants.
problem Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials.
method Combining recursive relations (Virasoro constraints) and asymmetric simple random walk jump probabilities.
result Confirm predictions about Masur-Veech volumes and area Siegel-Veech constants.
Computes constants for specific geometric structures.
problem Calculating constants for specific geometric structures.
method Analyzes saddle connections and Prym eigenforms.
result Computed Siegel-Veech constants for real quadratic orders.
We show that the subgroup of the knot concordance group generated by links of isolated complex singularities intersects the subgroup of algebraically slice knots in an infinite rank subgroup.
New examples of Howson groups that are not strongly Howson found.
problem Understanding the difference between Howson and strongly Howson groups.
method Constructing specific examples of groups to demonstrate the distinction.
result First examples of Howson groups that are not strongly Howson.
We consider the problem of existence of constant scalar curvature Kaehler metrics on complete intersections of sections of vector bundles. In particular we give general formulas relating the Futaki invariant of such a manifold to the weight of sections defining it and to the Futaki invariant of the ambient manifold. As…
Solves geometric Cauchy problem for submanifolds with constant rank.
problem Finding submanifolds with constant rank in a given distribution.
method Constructive approach to solve the geometric Cauchy problem.
result A solution exists and is unique in a neighborhood of the submanifold.
Upper bounds for Steklov eigenvalues derived from intersection indices.
problem Finding upper bounds for Steklov eigenvalues of submanifolds in Euclidean space.
method Using intersection indices of submanifolds and their boundaries.
result Explicit upper bounds involving intersection index, volume, and dimensional constants.
Let R be an algebraic curvature tensor for a non-degenerate inner product of signature(p,q) where q>4. If π is a spacelike 2 plane, let R(π) be the associated skew-symmetric curvature operator. We classify the algebraic curvature tensors so R(-) has constant rank 2 and show these are geometrically realizable by hyp…
New invariant real rank identifies constant real Lie algebroids.
problem Characterizing complex Lie algebroids with constant real rank.
method Introducing real rank and minimal complex subalgebroid.
result Local splitting and characterization of complex Lie algebroids.
Formula for Masur-Veech volumes in quadratic differentials with odd zeros.
problem Calculating volumes of specific quadratic differential strata.
method Intersection theory, topological recursion, Hodge integrals.
result Conjectural formula for volumes proved for odd zero orders.
Using the thermodynamics formalism, we introduce a notion of intersection for projective Anosov representations, show analyticity results for the intersection and the entropy, and rigidity results for the intersection. We use the renormalized intersection to produce a Out(Γ)-invariant Riemannian metric on the smooth …
The paper proves constant rank theorems for special Lagrangian equations.
problem Understanding saddle solutions and Liouville type results for special Lagrangian equations.
method Argument based on saddle solutions and Liouville type results for the special Lagrangian equation.
result Obtained constant rank theorems for saddle solutions to the special Lagrangian equation and the quadratic Hessian equation.
We introduce a singular chain intersection homology theory which generalizes that of King and which agrees with the Deligne sheaf intersection homology of Goresky and MacPherson on any topological stratified pseudomanifold, compact or not, with constant or local coefficients, and with traditional perversities or superp…
New invariant csm simplifies computing geometric invariants of recursive group orbits.
problem Computing geometric invariants of recursive group orbits is hard.
method Introduced new invariant csm and used it to compute invariants explicitly. result Explicit formulas for local Euler obstructions and sectional Euler characteristics.
New lower bounds show learning intersections of halfspaces is hard even for a few halfspaces.
problem Learning intersections of halfspaces in polynomial time under standard assumptions.
method Unified connection to parallel pancakes distribution for proving hardness.
result Learning ω(loglogN) halfspaces in dimension N requires super-polynomial time under standard assumptions. Computes intersection cohomology of moduli space of Higgs bundles on a genus 2 curve.
problem Computing the intersection cohomology of the moduli space of Higgs bundles.
method Constructs a semismall desingularization and uses the decomposition theorem to compute the cohomology.
result Proves the mixed Hodge structure on the intersection cohomology is pure.
The study calculates the Smith-Thom deficiency of Hilbert squares and provides conditions for maximality.
problem Calculating the Smith-Thom deficiency of Hilbert squares and conditions for maximality.
method Using Mayer-Vietoris mapping and rank calculations.
result Established necessary and sufficient conditions for maximality of Hilbert squares in projective complete intersections.
The homology cobordism group of homology cylinders is a generalization of the mapping class group and the string link concordance group. We study this group and its filtrations by subgroups by developing new homomorphisms. First, we define extended Milnor invariants by combining the ideas of Milnor's link invariants an…
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.
Researchers introduce new energies to study constant scalar curvature metrics.
problem Understanding constant scalar curvature metrics on compact Kähler manifolds.
method Introduced a family of Kβ energies using Berman's quantization and intersection theory. Combined with non-Archimedean techniques, provided a uniform Yau-Tian-Donaldson correspondence. result Uniform Yau-Tian-Donaldson correspondence characterizes the existence of a unique constant scalar curvature Kähler metric.
The modified J-flow with Calabi ansatz shows convergence or blow-up behavior based on topological constants.
problem Analyzing the behavior of the modified J-flow with Calabi ansatz.
method Using the Calabi symmetry and studying the singularities of the flow.
result The modified J-flow with Calabi ansatz converges to a solution away from a variety, and blows up along the variety.
For the free group FN of finite rank N≥2 we construct a canonical Bonahon-type continuous and Out(FN)-invariant \emph{geometric intersection form} \[ <, >: \bar{cv}(F_N)\times Curr(F_N)\to \mathbb R_{\ge 0}. \] Here cvˉ(FN) is the closure of unprojectivized Culler-Vogtmann's Outer space cv(FN)…
New fault-tolerant quantum gates for homological LDPC codes with constant or almost-constant rate.
problem Fault-tolerant quantum computing for homological LDPC codes with constant or almost-constant encoding rate.
method Derive generic formula for transversal and logical gates acting on 3-manifolds, using higher symmetries and cup product cohomology.
result Parallelizable logical gates for homological LDPC codes with constant or almost-constant rate.
Quadratic growth of intersecting curves on surfaces resolved.
problem Understanding the largest size of intersecting simple closed curves on surfaces.
method Introduced almost nibs, flowers, and stem systems to analyze curve intersections.
result The size of intersecting curves grows quadratically with the surface's Euler characteristic.
Study the intersection of a hyperplane with a lightcone in Minkowski spacetime.
problem Understanding the anisotropic criterion for formation of trapped surfaces in vacuum.
method Investigated the intrinsic and extrinsic geometry of the intersection of a hyperplane with a lightcone in Minkowski spacetime.
result Found that the intersection has constant positive, zero, or negative Gaussian curvature depending on the hyperplane's type.
Research provides obstructions to Stein fillings of certain rational homology spheres.
problem Obstructing Stein fillings of specific rational homology spheres.
method Uses Pin(2)-monopole Floer homology.
result Provides obstructions to the intersection forms of Stein fillings.
We extend asymptotic formulas for saddle connections on translation surfaces.
problem Counting saddle connections on translation surfaces with large genus.
method Recursive formulas and asymptotic analysis for all strata and multiplicities.
result Asymptotics for all saddle connections on translation surfaces of growing genus.