Hamilton proved a theorem about focusing light rays, leading to new mathematical concepts.
problem Focusing light rays to a single point using optical instruments.
method Original proof and symplectic geometry proof of the Malus-Dupin theorem.
result A family of light rays can be focused to a single point if they are rectangular before entering the instrument.
Geometrization Theorem solves complex geometry problems.
problem Complex geometry problems in differential geometry.
method Based on Hamilton's program, proved by Grigory Perelman.
result Generalized Poincaré's Conjecture.
Hamiltonian cycles found in toroidal maps.
problem Hamiltonicity of doubly semi-equivelar maps on the torus.
method Analyzing 2-uniform tilings of the plane to derive Hamiltonian cycles.
result Every doubly semi-equivelar map on the torus contains a Hamiltonian cycle.
Characterizes diagrams achieving Morton-Franks-Williams inequality for positive knots and links.
problem Understanding when the Morton-Franks-Williams inequality holds for positive knots and links.
method Combinatorial characterisation and generating examples.
result Examples of diagrams achieving crossing number, braid index, and maximal self-linking number.
Using alternating Heegaard diagrams, we construct some 3-manifolds which admit diffeomorphisms such that the non-wandering sets of the diffeomorphisms are composed of Smale-Williams solenoid attractors and repellers, an interesting example is the truncated-cube space. In addition, we prove that if the nonwandering set …
This paper classifies expanding attractors and non-transitive Anosov flows on specific knot and manifold spaces.
problem Classifying expanding attractors and non-transitive Anosov flows on specific knot and manifold spaces.
method Using the derived Anosov (DA) expanding attractor and the Franks-Williams manifold, the paper proves the uniqueness of these structures.
result The DA expanding attractor and the Franks-Williams non-transitive Anosov flow are the unique structures supported by N0 and M0 respectively. We give conditions on a knot on which the Morton-Franks-Williams inequality is not sharp. As applications, we show infinitely many examples of knots where the inequality is not sharp and also prove (by giving examples) that the deficit of the inequality can be arbitrarily large.
Hierarchical clustering uses OWA operators to generalize linkage methods and avoid dendrogram inversions.
problem Avoiding unaesthetic inversions in hierarchical clustering dendrograms.
method OWA-based linkages combined with the Lance-Williams formula and conditions on weight generators.
result Conditions for weight generators to produce dendrograms without inversions.
A metric is provided for the space of submanifolds.
problem Defining a metric for the space of submanifolds of Rn. method Using the scanning map of a topological sheaf on manifolds, an explicit metric is given. Another metric is derived using the Hausdorff distance.
result An explicit metric is provided for the space of submanifolds of Rn. We generalize the Morton-Franks-Williams inequality to the colored sl(N) link homology defined in arXiv:0907.0695, which gives infinitely many new bounds for the braid index and the self linking number. A key ingredient of our proof is a composition product for the general MOY graph polynomial, which gener…
We study the Morton-Franks-Williams inequality for closures of simple braids (also known as positive permutation braids). This allows to prove, in a simple way, that the set of simple braids is a orthonormal basis for the inner product of the Hecke algebra of the braid group defined by Kálmán, who first obtained this r…
Short note on braid index and quasipositivity of certain pretzel knots.
problem Calculating braid index and identifying quasipositive status for specific pretzel knots.
method Used Morton-Franks-Williams inequalities and Khovanov-Rozansky concordance homomorphisms.
result Determined braid index and identified quasipositivity for knots with even crossings in one strand.
Given a TQFT in dimension d+1, and an infinite cyclic covering of a closed (d+1)-dimensional manifold M, we define an invariant taking values in a strong shift equivalence class of matrices. The notion of strong shift equivalence originated in R. Williams' work in symbolic dynamics. The Turaev-Viro module associated to…
Thurston's influence on French math traced and problems solved.
problem Major problems in French math traced back to Thurston.
method Overview and survey of Thurston's influence and results.
result French math problems rooted in Thurston's work.
New minimal link diagrams found, including torus links and homogeneous ones.
problem Finding minimal link diagrams with new classes.
method Morton-Franks-Williams inequality approach.
result New classes of minimal link diagrams, including previously unproven ones.
We find all Heegaard diagrams with the property "alternating" or "weakly alternating" on a genus two orientable closed surface. Using these diagrams we give infinitely many genus two 3--manifolds, each admits an automorphism whose non-wondering set consists of two Williams solenoids, one attractor and one repeller. The…
We give an overview over the higher torsion invariants of Bismut-Lott, Igusa-Klein and Dwyer-Weiss-Williams, including some more or less recent developments.
Computes homotopy groups of diffeomorphism spaces for high-dimensional manifolds.
problem Determining homotopy groups of diffeomorphism spaces for high-dimensional manifolds.
method Computes rational homotopy groups of classifying spaces of diffeomorphisms.
result Determines rational pseudoisotopy stable range for compact spin manifolds.
This is a chapter that is to appear in the "Handbook of Knot Theory", edited by William W. Menasco and Morwen B. Thistlethwaite.
Study parametrized cobordism categories for smooth bundles, proving index theorem.
problem Characterize bivariant transformations in cobordism categories.
method Define parametrized cobordism categories and use bivariant theories.
result Prove Dwyer-Weiss-Williams index theorem for parametrized A-theory.
Abstract: From complex financial models to simpler Hamilton-Jacobi equations.
problem Complex financial models for multi-dimensional Black-Scholes.
method Linked Hamilton-Jacobi equations to simplify financial models.
result Simplified financial models using Hamilton-Jacobi equations.
Semi-Equivelar maps are generalizations of Archimedean solids to the surfaces other than 2-sphere. There are eight semi-equivelar maps of types {33,42}, {32,4,3,4}, {6,3,6,3}, {34,6}, {4,82}, {3,122}, {4,6,12}, {6,4,3,4} exist on the torus. In this article we show the e…
Experimentally identified 22 L-space knots with tunnel number >1, some having high genus and braid index.
problem Identifying L-space knots with tunnel number greater than 1.
method Cataloging hyperbolic manifolds, using SnapPy and KLO to find knot presentations as closures of positive braids.
result Found 9 asymmetric L-space knot complements with tunnel number 2, and 22 with tunnel number 1.
We show that 3-braid links with given (non-zero) Alexander or Jones polynomial are finitely many, and can be effectively determined. We classify among closed 3-braids strongly quasipositive and fibered ones, and show that 3-braid links have a unique incompressible Seifert surface. We also classify the positive braid wo…
Paper proves Hamilton's pinching theorem using mean curvature flow.
problem Hamilton's pinching theorem in extrinsic geometry.
method Mean curvature flow approach.
result Proof of Hamilton's pinching theorem.
The paper establishes a new pseudoisotopy result for embedding spaces, leading to computations of homotopy groups of long knots.
problem Computing homotopy groups of spaces of long knots in high codimension.
method Using pseudoisotopy results and algebraic K-theory, the paper describes the difference in homotopy types of block and ordinary embeddings of a codimension at least three embedding.
result The homotopy type of spaces of long knots of codimension at least 3 is determined explicitly, including torsion information.
Proves Hamilton's theorem using mean curvature flow.
problem Compactness of pinched hypersurfaces with bounded curvature.
method Mean curvature flow to prove Hamilton's theorem.
result Rigorous proof of Hamilton's theorem.
Geometric models for Lie--Hamilton systems on \(\mathbb{R}^2\) are described.
problem Analyzing Lie--Hamilton systems on \(\mathbb{R}^2\).
method Two geometric models: 1) restriction to symplectic leaves, 2) projection onto quotient space.
result Natural framework for Lie--Hamilton systems on \(\mathbb{R}^2\).
Homological stability proved for handlebody mapping class groups.
problem Homological stability for handlebody mapping class groups.
method Categorical framework developed by Randal-Williams and Wahl, allowing for any number of marked discs and boundary points.
result Homology of handlebody groups stabilizes with respect to genus and number of marked discs for all finite degree coefficient systems.
We give criteria for an invariant of lens space links to bound the maximal self-linking number in certain tight contact lens spaces. As a corollary we extend the Franks-Williams-Morton inequality to the setting of lens spaces.
Reduction theory has played a major role in the study of Hamiltonian systems. On the other hand, the Hamilton-Jacobi theory is one of the main tools to integrate the dynamics of certain Hamiltonian problems and a topic of research on its own. Moreover, the construction of several symplectic integrators rely on approxim…
Diffieties formalize geometrically the concept of differential equations. We introduce and study Hamilton-Jacobi diffieties. They are finite dimensional subdiffieties of a given diffiety and appear to play a special role in the field theoretic version of the geometric Hamilton-Jacobi theory.
Survey reviews Hamilton-Jacobi theory in various geometric settings, focusing on Jacobi and Leibniz identities.
problem Analyzing Hamilton-Jacobi theory across different geometric backgrounds.
method Geometric review of Hamilton-Jacobi theory, focusing on Jacobi and Leibniz identities.
result Novel Hamilton-Jacobi equation for conformal Hamiltonian vector fields.
In this paper we develop a Hamilton-Jacobi theory in the setting of almost Poisson manifolds. The theory extends the classical Hamilton-Jacobi theory and can be also applied to very general situations including nonholonomic mechanical systems and time dependent systems with external forces.
Develops Lagrange-Hamilton geometry for COVID-19 disease dynamics.
problem Modeling the spread of COVID-19 disease.
method Least squares variational method, nonlinear connections, d-torsions, Lagrangian Yang-Mills.
result Jacobi stability of the dynamical system.
New graph Hamiltonicity via cohomology of Artin groups.
problem Characterizing Hamiltonicity in graphs using cohomology.
method Defining a new graph from matrices and analyzing cohomology.
result New graph Hamiltonicity characterization via cohomology.
We recreate an unpublished proof of William Thurston from the early 1970's that any smooth 2-plane field on a manifold of dimension at least 4 is homotopic to the tangent plane field of a foliation.
Paper proves Hamilton-Tian conjecture using partial C0-estimate.
problem Existence of Kähler-Einstein metrics on Fano manifolds.
method Using Liu-Székelyhidi's partial C0-estimate for polarized Kähler metrics with Ricci bounded below.
result Proves Hamilton-Tian conjecture for Kähler-Ricci flow.
Study describes how compact Ricci solitons degenerate as cone angles approach zero.
problem Understanding degenerations of compact Ricci solitons as cone angles approach zero.
method Completely describes the degenerations of compact Ricci solitons, including the Gromov--Hausdorff limit of cigar solitons from conical teardrop solitons.
result Gromov--Hausdorff limit of cigar solitons from conical teardrop solitons.
Paper confirms Hamilton-Tian conjecture for specific Sasakian manifolds.
problem Hamilton-Tian conjecture for specific Sasakian manifolds.
method Sasaki-Ricci flow, compact transverse Fano Sasakian 5-manifolds, klt foliation singularities.
result Confirmed Hamilton-Tian conjecture for compact transverse Fano Sasakian 5-manifolds.
Machine learning identifies Shakespeare and Fletcher's contributions to Henry VIII.
problem Determining the relative contributions of Shakespeare and Fletcher in Henry VIII.
method Combined analysis of vocabulary and versification with machine learning techniques.
result Supports canonical division and new modifications of Henry VIII's authorship.
We describe the Williams zeta functions and the twist zeta functions of sub-Lorenz templates generated by renormalizable Lorenz maps, in terms of the corresponding zeta-functions of the sub-Lorenz templates generated by the renormalized map and by the map that determines the renormalization type.
We give examples of closed orientable graph 3-manifolds with fundamental group which is not a subgroup of GL(4,k) for any field k. This answers a question in the Kirby problem list from 1977 which is credited to the late William Thurston.
Reduces Hamilton-Jacobi theory for nonholonomic systems with symmetries.
problem Hamilton-Jacobi theory for nonholonomic systems with symmetries.
method Reduction procedure for Hamilton-Jacobi equation.
result Reconstructs solutions in the unreduced picture from reduced equations.
New theory extends Hamilton-Jacobi for contact systems, ensuring integrability.
problem Integrability of contact Hamiltonian systems.
method Developed a Hamilton-Jacobi theory for fibered phase spaces, applied to contact systems, studied HJE solutions.
result Complete pseudo-isotropic solutions ensure integrability by quadratures for contact systems.
This paper studies Hamilton-Jacobi equations for magnetic systems with constraints.
problem Understanding dynamics of magnetic systems with geometric constraints.
method Developed Hamilton-Jacobi equations for magnetic systems with nonholonomic constraints.
result Revealed relationships between magnetic structures, constraints, and dynamics.
Proves estimates for Kähler-Ricci flow solutions.
problem Positive solutions to Kähler-Ricci flow.
method Matrix Li-Yau-Hamilton estimates coupled with flow.
result Monotonicity formula derived.
The Hamilton-Jacobi equation for a Hamiltonian section on a Lie affgebroid is introduced and some examples are discussed.