Quandle identities give rise to subcomplexes in homology.
problem Understanding the relationship between quandle identities and their homological counterparts.
method Examining subcomplexes constructed from quandle identities and their invariance under Reidemeister moves.
result Quandle identities give rise to 2-cycles in homology, and these cycles can be extended to form abelian extensions.
We prove that tangent cones to 2-dimensional calibrated cycles are unique. Using this result we prove a rate of convergence for the mass of the blow-up of a calibrated integral 2-cycle towards the limiting density. With the same techniques, we can also prove such a rate for J-holomorphic maps between almost complex man…
We consider any pseudo holomorphic integral 2-cycle in an arbitrary almost complex manifold and perform a blow up analysis at an arbitrary point. Building upon a pseudo algebraic blow up (previously introduced by the author) we prove a geometric rate of decay for the mass ratio towards the limiting density, with an exp…
We explore a relationship between topological properties of orbits of 2-cycles in the symplectomorphism group Symp(M) and the existence of rational curves in M. Under the absence of rational curves hypothesis, we show that evaluation map vanishies on the second homotopy group and obtain a Gottlieb-type vanishing theore…
Constructs special Lagrangian 3-spheres in non-Kähler compact threefolds.
problem Understanding transitions between Kähler and non-Kähler geometries.
method Analyzes topological transitions of Calabi-Yau threefolds to construct special Lagrangian cycles.
result Special Lagrangian 3-spheres emerge from non-Kähler geometries, exchanging holomorphic 2-cycles for 3-cycles.
We introduce an invariant linked to some foundational questions in geometric measure theory and provide bounds on this invariant by decomposing an arbitrary cycle into uniformly rectifiable pieces. Our invariant measures the difficulty of cutting a nonorientable closed manifold or mod-2 cycle in Rn into ori…
New infinite family of 2-complexes intrinsically linked in 4D.
problem Intrinsic linking of 2-complexes in 4D.
method Examining suspensions of graphs containing K6 as a minor.
result Embeddings of suspensions contain intrinsically linked cycles.
Novikov's problem of semiclassical orbits of quasi-electrons in a normal metal leads to a correspondance between 3-ply periodic functions in R and fractals in R P^2. These fractals are the complement of infinitely many open sets labeled by integer 2-cycles of T^3. Here we present a characterization of the fractal point…
Positivity of intersections in 4-manifolds leads to taming symplectic structures.
problem Taming symplectic structures in almost complex 4-manifolds.
method Proof of positivity of intersections of pseudoholomorphic curves.
result Positivity of intersections is stable and leads to taming symplectic structures.
Study 2-complexes' homology properties and torsion growth.
problem Quantitative connections between 1-cycle filling inequalities and homology complexities.
method Geometric lower bounds on first homology of finite covers.
result Geometric lower bound on first homology size of finite covers.
We consider the classical problem of a position of n-dimensional manifold M in R^{n+2}. We show that we can define the fundamental (n+1)-cycle and the shadow fundamental (n+2)-cycle for a fundamental quandle of a knotting M to R^{n+2}. In particular, we show that for any fixed quandle, quandle coloring, and shadow quan…
The study confirms Gromov's speculation and provides bounds for taming symplectic structures.
problem Understanding the relationship between taming symplectic structures and the area of pseudoholomorphic curves.
method Analyzes the numerical cone of taming symplectic structures and characterizes coarsely holomorphic curves.
result An almost complex manifold with an area bound admits a taming symplectic structure, confirming Gromov's speculation.
Paper studies Iwasawa invariants for 3-manifolds, proving a formula similar to Kida's.
problem Analogizing Iwasawa invariants to 3-dimensional topology.
method Using p-adic representations of a finite group and parallel to Iwasawa's second proof. result Proves an analogue of Kida's formula for λ-invariants in p-extensions of Zp-fields for 3-manifolds. Formulates quantum jet bundles over noncommutative algebras with connections and braiding.
problem Defining jet bundles over noncommutative algebras with connections and braiding.
method Formalizes jet bundles over noncommutative algebras with flat connections and braiding tensor obeying Yang-Baxter equation.
result Examples include permutation groups, matrix algebras, and quantum spacetime models.
The paper analyzes flows related to Higgs energies on manifolds.
problem Analyzing flows related to Higgs energies on manifolds.
method Developing asymptotic analysis for gradient flow of self-dual U(1)-Higgs energies. result Solutions converge to codimension-two mean curvature flows.
The abstract constructs Frobenius manifolds from stability conditions on quiver categories.
problem Constructing Frobenius manifolds from stability conditions on quiver categories.
method Using the invariants counting semistable objects in the 3CY triangulated category D(Q), a family of semisimple Frobenius manifold structures is constructed. result The construction yields different branches of Frobenius manifolds for each mutation of the quiver.
Solves a triangulation problem by showing minimum tetrahedra equals minimum integral 3-chain.
problem Finding the minimum number of tetrahedra to extend a triangulation of a 2-sphere to a 3-ball.
method Relates the minimum number of tetrahedra to the minimum integral 3-chain norm, proving them equal and showing how to achieve the minimum.
result The minimum number of tetrahedra needed to extend a triangulation of a 2-sphere to a 3-ball equals the minimum integral 3-chain norm.
Let M3 be an oriented 3-manifold. We investigate when one of the fibers or a combination of fiber components, Fbest, of a \emph{harmonic} map f:M3→S1 with Morse-type singularities delivers the Thurston norm χ−([Fbest]) of its homology class [Fbest]∈H2(M3;Z). In particular, for a map …
The paper studies minimal submanifolds from the abelian Higgs model, proving convergence of energy measures and currents.
problem Existence and properties of minimal submanifolds from the abelian Higgs model.
method Analyzes rescalings of the self-dual Yang-Mills-Higgs energy, showing convergence of energy measures and currents.
result Provides a variational construction of nontrivial critical points and proves the existence of stationary integral (n-2)-varifolds.
Modeling bank leverage dynamics using dynamical systems and neural networks.
problem Understanding leverage dynamics in financial systems.
method Dynamical systems, deep neural networks, adaptive expectation scheme.
result Chaotic behavior in leverage dynamics for a significant fraction of banks.
New algorithms deform and contract discrete manifolds into spheres.
problem Deforming and contracting discrete manifolds into spheres.
method Use triangulation techniques to clarify algorithms for PL complexes.
result Proves theorem for simply-connected closed 3-manifolds.
The paper shows how Yang-Mills-Higgs energies converge to the (n−2)-area functional.
problem Understanding the convergence of Yang-Mills-Higgs energies to the (n−2)-area functional. method Analyzing the convergence of critical points of Yang-Mills-Higgs energies to minimal submanifolds and proving Γ-convergence. result Yang-Mills-Higgs energies converge to the (n−2)-area functional as εo0.