The paper proves rigidity of Seiberg-Witten invariants for spin 4-manifold families.
problem Rigidity of Seiberg-Witten invariants for spin 4-manifold families.
method Mechanism of rigidity theorem and 10/8-type inequality.
result Existence of non-smoothable topological families of 4-manifolds.
New families of embeddings in 4-manifolds, topologically trivial but smoothly non-trivial.
problem Constructing non-trivial smooth embeddings of 3-manifolds in 4-manifolds.
method Parameterized families of embeddings, using high-dimensional spheres.
result Embeddings of homology spheres and any 3-manifold in blown-up K3 surfaces.
Paper constructs continuous families of topological Morse functions.
problem Existence and deformability of topological Morse functions.
method Simple construction of continuous families of topological Morse functions.
result Gives a construction of continuous families of topological Morse functions.
This paper identifies all topological symmetry groups for Heawood family graphs.
problem Understanding symmetries of spatial graphs in 3D space.
method Analyzing automorphisms of graphs embedded in S3. result All graphs in the Heawood family are intrinsically chiral.
The paper explores the topology and curvature of isoparametric families in spheres.
problem Investigating the topology and curvature of isoparametric families in spheres.
method The paper investigates the topology and curvature of isoparametric families in spheres using homotopy, homeomorphism, diffeomorphism types, parallelizability, and Lusternik-Schnirelmann category.
result The paper determines conditions for non-negative sectional curvatures and positive Ricci curvatures in isoparametric families.
This paper classifies topological symmetry groups for Petersen family graphs.
problem Understanding symmetries of graphs embedded in 3D space.
method Examined all embeddings of Petersen family graphs in S3 and classified their topological symmetry groups. result Identified all possible groups that can be realized as topological symmetry groups for each graph in the Petersen family.
Study on Alexander polynomials in braids, linking number theory and topology.
problem Distribution of Alexander polynomials in specific families of braids.
method Exploration of arithmetic invariants and analogies with number theory.
result New directions in arithmetic topology and statistics.
Teaches Dirac operators for geometry and topology.
problem Interactions between geometry and topology.
method Families of Dirac operators and index theorems.
result Applications to metrics of positive scalar curvature and the three-dimensional Weinstein conjecture.
We show that the topological classification and the smooth classification are generically the same for certain families of plane curves in a semi-local case(the double local case). Especially we give the normal form of transversely jointed two families of plane curves with second order contact at the envelope.
Paper proves families of singularities can be topologically trivialized.
problem Understanding behavior of singularities under small perturbations.
method Establishes sufficient conditions for embedded topological trivialization.
result New instances of topological stability, including μ-constant deformations. Gerbes encode spectral gaps in topological insulators.
problem Capturing spectral gaps in topological insulators.
method Geometric encoding through 'Real' gerbes.
result Gerbes precisely capture spectral gaps.
New examples of degenerating metrics on R^4 found.
problem Degenerating Poincaré-Einstein metrics on R^4.
method Riemannian ansatz of Debever and Deimaniański, constructing families.
result Continuous families of metrics with cusps and unexpected degenerations.
New communication topologies improve deep reinforcement learning efficiency.
problem Optimizing communication topology for faster and more robust learning in deep reinforcement learning.
method Introduced alternative network topologies (Erdos-Renyi random graphs) and compared their performance with fully-connected and star topologies.
result Erdos-Renyi random graphs outperform fully-connected networks in deep reinforcement learning tasks.
In this article we determine, for an infinite family of maps on the plane, the topology of the surface on which the minimal regular covering occurs. This infinite family includes all Archimedean maps.
This work generalizes a formula linking Seiberg-Witten prepotential and topological recursion.
problem Analyzing the relationship between Seiberg-Witten curves and topological recursion.
method Analytical approach using Seiberg-Witten family of curves.
result A generalized formula relating Seiberg-Witten prepotential to the genus zero part of topological recursion on a Seiberg-Witten curve.
We construct an infinite family of topologically slice 2--component boundary links ℓi, none of which is smoothly concordant to a split link, such that g4(ℓi)=i.
This paper proposes an axiomatic for Cyclic Foam Topological Field theories. That is Topological Field theories, corresponding to String theories, where particles are arbitrary graphs. World surfaces in this case are two-manifolds with one-dimensional singularities. We proved that Cyclic Foam Topological Field theories…
Study of section conjecture analogues over complex numbers and Kodaira fibrations.
problem Investigate Grothendieck's section conjecture over complex numbers and Kodaira fibrations.
method Topological and Hodge theoretic analogues of the section conjecture over complex numbers, studied in the context of Kodaira fibrations and families of Jacobians.
result Both topological and Hodge-theoretic analogues of the injectivity part of the section conjecture hold for families of curves, but the topological analogue of the surjectivity part does not hold in general.
Constructing exponential families from statistical manifolds.
problem The central problem of constructing exponential families from statistical manifolds.
method Constructive approach proving every compact statistical manifold admits a foliation of Hessian manifolds.
result Compact orientable leaves are either finite quotients of flat torus or mapping torus with periodic monodromy.
The following numerical control over the topological equivalence is proved: two complex polynomials in n=3 variables and with isolated singularities are topologically equivalent if one deforms into the other by a continuous family of polynomial functions fs:Cn→C with isolated sin…
Extends topological theorems to new mapping families.
problem Limitations of existing topological theorems.
method Replaces large subset requirements with smaller subset requirements.
result Extends theorems to m-sphere to n-space mappings with m<n.
Study shows infinite kernels in topological monodromy for curve families.
problem Understanding kernels of topological monodromy representations.
method Extending Kuno's arguments and using Carlson-Toledo techniques.
result Kernels are infinite for certain linear systems on surfaces.
Optimal curves minimize crossings on surfaces.
problem Minimizing crossings on congruence surfaces.
method Examining systoles and their intersections.
result Modular systoles have minimal crossing numbers.
Study of projective Fraïssé limits of trees with confluent epimorphisms.
problem Finite trees with monotone epimorphisms do not amalgamate.
method Developed new mappings and properties of continua to apply to topological graphs.
result Topological realization of the Fraïssé limit of finite trees with ramification vertices of order at most 3 is the Wa\. zewski dendrite D3. We classify topologically trivial Legendrian Θ-graphs and identify the complete family of nondestabilizeable Legendrian realizations in this topological class. In contrast to all known results for Legendrian knots, this is an infinite family of graphs. We also show that any planar graph that contains a subdivision of…
In the paper we formulate and derive the family blowup formula of family Seiberg-Witten invariants. The formula has been used in the enumerative application of counting singular curves on algebraic surfaces. We first give a topological derivation of the formula by using family index theorem. Then we define the algebrai…
For fixed large genus, we construct families of complete immersed minimal surfaces in R3 with four ends and dihedral symmetries. The families exist for all large genus and at an appropriate scale degenerate to the plane.
Consider two families of closed oriented curves in a d-manifold. At each point of intersecction of a curve of one family with a curve of the other family, form a new closed curve by going around the first curve and then going around the second. Typically, an i-dimensional family and a j-dimensional family will produce …
New proofs of h-principles in contact 3-manifolds.
problem Characterizing contractible bypasses in contact 3-manifolds.
method Topological characterization and disjoint bypass construction.
result New proofs of h-principles in overtwisted contact 3-manifolds.
Study vineyards linking TDA and knot theory, showing rich topological features.
problem Linking computational topology and knot theory through vineyards.
method Construct periodic functions to study evolution of persistence diagrams.
result Vineyards are as rich as possible in topological terms.
Using the same method we provide negative answers to the following questions: Is it possible to find real equations for complex polynomials in two variables up to topological equivalence (Lee Rudolph) ? Can two topologically equivalent polynomials be connected by a continuous family of topologically equivalent polynomi…
The paper proves the existence of a continuous family of translating surfaces under a specific curvature flow.
problem Existence of convex translating surfaces under flow by α-th power of Gauss curvature. method Constructing a family of translating surfaces using Jacobi fields and analyzing their effective growth rates.
result The family of translating surfaces is a topological manifold with quantitative convergence rates.
Study shows certain toric arrangements have minimal topological complements.
problem Understanding the topology of toric arrangements.
method Associated a matrix to toric arrangements and analyzed those with maximal rank.
result Complement manifolds of these toric arrangements are diffeomorphic to centered ones.
Topology classifies bipolar surfaces; they are not embedded.
problem Classifying bipolar surfaces in the 5-sphere.
method Topological classification and area bounds calculation.
result Bipolar surfaces are not embedded.
New topologies for star-shaped sets without boundedness.
problem Defining convergence for unbounded star-shaped sets.
method Introducing radial distance functionals and new topologies.
result New topologies τWr and τAWr for star-shaped sets. Develops a categorified excision principle for elliptic symbol families.
problem Expressing the index class in K-theory using differential-topological data.
method Categorical index calculus, excision principle.
result Allows comparison of categorified index problems on different manifolds.
This thesis explores bifoliated planes and their group actions from Anosov flows.
problem Understanding bifoliated planes and their group actions from Anosov flows.
method Analyzes bifoliated planes associated with Anosov flows and describes their properties.
result Shows left-orderability of groups acting on bifoliated planes and describes bifoliated planes from specific families of Anosov flows.
A new model learns networked data with topology and features.
problem Learning from networked data with topology and features.
method Networked exponential families and network Lasso for efficient learning.
result Efficiently learns networked data with topology and features.
Recently the first author studied the bifurcation of critical points of families of functionals on a Hilbert space, which are parametrised by a compact and orientable manifold having a non-vanishing first integral cohomology group. We improve this result in two directions: topologically and analytically. From the analy…
CIFs replace single bijections with continuous families to avoid topological limitations.
problem Normalising flows struggle with targets with complex topologies.
method Propose Continuously Indexed Flows (CIFs) replacing single bijections with a continuous family.
result CIFs avoid topological limitations and perform better empirically.
New puzzles from geometry and topology.
problem Exploring puzzles from geometric and topological concepts.
method Construction from square-tiled shapes, discussion of underlying mathematics.
result Puzzles naturally associated to puzzle spaces.
The study proves limitations on isospectral hyperbolic surfaces with discrete length spectra.
problem Characterizing isospectral hyperbolic surfaces with discrete length spectra.
method Utilizing Sunada's method and topological self-duplicating ends, the study explores isospectral families and their cardinality.
result Finite groups can be realized as full isometry groups of hyperbolic structures with discrete spectrum on surfaces with self-duplicating ends.
We present a new, far simpler family of counter-examples to Kushnirenko's Conjecture. Along the way, we illustrate a computer-assisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the nature of optimal upper bounds in real fewnomial theory. We use a powerful recen…
3D topological models link to HOMFLYPT homology via braids.
problem Understanding topological invariants of links and braids.
method 3D topological B-models with Hilbert schemes of points.
result Hilbert space of braids corresponds to HOMFLYPT homology.
The study finds a way to create minimal surfaces with specific shapes in 3-manifolds.
problem Creating minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature.
method Develops a min-max theory and shows deformability of surfaces in a generic metric.
result Establishes a theorem for producing minimal surfaces with prescribed genus.
Generically, topological insulators have conical points leading to Dirac-like currents.
problem Understanding the conical structure of degeneracies in topological phases of matter.
method Analyzing Hermitian matrices with three parameters to show conical points.
result Adiabatic deformations of topological insulators result in Dirac-like currents whose total conductivity equals the chiral number of conical points.
In this paper we exhibit infinite families of embedded tori in 4-manifolds that are topologically isotopic but smoothly distinct. The interesting thing about these tori is that they are topologically trivial in the sense that each bounds a topologically embedded solid handlebody. This implies that there are stably ribb…
Constructs exotic proper 2-knots from open 2-handles.
problem Proving topological equivalence of exotic proper knotted surfaces.
method Flexible construction of exotic open 2-handles, distinguishing through genus functions and end Floer homology.
result Constructs infinite families of irreducible exotic proper knotted surfaces.