Cactus doodles are geometric objects derived from cactus groups.
problem Understanding geometric structures related to algebraic groups.
method Defining cactus doodles via local moves on plane curves and relating them to cactus groups.
result Established properties of cactus doodles and their connection to cactus groups.
Affine cactus groups are CAT(0) and hyperbolic.
problem Characterizing geometric properties of affine cactus groups.
method Analyzing CAT(0) and hyperbolic properties through group theory.
result Affine cactus groups of degree three are hyperbolic.
Classifies doodles into prime and super prime types, describing them with doodle codes.
problem Classifying doodles into prime and super prime types.
method Using doodle codes to describe complementary regions and enumerate doodle diagrams.
result Super prime doodles have a Hamiltonian circuit.
Explains the pure cactus group of degree three and its relation to four points on a circle.
problem Understanding the relationship between cactus groups and configuration spaces.
method Provides an explicit description of the pure cactus group of degree three and its connection to the configuration space of four points on a circle.
result Explicitly describes the relationship between the pure cactus group of degree three and the configuration space of four points on the circle.
Alexander invariant created for doodles, vanishes on unlinked doodles.
problem Creating an Alexander type invariant for doodles.
method Deformation of Tits representation and Chebyshev polynomials of second kind.
result Invariant vanishes on unlinked doodles with more than one component.
Summary of pure cactus groups and circle points.
problem Understanding pure cactus groups and configuration spaces.
method Summarizes previous work on the topic.
result Summary of results from previous papers and thesis.
Doodles were introduced in [R. Fenn and P. Taylor, Introducing doodles, Topology of low-dimensional manifolds, pp. 37--43, Lecture Notes in Math., 722, Springer, Berlin, 1979] but were restricted to embedded circles in the 2-sphere. Khovanov, [M. Khovanov, Doodle groups, Trans. Amer. Math. Soc. 349 (1997), 2297--2315],…
Doodles link to commutator identities in a 2-sphere.
problem Understanding commutator identities in free groups via doodles.
method Analyzing doodles with proper noose systems and establishing bijections.
result A bijection between doodles and commutator identities.
Paper introduces skew-symmetric matrices for virtual doodle classification.
problem Classifying virtual doodles and distinguishing non-classical doodles.
method Use skew-symmetric augmented matrices and homology intersection number.
result Characterization of virtualization of classical doodles.
Complete invariant defined for doodles on a sphere.
problem Doodles on a 2-sphere.
method Coefficients in series of chord diagrams.
result Finite type invariants of order at most 2n.
Presented a simple group presentation for degree four cactus group.
problem Presented a simple group presentation for the pure cactus group of degree four.
method Action on hyperbolic plane, Dirichlet polygon construction.
result Isomorphic to the fundamental group of connected sum of five real projective planes.
We discuss Gauss codes of virtual diagrams and virtual doodles. The notion of a left canonical Gauss code is introduced and it is shown that oriented virtual doodles are uniquely presented by left canonical Gauss codes.
A virtual doodle is an equivalence class of virtual diagrams under an equivalence relation generated by flat version of classical Reidemesiter moves and virtual Reidemsiter moves such that Reidemeister moves of type 3 are forbidden. In this paper we discuss colorings of virtual diagrams using an algebra, called a doodl…
Paper defines doodles on closed surfaces, unifying classical and virtual theories.
problem Classifying doodles on closed surfaces, especially non-orientable ones.
method Introducing twisted virtual doodles, defining twin groups, and proving Alexander- and Markov-type theorems.
result Unified theory of doodles, showing trivial center and residually finite properties.
Paper shows how to represent Milnor's triple linking number using chord diagrams and doodle invariants.
problem Tackles the representation of Milnor's triple linking number.
method Establishes an analogous description for Milnor's triple linking number using counts of chord diagrams and doodle invariants.
result Shows that Milnor's triple linking number can be represented in terms of chord diagrams and doodle invariants.
Computes invariants distinguishing between immersions and embeddings of doodles and blobs on surfaces.
problem Distinguishing between immersions and embeddings of doodles and blobs on surfaces.
method Regular embeddings, bordisms, and exact sequences of abelian groups.
result Exact sequence describing bordisms of immersions and embeddings of doodles on A=RimesI. In 1997 M.~Khovanov proved that any doodle can be presented as closure of twin, this result is analogue of classical Alexander's theorem for braids and links. We give a description of twins that have equivalent closures, this theorem is analogue of classical Markov theorem.
Cactus improves auto-regressive decoding speed without sacrificing quality.
problem Accelerating auto-regressive decoding while maintaining output quality.
method Formalizes speculative sampling as constrained optimization and proposes Cactus for controlled divergence from the verifier distribution.
result Empirically validated effectiveness across various benchmarks.
DOODL learns shared spectral dynamics across related dynamical systems.
problem Learning independent dynamical operators for each system limits discovery of shared structure.
method DOODL learns a dictionary of characteristic spectral dynamics on a manifold of related systems.
result DOODL achieves errors one to two orders of magnitude lower than independent operator estimation methods.
To make music composition more approachable, we designed the first AI-powered Google Doodle, the Bach Doodle, where users can create their own melody and have it harmonized by a machine learning model Coconet (Huang et al., 2017) in the style of Bach. For users to input melodies, we designed a simplified sheet-music ba…
Graph theory criterion for Hodge theory to match linearly.
problem Matching nonlinear Hodge theory to linear on graphs.
method Minimizing edge potentials and solving nonlinear coclosed equations.
result Nonlinear selector agrees with Hodge projector on cactus graphs.
The paper classifies 10 antipodal pairings of self-dual maps.
problem Understanding the antipodal pairings of strongly involutive polyhedra.
method Classification of self-dual pairings and construction of polyhedra.
result Determination of 10 antipodal pairings among 24 self-dual pairings.
Proves Alexander and Markov theorems for higher genus virtual doodles.
problem Classifying isotopy classes of immersed circles on surfaces.
method Introduces virtual twin groups to extend twin groups and prove theorems for higher genus.
result Proves Alexander and Markov theorems for higher genus virtual doodles.
The main goal of this paper is a calculation of the integral (co)homology of the group of symmetric automorphisms of a free product. We proceed by giving a geometric interpretation of symmetric automorphisms via a moduli space of certain diagrams, which we name cactus products. To describe this moduli space a theory of…
Given a plane curve γ:S1→R2, we consider the problem of determining the minimal number I(γ) of inflections which curves $\mbox{diff}(γ)$ may have, where $\mbox{diff}$ runs over the group of diffeomorphisms of R2. We show that if γ is an immersed curve with D(γ) double points and no othe…
The BNSR-invariants of a group G are a sequence Σ1(G)⊇Σ2(G)⊇⋯ of geometric invariants that reveal important information about finiteness properties of certain subgroups of G. We consider the symmetric automorphism group ΣAutn and pure symmetric automorphism group PΣAutn of the free…
We investigate using reinforcement learning agents as generative models of images (extending arXiv:1804.01118). A generative agent controls a simulated painting environment, and is trained with rewards provided by a discriminator network simultaneously trained to assess the realism of the agent's samples, either uncond…
Let M be a closed, oriented manifold of dimension d. Let LM be the space of smooth loops in M. Chas and Sullivan recently defined a product on the homology H∗(LM) of degree −d. They then investigated other structure that this product induces, including a Batalin -Vilkovisky structure, and a Lie algebra str…
This paper explores Brunnian twin groups and their properties.
problem Alexander-Markov correspondence for isotopy classes of immersed circles.
method Diagrammatic representations and combinatorial structures.
result Brunnian twin groups are free groups when more than two strands are involved.
It was proven by González-Meneses, Manchón and Silvero that the extreme Khovanov homology of a link diagram is isomorphic to the reduced (co)homology of the independence simplicial complex obtained from a bipartite circle graph constructed from the diagram. In this paper we conjecture that this simplicial complex is al…
The paper solves isomonodromy problems and describes limits of Stokes matrices.
problem Solving isomonodromy problems and describing limits of Stokes matrices.
method Analyzes the boundary and monodromy problems of isomonodromy equations, derives explicit expressions for Stokes matrices, and describes limits of Stokes matrices as irregular data degenerates.
result Derives explicit expressions for Stokes matrices and describes limits of Stokes matrices as irregular data degenerates.