Newly discovered 5 triangle-free intrinsically knotted graphs with 22 edges.
problem Identifying intrinsically knotted graphs with specific properties.
method Analyzing graphs with 22 edges, using specific graph operations and properties.
result There are exactly five triangle-free intrinsically knotted graphs with 22 edges.
The study proves conjecture for specific Artin groups.
problem Proving conjecture about Artin groups' properties.
method Analyzing Artin groups associated to triangle-free graphs and cones over square-free bipartite graphs.
result Proves conjecture for specific Artin groups.
Algorithm reconstructs triangle-free networks from data, certifying correctness.
problem Reconstructing triangle-free dynamic networks from observational data.
method Developed an algorithm for triangle-free networks, providing guarantees on correctness.
result Algorithm either certifies correctness or outputs a sparser graph with no false positives.
We consider the problem of estimating undirected triangle-free graphs of high dimensional distributions. Triangle-free graphs form a rich graph family which allows arbitrary loopy structures but 3-cliques. For inferential tractability, we propose a graphical Fermat's principle to regularize the distribution family. Suc…
Study on Coxeter groups' boundary planarity, finding exceptions.
problem Planarity of Coxeter groups' boundaries under right angles.
method Characterization of defining graphs and analysis of boundaries.
result Non-planarity of defining graphs does not always imply non-planarity of boundaries.
New groups act on cube complexes without compact cubulation.
problem Triangle-free Artin groups without compact cubulation.
method Proved proper actions on CAT(0) cube complexes.
result First examples of non-cocompactly cubulated groups.
New representations of pure braid groups defined from A2 web spaces.
problem Understanding the structure of pure braid groups.
method Defining representations of pure braid groups using A2 web spaces and calculating matrix representations. result Matrix representations of ρn about the standard generators of P2k calculated. A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and Oh, and, independently, Barsotti and Mattman, showed that K7 and the 13 graphs obtained from K7…
Estimates graph curvature and diameter using Laplacian eigenvalues.
problem Estimating graph curvature and diameter using Laplacian eigenvalues.
method Combination of gradient estimates and strong nodal domain walks.
result Li-Yau type eigenvalue-diameter estimate for signed graphs.
Groups on CAT(0) cube complexes grow exponentially uniformly.
problem Uniform exponential growth of groups acting on CAT(0) cube complexes.
method Study groups acting without global fixed points on CAT(0) square complexes.
result Groups with uniform exponential growth or stabilize Euclidean subcomplexes.
Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic l1-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexe…
Study on right-angled Coxeter groups and their geometric properties.
problem Characterizing the coarse geometry of right-angled Coxeter groups.
method Analyzing graph properties and applying geometric group theory.
result Proves properties of right-angled Coxeter groups, including quasi-isometry and divergence.
Artin groups of hyperbolic type are boundary amenable and have rigid properties.
problem Characterizing rigidity and measure equivalence properties of Artin groups.
method Analyzing boundary amenability, measure equivalence, and fixed set graphs.
result Measure equivalent Artin groups of hyperbolic type have isomorphic fixed set graphs.