PD_3-groups split as HNN extensions, revealing homology class properties.
problem Understanding PD_3-groups splitting as HNN extensions.
method Analyzing PD_3-groups splitting as HNN extensions and examining homology classes.
result The Poincaré dual of the homology class of a PD_3-group in the kernel of an epimorphism is revealed.
We prove the Tits alternative for an almost coherent PD(3) group which is not virtually properly locally cyclic. In particular, we show that an almost coherent PD(3) group which cannot be generated by fewer than four elements always contains a rank 2 free group.
Study extends Elkalla's work on subnormal subgroups to PD3-groups, but L2-Betti numbers need verification.
problem Verifying L2-Betti numbers for PD3-groups and group pairs. method Algebraic arguments extending Elkalla's work, but reliant on unproven L2-Betti number hypothesis. result Need further research on L2-Betti numbers for general PD3-groups. Study subgroups of pro-p PD^3 groups, finding specific conditions.
problem Characterize subgroups of pro-p PD^3 groups. method Analyzes properties of subnormal and finitely presented subgroups.
result Conditions on subgroups of pro-p PD^3 groups. The abstract discusses conjectures about Chern-Simons invariants of 3-manifolds.
problem Conjectures about the reciprocity of Chern-Simons invariants of 3-manifolds.
method Supporting evidence through Galois descent of a K3-group. result The conjectures hold under the condition of Galois descent of a K3-group. Characterizes groups of branched twist-spun knots.
problem Understanding the groups of branched twist-spun knots.
method Characterization through 3-manifold groups and conjectural algebraic approach.
result Each group is the group of at most finitely many branched twist spins.
Homotopy types of 4-manifolds tied to their fundamental groups.
problem Determining the homotopy type of 4-manifolds based on their fundamental groups.
method Uses the fundamental group, second homotopy group, first Stiefel-Whitney class, and equivariant intersection pairing.
result Homotopy type of 4-manifolds is determined by given group properties.
Study pro-p completions of orientable PD_n groups, proving best results in three cases.
problem Understanding pro-p completions of orientable PD_n groups. method Examined four cases of orientable PD_3-groups and some PD_n groups (n≤5), providing examples and proving best results in three cases.
result Best results in three out of four cases of orientable PD_3-groups.
This paper characterizes semigenerated Carnot groups and applies it to rectifiability of perimeter sets.
problem Characterizing semigenerated Carnot groups and their applications to rectifiability.
method Algebraic approach focusing on semigroup generation and Engel-type quotients.
result Complete characterization of semigeneration in Carnot groups of step 3 and sufficient criteria for semigeneration in Carnot groups of arbitrary step.
Study of skateboard flips as continuous curves in SO(3) group.
problem Characterize skateboard flip tricks as continuous motions.
method Model flips as curves in SO(3), analyze lifts to S3, derive formulas. result There are only four distinct flip tricks up to continuous deformation.
We construct a non-abelian extension Γ of S1 by $\cy 3 \times \cy 3$, and prove that Γ acts freely and smoothly on S5×S5. This gives new actions on S5×S5 for an infinite family $\cP$ of finite 3-groups. We also show that any finite odd order subgroup of the exceptional Lie group $G_…
Researchers derive q-series for SO(3) and OSp(1∣2) groups.
problem Deriving q-series for SO(3) and OSp(1∣2) groups. method Change of variable relating SU(2) link invariants to SO(3) and OSp(1∣2) link invariants. result Explicit q-series for SO(3) and OSp(1∣2) groups. Random groups with high density have Property (T).
problem Determining the conditions for Property (T) in random groups.
method Analyzing k-gonal random groups and their limits. result Property (T) holds with high probability for high densities.
Study groups admitting unbounded quasimorphisms to R with coarsely-connected quasikernel.
problem Understanding PD3 groups and their properties. method Coarse generalization of Shapiro's lemma, homological isoperimetric inequalities, and Margolis's coarse homological algebra.
result Groups admitting unbounded quasimorphisms to R with coarsely-connected quasikernel are either torus or Klein-bottle bundles over S^1, or quasiisometric to Riemannian manifolds.
Study uniformly differentiable graphs in Carnot groups, proving area formulas.
problem Characterize uniformly differentiable intrinsic graphs in Carnot groups.
method Characterize uniform intrinsic differentiability via Hölder properties of projections of vector fields.
result Explicit area formula for uniformly intrinsically differentiable maps in Carnot groups.
In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-gr…
We show that the Fibered Isomorphism Conjecture (FIC) of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for the fundamental groups of a large class of 3-manifolds. We also prove that if the FIC is true for irreducible 3-manifold groups then it is true for all 3-manifold groups. …
New methods validate a hypothesis explaining how neural nets generalize well.
problem Why over-parameterized nets generalize well despite memorizing training data.
method Developed new algorithms to suppress weak gradient directions without per-example gradients.
result Validated a hypothesis about gradient directions and their role in generalization.
The thesis examines the relationship between three-manifold invariants and knot theory, finding equalities and patterns.
problem Examining the relationship between three-manifold invariants and knot theory.
method Analytic continuation and quiver representation theory.
result Found equalities and patterns in knot theory and quiver representation.