Apolipoprotein E allele 4 effects on Single-Subject Gray Matter Networks in Mild Cognitive Impairment.

There is evidence that gray matter networks are disrupted in Mild Cognitive Impairment (MCI) and associated with cognitive impairment and faster disease progression. However, it remains unknown how these alterations are related to the presence of Apolipoprotein E isoform E4 (ApoE4), the most prominent genetic risk factor for late-onset Alzheimer's disease (AD). To investigate this topic at the individual level, we explore the impact of ApoE4 and the disease progression on the Single-Subject Gray Matter Networks (SSGMNets) using the graph theory approach. Our data sample comprised 200 MCI patients selected from the ADNI database, classified as non-Converters and Converters (will progress into AD). Each group included 50 ApoE4-positive ('Carriers', ApoE4 + ) and 50 ApoE4-negative ('non-Carriers', ApoE4-). The SSGMNets were estimated from structural MRIs at two-time points: baseline and conversion. We investigated whether altered network topological measures at baseline and their rate of change (RoC) between baseline and conversion time points were associated with ApoE4 and disease progression. We also explored the correlation of SSGMNets attributes with general cognition score (MMSE), memory (ADNI-MEM), and CSF-derived biomarkers of AD (Aβ42, T-tau, and P-tau). Our results showed that ApoE4 and the disease progression modulated the global topological network properties independently but not in their RoC. MCI converters showed a lower clustering index in several regions associated with neurodegeneration in AD. The SSGMNets' topological organization was revealed to be able to predict cognitive and memory measures. The findings presented here suggest that SSGMNets could indeed be used to identify MCI ApoE4 Carriers with a high risk for AD progression

Hierarchically hyperbolic spaces I: curve complexes for cubical groups

In the context of CAT(0) cubical groups, we develop an analogue of the theory of curve complexes and subsurface projections. The role of the subsurfaces is played by a collection of convex subcomplexes called a \emph{factor system}, and the role of the curve graph is played by the \emph{contact graph}. There are a number of close parallels between the contact graph and the curve graph, including hyperbolicity, acylindricity of the action, the existence of hierarchy paths, and a Masur--Minsky-style distance formula. We then define a \emph{hierarchically hyperbolic space}; the class of such spaces includes a wide class of cubical groups (including all virtually compact special groups) as well as mapping class groups and Teichm\"{u}ller space with any of the standard metrics. We deduce a number of results about these spaces, all of which are new for cubical or mapping class groups, and most of which are new for both. We show that the quasi-Lipschitz image from a ball in a nilpotent Lie group into a hierarchically hyperbolic space lies close to a product of hierarchy geodesics. We also prove a rank theorem for hierarchically hyperbolic spaces; this generalizes results of Behrstock--Minsky, Eskin--Masur--Rafi, Hamenst\"{a}dt, and Kleiner. We finally prove that each hierarchically hyperbolic group admits an acylindrical action on a hyperbolic space. This acylindricity result is new for cubical groups, in which case the hyperbolic space admitting the action is the contact graph; in the case of the mapping class group, this provides a new proof of a theorem of Bowditch.Comment: To appear in "Geometry and Topology". This version incorporates the referee's comment

Quasiflats in hierarchically hyperbolic spaces

The rank of a hierarchically hyperbolic space is the maximal number of unbounded factors in a standard product region. For hierarchically hyperbolic groups, this coincides with the maximal dimension of a quasiflat. Examples for which the rank coincides with familiar quantities include: the dimension of maximal Dehn twist flats for mapping class groups, the maximal rank of a free abelian subgroup for right-angled Coxeter and Artin groups, and, for the Weil--Petersson metric, the rank is the integer part of half the complex dimension of Teichm\"{u}ller space. We prove that any quasiflat of dimension equal to the rank lies within finite distance of a union of standard orthants (under a mild condition satisfied by all natural examples). This resolves outstanding conjectures when applied to various examples. For mapping class group, we verify a conjecture of Farb; for Teichm\"{u}ller space we answer a question of Brock; for CAT(0) cubical groups, we handle special cases including right-angled Coxeter groups. An important ingredient in the proof is that the hull of any finite set in an HHS is quasi-isometric to a CAT(0) cube complex of dimension bounded by the rank. We deduce a number of applications. For instance, we show that any quasi-isometry between HHSs induces a quasi-isometry between certain simpler HHSs. This allows one, for example, to distinguish quasi-isometry classes of right-angled Artin/Coxeter groups. Another application is to quasi-isometric rigidity. Our tools in many cases allow one to reduce the problem of quasi-isometric rigidity for a given hierarchically hyperbolic group to a combinatorial problem. We give a new proof of quasi-isometric rigidity of mapping class groups, which, given our general quasiflats theorem, uses simpler combinatorial arguments than in previous proofs.Comment: 58 pages, 6 figures. Revised according to referee comments. This is the final pre-publication version; to appear in Duke Math. Jou

A SAT-based Resolution of Lam's Problem

In 1989, computer searches by Lam, Thiel, and Swiercz experimentally resolved Lam's problem from projective geometry\unicode{x2014}the long-standing problem of determining if a projective plane of order ten exists. Both the original search and an independent verification in 2011 discovered no such projective plane. However, these searches were each performed using highly specialized custom-written code and did not produce nonexistence certificates. In this paper, we resolve Lam's problem by translating the problem into Boolean logic and use satisfiability (SAT) solvers to produce nonexistence certificates that can be verified by a third party. Our work uncovered consistency issues in both previous searches\unicode{x2014}highlighting the difficulty of relying on special-purpose search code for nonexistence results.Comment: To appear at the Thirty-Fifth AAAI Conference on Artificial Intelligenc