Mohammed Abouzaid, Andrew J. Blumberg, Martin Hairer et al.
To assess the ability of current AI systems to correctly answer research-level mathematics questions, we share a set of ten math questions which have arisen naturally in the research process of the authors. The questions had not been shared publicly until now; the answers are known to the authors of the questions but will remain encrypted for a short time.
Andrew J. Blumberg, Mathieu Carrière, Jun Hou Fung et al.
We introduce a new algorithm for finding robust circular coordinates on data that is expected to exhibit recurrence, such as that which appears in neuronal recordings of C. elegans. Techniques exist to create circular coordinates on a simplicial complex from a dimension 1 cohomology class, and these can be applied to the Rips complex of a dataset when it has a prominent class in its dimension 1 cohomology. However, it is known this approach is extremely sensitive to uneven sampling density. Our algorithm comes with a new method to correct for uneven sampling density, adapting our prior work on averaging coordinates in manifold learning. We use rejection sampling to correct for inhomogeneous sampling and then apply Procrustes matching to align and average the subsamples. In addition to providing a more robust coordinate than other approaches, this subsampling and averaging approach has better efficiency. We validate our technique on both synthetic data sets and neuronal activity recordings. Our results reveal a topological model of neuronal trajectories for C. elegans that is constructed from loops in which different regions of the brain state space can be mapped to specific and interpretable macroscopic behaviors in the worm.
Andrew J. Blumberg, Prithwish Bhaumik, Stephen G. Walker
We study the problem of distinguishing between two distributions on a metric space; i.e., given metric measure spaces $({\mathbb X}, d, μ_1)$ and $({\mathbb X}, d, μ_2)$, we are interested in the problem of determining from finite data whether or not $μ_1$ is $μ_2$. The key is to use pairwise distances between observations and, employing a reconstruction theorem of Gromov, we can perform such a test using a two sample Kolmogorov--Smirnov test. A real analysis using phylogenetic trees and flu data is presented.
Sebastian Angel, Riad S. Wahby, Max Howald et al.
Malicious peripherals designed to attack their host computers are a growing problem. Inexpensive and powerful peripherals that attach to plug-and-play buses have made such attacks easy to mount. Making matters worse, commodity operating systems lack coherent defenses, and users are often unaware of the scope of the problem. We present Cinch, a pragmatic response to this threat. Cinch uses virtualization to attach peripheral devices to a logically separate, untrusted machine, and includes an interposition layer between the untrusted machine and the protected one. This layer regulates interaction with devices according to user-configured policies. Cinch integrates with existing OSes, enforces policies that thwart real-world attacks, and has low overhead.