Department of Mathematics

Viazovska proved that the [Formula: see text] lattice sphere packing is the densest sphere packing in [Formula: see text] dimensions. Her proof relies on two inequalities between functions defined in terms of modular and quasimodular forms. We give a direct proof of these inequalities that does not rely on computer calculations.

Populations of neurons represent sensory, motor, and cognitive variables via patterns of activity distributed across the population. The size of the population used to encode a variable is typically much greater than the dimension of the variable itself, and thus, the corresponding neural population activity occupies lower-dimensional subsets of the full set of possible activity states. Given population activity data with such lower-dimensional structure, a fundamental question asks how close the low-dimensional data lie to a linear subspace. The linearity or nonlinearity of the low-dimensional structure reflects important computational features of the encoding, such as robustness and generalizability. Moreover, identifying such linear structure underlies common data analysis methods such as Principal Component Analysis (PCA). Here, we show that for data drawn from many common population codes the resulting point clouds and manifolds are exceedingly nonlinear, with the dimension of the best-fitting linear subspace growing at least exponentially with the true dimension of the data. Consequently, linear methods like PCA fail dramatically at identifying the true underlying structure, even in the limit of arbitrarily many data points and no noise.

We develop the notion of a Kleinian Sphere Packing, a generalization of "crystallographic"(Apollonian-like) sphere packings defined in [A. Kontorovich and K. Nakamura, Geometry and arithmetic of crystallographic sphere packings, Proc. Natl. Acad. Sci. USA 116 2019, 2, 436-441]. Unlike crystallographic packings, Kleinian packings exist in all dimensions, as do "superintegral"such. We extend the Arithmeticity Theorem to Kleinian packings, that is, the superintegral ones come from ℚ -arithmetic lattices of simplest type. The same holds for more general objects we call Kleinian Bugs, in which the spheres need not be disjoint but can meet with dihedral angles π/m for finitely many m. We settle two questions from Kontorovich and Nakamura (2019): (i) that the Arithmeticity Theorem is in general false over number fields, and (ii) that integral packings only arise from non-uniform lattices.

We compare various notions of proper discontinuity for group actions. We also discuss fundamental domains and criteria for cocompactness.

We prove nonemptyness of domains of proper discontinuity of Anosov groups of affine Lorentzian transformations.

Establishing the host range for novel viruses remains a challenge. Here, we address the challenge of identifying non-human animal coronaviruses that may infect humans by creating an artificial neural network model that learns from spike protein sequences of alpha and beta coronaviruses and their binding annotation to their host receptor. The proposed method produces a human-Binding Potential (h-BiP) score that distinguishes, with high accuracy, the binding potential among coronaviruses. Three viruses, previously unknown to bind human receptors, were identified: Bat coronavirus BtCoV/133/2005 and Pipistrellus abramus bat coronavirus HKU5-related (both MERS related viruses), and Rhinolophus affinis coronavirus isolate LYRa3 (a SARS related virus). We further analyze the binding properties of BtCoV/133/2005 and LYRa3 using molecular dynamics. To test whether this model can be used for surveillance of novel coronaviruses, we re-trained the model on a set that excludes SARS-CoV-2 and all viral sequences released after the SARS-CoV-2 was published. The results predict the binding of SARS-CoV-2 with a human receptor, indicating that machine learning methods are an excellent tool for the prediction of host expansion events.