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1. Algebraic Representations for Volumetric Frame Fields

   https://doi.org/10.1145/3366786  

   Palmer, David; Bommes, David; Solomon, Justin (April 2020, ACM Transactions on Graphics)

   Field-guided parameterization methods have proven effective for quad meshing of surfaces; these methods compute smoothcross fieldsto guide the meshing process and then integrate the fields to construct a discrete mesh. A key challenge in extending these methods to three dimensions, however, isrepresentationof field values. Whereas cross fields can be represented by tangent vector fields that form a linear space, the 3D analog—an octahedral frame field—takes values in a nonlinear manifold. In this work, we describe the space of octahedral frames in the language of differential and algebraic geometry. With this understanding, we develop geometry-aware tools for optimization of octahedral fields, namely geodesic stepping and exact projection via semidefinite relaxation. Our algebraic approach not only provides an elegant and mathematically sound description of the space of octahedral frames but also suggests a generalization to frames whose three axes scale independently, better capturing the singular behavior we expect to see in volumetric frame fields. These newodeco frames, so called as they are represented by orthogonally decomposable tensors, also admit a semidefinite program–based projection operator. Our description of the spaces of octahedral and odeco frames suggests computing frame fields via manifold-based optimization algorithms; we show that these algorithms efficiently produce high-quality fields while maintaining stability and smoothness. 

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2. Isolations of geodesic planes in the frame bundle of a hyperbolic 3-manifold

   https://doi.org/10.1112/S0010437X22007928  

   Mohammadi, Amir; Oh, Hee (March 2023, Compositio Mathematica)

   We present a quantitative isolation property of the lifts of properly immersed geodesic planes in the frame bundle of a geometrically finite hyperbolic $$3$$ -manifold. Our estimates are polynomials in the tight areas and Bowen–Margulis–Sullivan densities of geodesic planes, with degree given by the modified critical exponents. 

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3. Constant-time quantum algorithm for homology detection in closed curves

   https://doi.org/10.21468/SciPostPhys.15.2.049  

   Nghiem Vu, Nhat Anh; Gu, Xianfeng David; Wei, Tzu-Chieh (January 2023, SciPost Physics)

   Given a loop or more generally 1-cycle r r of size L on a closed two-dimensional manifold or surface, represented by a triangulated mesh, a question in computational topology asks whether or not it is homologous to zero. We frame and tackle this problem in the quantum setting. Given an oracle that one can use to query the inclusion of edges on a closed curve, we design a quantum algorithm for such a homology detection with a constant running time, with respect to the size or the number of edges on the loop r r , requiring only a single usage of oracle. In contrast, classical algorithm requires \Omega(L) Ω ( L ) oracle usage, followed by a linear time processing and can be improved to logarithmic by using a parallel algorithm. Our quantum algorithm can be extended to check whether two closed loops belong to the same homology class. Furthermore, it can be applied to a specific problem in the homotopy detection, namely, checking whether two curves are not homotopically equivalent on a closed two-dimensional manifold. 

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4. Gabor frames from contact geometry in models of the primary visual cortex

   https://doi.org/10.46298/mna.9766  

   Liontou, Vasiliki; Marcolli, Matilde (June 2023, Mathematical Neuroscience and Applications)

   We analyze the interplay between contact geometry and Gabor filters signalanalysis in geometric models of the primary visual cortex. We show inparticular that a specific framed lattice and an associated Gabor system isdetermined by the Legendrian circle bundle structure of the $$3$$-manifold ofcontact elements on a surface (which models the V1-cortex), together with thepresence of an almost-complex structure on the tangent bundle of the surface(which models the retinal surface). We identify a scaling of the lattice, alsodictated by the manifold geometry, that ensures the frame condition issatisfied. We then consider a $$5$$-dimensional model where receptor profilesalso involve a dependence on frequency and scale variables, in addition to thedependence on position and orientation. In this case we show that a proposedprofile window function does not give rise to frames (even in a distributionalsense), while a natural modification of the same generates Gabor frames withrespect to the appropriate lattice determined by the contact geometry. 

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5. DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking

   Gabriele Corso; Hannes Stärk; Bowen Jing; Regina Barzilay; Tommi Jaakkola (January 2023, International Conference on Learning Representations (ICLR 2023))

   Predicting the binding structure of a small molecule ligand to a protein -- a task known as molecular docking -- is critical to drug design. Recent deep learning methods that treat docking as a regression problem have decreased runtime compared to traditional search-based methods but have yet to offer substantial improvements in accuracy. We instead frame molecular docking as a generative modeling problem and develop DiffDock, a diffusion generative model over the non-Euclidean manifold of ligand poses. To do so, we map this manifold to the product space of the degrees of freedom (translational, rotational, and torsional) involved in docking and develop an efficient diffusion process on this space. Empirically, DiffDock obtains a 38% top-1 success rate (RMSD<2A) on PDBBind, significantly outperforming the previous state-of-the-art of traditional docking (23%) and deep learning (20%) methods. Moreover, while previous methods are not able to dock on computationally folded structures (maximum accuracy 10.4%), DiffDock maintains significantly higher precision (21.7%). Finally, DiffDock has fast inference times and provides confidence estimates with high selective accuracy. 

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