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Abstract Several studies have now described instances where G-rich sequences in promoters and enhancers regulate gene expression through forming G-quadruplex (G4) structures. Relatedly, our group recently identified 301 long genomic stretches significantly enriched for minimal G4 motifs (LG4s) in humans and found the majority of these overlap annotated enhancers, and furthermore, that the promoters regulated by these LG4 enhancers are similarly enriched with G4-capable sequences. While the generally accepted model for enhancer:promoter specificity maintains that interactions are dictated by enhancer- and promoter-bound transcriptional activator proteins, the current study tested an alternative hypothesis: that LG4 enhancers interact with cognate promoters via a direct G4:G4 DNA-based mechanism. This work establishes the nuclear proximity of LG4 enhancer:promoter pairs, biochemically demonstrates the ability of individual LG4 single-stranded DNAs (ssDNAs) to directly interact target promoter ssDNAs, and confirms that these interactions, as well as the ability of LG4 enhancers to activate target promoters in culture, are mediated by G4 DNA. 

This paper describes a method for fast simplification of surface meshes. Whereas past methods focus on visual appearance, our goal is to solve equations on the surface. Hence, rather than approximate the extrinsic geometry, we construct a coarseintrinsic triangulationof the input domain. In the spirit of thequadric error metric (QEM), we perform greedy decimation while agglomerating global information about approximation error. In lieu of extrinsic quadrics, however, we store intrinsic tangent vectors that track how far curvature drifts during simplification. This process also yields a bijective map between the fine and coarse mesh, and prolongation operators for both scalar- and vector-valued data. Moreover, we obtain hard guarantees on element quality via intrinsic retriangulation---a feature unique to the intrinsic setting. The overall payoff is a black box approach to geometry processing, which decouples mesh resolution from the size of matrices used to solve equations. We show how our method benefits several fundamental tasks, including geometric multigrid, all-pairs geodesic distance, mean curvature flow, geodesic Voronoi diagrams, and the discrete exponential map. 

Sphere tracingis a fast and high-quality method for visualizing surfaces encoded by signed distance functions (SDFs). We introduce a similar method for a completely different class of surfaces encoded byharmonic functions, opening up rich new possibilities for visual computing. Our starting point is similar in spirit to sphere tracing: using conservativeHarnack boundson the growth of harmonic functions, we develop aHarnack tracingalgorithm for visualizing level sets of harmonic functions, including those that are angle-valued and exhibit singularities. The method takes much larger steps than naïve ray marching, avoids numerical issues common to generic root finding methods and, like sphere tracing, needs only perform pointwise evaluation of the function at each step. For many use cases, the method is fast enough to run real time in a shader program. We use it to visualize smooth surfaces directly from point clouds (via Poisson surface reconstruction) or polygon soup (via generalized winding numbers) without linear solves or mesh extraction. We also use it to visualize nonplanar polygons (possibly with holes), surfaces from architectural geometry, mesh exoskeletons, and key mathematical objects including knots, links, spherical harmonics, and Riemann surfaces. Finally we show that, at least in theory, Harnack tracing provides an alternative mechanism for visualizing arbitrary implicit surfaces. 

In the plane, thewinding numberis the number of times a curve wraps around a given point. Winding numbers are a basic component of geometric algorithms such as point-in-polygon tests, and their generalization to data with noise or topological errors has proven valuable for geometry processing tasks ranging from surface reconstruction to mesh booleans. However, standard definitions do not immediately apply on surfaces, where not all curves bound regions. We develop a meaningful generalization, starting with the well-known relationship between winding numbers and harmonic functions. By processing the derivatives of such functions, we can robustly filter out components of the input that do not bound any region. Ultimately, our algorithm yields (i) a closed, completed version of the input curves, (ii) integer labels for regions that are meaningfully bounded by these curves, and (iii) the complementary curves that do not bound any region. The main computational cost is solving a standard Poisson equation, or for surfaces with nontrivial topology, a sparse linear program. We also introduce special basis functions to represent singularities that naturally occur at endpoints of open curves. 

This course provides a first introduction to intrinsic triangulations and their use in mesh processing algorithms. As geometric data becomes more ubiquitous, e.g., in applications such as augmented reality or machine learning, there is a pressing need to develop algorithms that work reliably on low-quality data. Intrinsic triangulations provide a powerful framework for these problems, by de-coupling the mesh used to encode geometry from the one used for computation. The basic shift in perspective is to encode the geometry of a mesh not in terms of ordinary vertex positions, but instead only in terms of edge lengths. Intrinsic triangulations have a long history in mathematics, but only in recent years have been applied to practical geometric computing. The course begins by giving motivation for intrinsic triangulations in terms of recent problems in computer graphics, followed by an interactive coding session where participants can make first contact with the idea of intrinsic meshes. We then give some mathematical background, and describe key data structures (overlay, signpost, normal coordinates). Using this machinery, we translate algorithms from computational geometry and scientific computing into cutting-edge algorithms for curved surfaces. For instance, we look at mesh parameterization, vector field processing, finding geodesics, solving partial differential equations (PDEs), and more. We also discuss processing of nonmanifold meshes and point clouds; participants can explore these algorithms via interactive demos. We conclude with a discussion of open questions and opportunities for future work. 

This paper describes a numerically robust data structure for encoding intrinsic triangulations of polyhedral surfaces. Many applications demand a correspondence between the intrinsic triangulation and the input surface, but existing data structures either rely on floating point values to encode correspondence, or do not support remeshing operations beyond basic edge flips. We instead provide an integer-based data structure that guarantees valid correspondence, even for meshes with near-degenerate elements. Our starting point is the framework ofnormal coordinatesfrom geometric topology, which we extend to the broader set of operations needed for mesh processing (vertex insertion, edge splits,etc.). The resulting data structure can be used as a drop-in replacement for earlier schemes, automatically improving reliability across a wide variety of applications. As a stress test, we successfully compute an intrinsic Delaunay refinement and associated subdivision for all manifold meshes in the Thingi10k dataset. In turn, we can compute reliable and highly accurate solutions to partial differential equations even on extremely low-quality meshes. 

This paper describes a numerical method for surface parameterization, yielding maps that are locally injective and discretely conformal in an exact sense. Unlike previous methods for discrete conformal parameterization, the method is guaranteed to work for any manifold triangle mesh, with no restrictions on triangulatiothat each task can be formulated as a convex problem where the triangulation is allowed to change---we complete the picture by introducing the machinery needed to actually construct a discrete conformal map. In particular, we introduce a new scheme for tracking correspondence between triangulations based onnormal coordinates, and a new interpolation procedure based on layout in thelight cone.Stress tests involving difficult cone configurations and near-degenerate triangulations indicate that the method is extremely robust in practice, and provides high-quality interpolation even on meshes with poor elements.