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1. Tri-plane diagrams for simple surfaces in S4

   https://doi.org/10.1142/S0218216523500414  

   Allred, Wolfgang; Aragón, Manuel; Dooley, Zack; Goldman, Alexander; Lei, Yucong; Martinez, Isaiah; Meyer, Nicholas; Peters, Devon; Warrander, Scott; Wright, Ana; et al (June 2023, Journal of Knot Theory and Its Ramifications)

   Meier and Zupan proved that an orientable surface [Formula: see text] in [Formula: see text] admits a tri-plane diagram with zero crossings if and only if [Formula: see text] is unknotted, so that the crossing number of [Formula: see text] is zero. We determine the minimal crossing numbers of nonorientable unknotted surfaces in [Formula: see text], proving that [Formula: see text], where [Formula: see text] denotes the connected sum of [Formula: see text] unknotted projective planes with normal Euler number [Formula: see text] and [Formula: see text] unknotted projective planes with normal Euler number [Formula: see text]. In addition, we convert Yoshikawa’s table of knotted surface ch-diagrams to tri-plane diagrams, finding the minimal bridge number for each surface in the table and providing upper bounds for the crossing numbers. 

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2. Metric Thickenings and Group Actions

   https://doi.org/10.1142/S1793525320500569  

   Adams, Henry; Heim, Mark; Peterson, Chris (January 2021, Journal of Topology and Analysis)

   null (Ed.)

   Let [Formula: see text] be a group acting properly and by isometries on a metric space [Formula: see text]; it follows that the quotient or orbit space [Formula: see text] is also a metric space. We study the Vietoris–Rips and Čech complexes of [Formula: see text]. Whereas (co)homology theories for metric spaces let the scale parameter of a Vietoris–Rips or Čech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large, we instead consider intermediate scale parameters (neither tending to zero nor to infinity). As a particular case, we study the Vietoris–Rips and Čech thickenings of projective spaces at the first scale parameter where the homotopy type changes. 

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3. Space Complexity of Vertex Connectivity Oracles

   https://doi.org/10.1137/23M1564845  

   Pettie, Seth; Saranurak, Thatchaphol; Yin, Longhui (April 2026, SIAM Journal on Computing)

   Abstract. A [Formula: see text]- vertex connectivity oracle for an undirected graph [Formula: see text] is a data structure that, given [Formula: see text], reports [Formula: see text], where [Formula: see text] is the pairwise vertex connectivity between [Formula: see text]. There are three main measures of efficiency: construction time, query time, and space. Prior work of Izsak and Nutov [ Inform. Process. Lett., 112 (2012), pp. 39–43] produced a data structure of [Formula: see text] words, which can even be encoded as a [Formula: see text]-bit labeling scheme, that can answer vertex connectivity queries in [Formula: see text] time. The construction time is polynomial but unspecified. In this paper, we address the top three complexity measures. (1) Space: We prove that any [Formula: see text]-vertex connectivity oracle requires [Formula: see text] bits of space for any [Formula: see text]. This proves that the Iszak–Nutov data structure is optimal up to polylogarithmic factors for every [Formula: see text] and that the sparsifiers of Nagamochi and Ibaraki [ Algorithmica, 7 (1992), pp. 583–596] are optimal compression schemes for [Formula: see text]-vertex connectivity up to a logarithmic factor. In particular, whereas all edge connectivities can be efficiently compressed (as a weighted [Formula: see text]-edge Gomory–Hu tree), vertex connectivity admits no asymptotic compression: [Formula: see text] bits are necessary. We design a variation on Izsak and Nutov’s data structure that uses [Formula: see text] words of space. (2) Query time: We answer queries in [Formula: see text] time, improving on the [Formula: see text] time bound of Izsak and Nutov [ Inform. Process. Lett., 112 (2012), pp. 39–43]. The main idea is to build instances of [Formula: see text] data structures, with additional structure based on affine planes. This structure allows for query time that is linear in the output size, which evades some conditional lower bounds that are polynomial in the query set sizes [ 42 ,  51 ]. (3) Construction time: Our data structure can be constructed in the time of [Formula: see text] max-flow computations, namely, [Formula: see text] time, using the recent near-linear time flow algorithm of [ 13 ]. The main technical contribution here is a fast algorithm to compute a [Formula: see text]-approximate Gomory–Hu tree for element connectivity in the time of [Formula: see text] max-flow computations. Element connectivity is a notion that generalizes edge and vertex connectivity. 

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4. Dimension of Splines on Graphs in the Case of Degree Two and Smoothness One in Two Variables

   https://doi.org/10.1137/24M1680040  

   Nazir, Shaheen; Schilling, Anne; Tymoczko, Julianna (June 2026, SIAM Journal on Applied Algebra and Geometry)

   Abstract. Continuous spline functions are defined as piecewise polynomials on the faces of a polyhedral complex that agree on the intersections of two faces. Splines are used in approximation theory and numerical analysis, with applications in data interpolation, to create smooth curves in computer graphics and to find numerical solutions to partial differential equations. Gilbert, Tymoczko, and Viel [ Pacific J. Math., 281 (2016), pp. 333–364] generalized the classical splines combinatorially and algebraically: a generalized spline is a vertex labeling of a graph [Formula: see text] by elements of the ring so that the difference between the labels of any two adjacent vertices lies in the ideal generated by the corresponding edge label. We study the generalized splines on the planar graphs whose edges are labeled by two-variable polynomials of the form [Formula: see text] and whose vertices are labeled by polynomials of degree at most two. In this paper we, address the upper-bound conjecture for the dimension of degree-2 splines of smoothness 1. The dimension is expressed in terms of the rank of the extended cycle basis matrix. We also provide a combinatorial algorithm on graphs to compute the rank by contracting certain subgraphs. 

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5. On the structure of limiting flocks in hydrodynamic Euler Alignment models

   https://doi.org/10.1142/S0218202519500507  

   Leslie, Trevor M.; Shvydkoy, Roman (December 2019, Mathematical Models and Methods in Applied Sciences)

   The goal of this paper is to study limiting behavior of a self-organized continuous flock evolving according to the 1D hydrodynamic Euler Alignment model. We provide a series of quantitative estimates that show how far the density of the limiting flock is from a uniform distribution. The key quantity that controls density distortion is the entropy [Formula: see text], and the measure of deviation from uniformity is given by a well-known conserved quantity [Formula: see text], where [Formula: see text] is velocity and [Formula: see text] is the communication operator with kernel [Formula: see text]. The cases of Lipschitz, singular geometric, and topological kernels are covered in the study. 

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