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1. The contraction category of graphs

   https://doi.org/10.1090/ert/616  

   Proudfoot, Nicholas ; Ramos, Eric ( June 2022 , Representation Theory of the American Mathematical Society)

   We study the category whose objects are graphs of fixed genus and whose morphisms are contractions. We show that the corresponding contravariant module categories are Noetherian and we study two families of modules over these categories. The first takes a graph to a graded piece of the homology of its unordered configuration space and the second takes a graph to an intersection homology group whose dimension is given by a Kazhdan–Lusztig coefficient; in both cases we prove that the module is finitely generated. This allows us to draw conclusions about torsion in the homology groups of graph configuration spaces, and about the growth of Betti numbers of graph configuration spaces and Kazhdan–Lusztig coefficients of graphical matroids. We also explore the relationship between our category and outer space, which is used in the study of outer automorphisms of free groups. 

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2. Filtered Frobenius Algebras in Monoidal Categories

   https://doi.org/10.1093/imrn/rnac314  

   Walton, Chelsea ; Yadav, Harshit ( November 2022 , International Mathematics Research Notices)

   We develop filtered-graded techniques for algebras in monoidal categories with the main goal of establishing a categorical version of Bongale’s 1967 result: a filtered deformation of a Frobenius algebra over a field is Frobenius as well. Toward the goal, we first construct a monoidal associated graded functor, building on prior works of Ardizzoni and Menini, Galatius et al., and Gwillian and Pavlov. Next, we produce equivalent conditions for an algebra in a rigid monoidal category to be Frobenius in terms of the existence of categorical Frobenius form; this builds on work of Fuchs and Stigner. These two results of independent interest are then used to achieve our goal. As an application of our main result, we show that any exact module category over a symmetric finite tensor category $\mathcal {C}$ is represented by a Frobenius algebra in $\mathcal {C}$. Several directions for further investigation are also proposed.

    

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3. Lim Ulrich sequences and Boij-Söderberg cones

   https://doi.org/10.1017/fms.2023.108  

   Iyengar, Srikanth B ; Ma, Linquan ; Walker, Mark E ( January 2023 , Forum of Mathematics, Sigma)

   This paper extends the results of Boij, Eisenbud, Erman, Schreyer and Söderberg on the structure of Betti cones of finitely generated graded modules and finite free complexes over polynomial rings, to all finitely generated graded rings admitting linear Noether normalizations. The key new input is the existence of lim Ulrich sequences of graded modules over such rings.

    

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4. Geometric transformation of finite element methods: Theory and applications

   https://doi.org/10.1016/j.apnum.2023.07.002  

   Holst, Michael ; Licht, Martin ( October 2023 , Applied Numerical Mathematics)

   We present a new technique to apply finite element methods to partial differential equations over curved domains. A change of variables along a coordinate transformation satisfying only low regularity assumptions can translate a Poisson problem over a curved physical domain to a Poisson problem over a polyhedral parametric domain. This greatly simplifies both the geometric setting and the practical implementation, at the cost of having globally rough non-trivial coefficients and data in the parametric Poisson problem. Our main result is that a recently developed broken Bramble-Hilbert lemma is key in harnessing regularity in the physical problem to prove higher-order finite element convergence rates for the parametric problem. Numerical experiments are given which confirm the predictions of our theory. 

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5. Topology and Local Geometry of the Eden Model

   https://doi.org/10.1007/s00454-022-00474-w  

   Manin, Fedor ; Roldán, Érika ; Schweinhart, Benjamin ( February 2023 , Discrete & Computational Geometry)

   The Eden cell growth model is a simple discrete stochastic process which produces a “blob” (aggregation of cells) in $$\mathbb {R}^d$$$R^d$: start with one cube in the regular grid, and at each time step add a neighboring cube uniformly at random. This process has been used as a model for the growth of aggregations, tumors, and bacterial colonies and the healing of wounds, among other natural processes. Here, we study the topology and local geometry of the resulting structure, establishing asymptotic bounds for Betti numbers. Our main result is that the Betti numbers at timetgrow at a rate between$$t^{(d-1)/d}$$$t^{(d-1)/d}$and$$P_d(t)$$$P_d\left(t\right)$, where$$P_d(t)$$$P_d\left(t\right)$is the size of the site perimeter. Assuming a widely believed conjecture, this establishes the rate of growth of the Betti numbers in every dimension. We also present the results of computational experiments on finer aspects of the geometry and topology, such as persistent homology and the distribution of shapes of holes.

    

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