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1. Ecological models: higher complexity in, higher feasibility out

   https://doi.org/10.1098/rsif.2020.0607  

   AlAdwani, Mohammad; Saavedra, Serguei (November 2020, Journal of The Royal Society Interface)

   null (Ed.)

   Finding a compromise between tractability and realism has always been at the core of ecological modelling. The introduction of nonlinear functional responses in two-species models has reconciled part of this compromise. However, it remains unclear whether this compromise can be extended to multispecies models. Yet, answering this question is necessary in order to differentiate whether the explanatory power of a model comes from the general form of its polynomial or from a more realistic description of multispecies systems. Here, we study the probability of feasibility (the existence of at least one positive real equilibrium) in complex models by adding higher-order interactions and nonlinear functional responses to the linear Lotka–Volterra model. We characterize complexity by the number of free-equilibrium points generated by a model, which is a function of the polynomial degree and system’s dimension. We show that the probability of generating a feasible system in a model is an increasing function of its complexity, regardless of the specific mechanism invoked. Furthermore, we find that the probability of feasibility in a model will exceed that of the linear Lotka–Volterra model when a minimum level of complexity is reached. Importantly, this minimum level is modulated by parameter restrictions, but can always be exceeded via increasing the polynomial degree or system’s dimension. Our results reveal that conclusions regarding the relevance of mechanisms embedded in complex models must be evaluated in relation to the expected explanatory power of their polynomial forms. 

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2. Higher rank hyperbolicity

   https://doi.org/10.1007/s00222-020-00955-w  

   Kleiner, Bruce; Lang, Urs (August 2020, Inventiones mathematicae)

   null (Ed.)
3. Flagged higher categories

   https://doi.org/10.1090/conm/718/14489  

   Ayala, David; Francis, John (January 2018, Contemporary mathematics)

   null (Ed.)
4. The higher Du Bois and higher rational properties for isolated singularities

   https://doi.org/10.1090/jag/824  

   Friedman, Robert; Laza, Radu (July 2024, Journal of Algebraic Geometry)

   Higher rational and higher Du Bois singularities have recently been introduced as natural generalizations of the standard definitions of rational and Du Bois singularities. In this note, we discuss these properties for isolated singularities, especially in the locally complete intersection (lci) case. First, we reprove the fact that a $k$-rational isolated singularity is $k$-Du Bois without any lci assumption. For isolated lci singularities, we give a complete characterization of the $k$-Du Bois and $k$-rational singularities in terms of standard invariants of singularities. In particular, we show that $k$-Du Bois singularities are $(k-<\#comment/>1)$-rational for isolated lci singularities. In the course of the proof, we establish some new relations between invariants of isolated lci singularities and show that many of these vanish. The methods also lead to a quick proof of an inversion of adjunction theorem in the isolated lci case. Finally, we discuss some results specific to the hypersurface case. 

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5. Higher-Order Weyl Semimetals

   https://doi.org/10.1103/PhysRevLett.125.266804  

   Ghorashi, Sayed Ali; Li, Tianhe; Hughes, Taylor L. (December 2020, Physical Review Letters)

   null (Ed.)