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1. The Dimension of the Disguised Toric Locus of a Reaction Network

   https://doi.org/10.1111/sapm.70071  

   Craciun, Gheorghe; Deshpande, Abhishek; Jin, Jiaxin (June 2025, Studies in Applied Mathematics)

   ABSTRACT Mathematical models of reaction networks are ubiquitous in applications, especially in chemistry, biochemistry, chemical engineering, ecology, and population dynamics. Under the standard assumption ofmass‐action kinetics, reaction networks give rise to general dynamical systems with polynomial right‐hand side. These depend on many parameters that are difficult to estimate and can give rise to complex dynamics, including multistability, oscillations, and chaos. On the other hand, a special class of reaction systems calledcomplex‐balanced systemsare known to exhibit remarkably stable dynamics; in particular, they have unique positive fixed points and no oscillations or chaotic dynamics. One difficulty, when trying to take advantage of the remarkable properties of complex‐balanced systems, is that the set of parameters where a network satisfies complex balance may have positive codimension and therefore zero measure. To remedy this we are studyingdisguised complex balanced systems(also known asdisguised toric systems), which may fail to be complex balanced with respect to an original reaction network , but are actually complex balanced with respect to some other network , and therefore enjoy all the stability properties of complex‐balanced systems. This notion is especially useful when the set of parameter values for which the network gives rise to disguised toric systems (i.e., thedisguised toric locusof ) has codimension zero. Our primary focus is to compute the exact dimension (and therefore the codimension) of this locus. We illustrate the use of our results by applying them to Thomas‐type and circadian clock models. 

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2. Nearly Periodic Maps and Geometric Integration of Noncanonical Hamiltonian Systems

   https://doi.org/10.1007/s00332-023-09891-4  

   Burby, J_W; Hirvijoki, E.; Leok, M. (February 2023, Journal of Nonlinear Science)

   Abstract M. Kruskal showed that each continuous-time nearly periodic dynamical system admits a formalU(1)-symmetry, generated by the so-called roto-rate. When the nearly periodic system is also Hamiltonian, Noether’s theorem implies the existence of a corresponding adiabatic invariant. We develop a discrete-time analog of Kruskal’s theory. Nearly periodic maps are defined as parameter-dependent diffeomorphisms that limit to rotations along aU(1)-action. When the limiting rotation is non-resonant, these maps admit formalU(1)-symmetries to all orders in perturbation theory. For Hamiltonian nearly periodic maps on exact presymplectic manifolds, we prove that the formalU(1)-symmetry gives rise to a discrete-time adiabatic invariant using a discrete-time extension of Noether’s theorem. When the unperturbedU(1)-orbits are contractible, we also find a discrete-time adiabatic invariant for mappings that are merely presymplectic, rather than Hamiltonian. As an application of the theory, we use it to develop a novel technique for geometric integration of non-canonical Hamiltonian systems on exact symplectic manifolds. 

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3. Lavaurs algorithm for cubic symmetric polynomials

   https://doi.org/10.1017/etds.2024.126  

   BLOKH, ALEXANDER; OVERSTEEGEN, LEX G; SELINGER, NIKITA; TIMORIN, VLADLEN; VEJANDLA, SANDEEP CHOWDARY (August 2025, Ergodic Theory and Dynamical Systems)

   Abstract As discovered by W. Thurston, the action of a complex one-variable polynomial on its Julia set can be modeled by a geodesic lamination in the disk, provided that the Julia set is connected. It also turned out that the parameter space of such dynamical laminations of degree two gives a model for the bifurcation locus in the space of quadratic polynomials. This model is itself a geodesic lamination, the so calledquadratic minor laminationof Thurston. In the same spirit, we consider the space of allcubic symmetric polynomials$$f_\unicode{x3bb} (z)=z^3+\unicode{x3bb} ^2 z$$in three articles. In the first one, we construct thecubic symmetric comajor laminationtogether with the corresponding quotient space of the unit circle. As is verified in the third paper, this yields a monotone model of thecubic symmetric connectedness locus, that is, the space of all cubic symmetric polynomials with connected Julia sets. In the present paper, the second in the series, we develop an algorithm for generating the cubic symmetric comajor lamination analogous to the Lavaurs algorithm for constructing the quadratic minor lamination. 

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4. Intersection complexes and unramified 𝐿-factors

   https://doi.org/10.1090/jams/990  

   Sakellaridis, Yiannis; Wang, Jonathan (July 2022, Journal of the American Mathematical Society)

   Let X X be an affine spherical variety, possibly singular, and L + X \mathsf L^+X its arc space. The intersection complex of L + X \mathsf L^+X , or rather of its finite-dimensional formal models, is conjectured to be related to special values of local unramified L L -functions. Such relationships were previously established in Braverman–Finkelberg–Gaitsgory–Mirković for the affine closure of the quotient of a reductive group by the unipotent radical of a parabolic, and in Bouthier–Ngô–Sakellaridis for toric varieties and L L -monoids. In this paper, we compute this intersection complex for the large class of those spherical G G -varieties whose dual group is equal to G ˇ \check G , and the stalks of its nearby cycles on the horospherical degeneration of X X . We formulate the answer in terms of a Kashiwara crystal, which conjecturally corresponds to a finite-dimensional G ˇ \check G -representation determined by the set of B B -invariant valuations on X X . We prove the latter conjecture in many cases. Under the sheaf–function dictionary, our calculations give a formula for the Plancherel density of the IC function of L + X \mathsf L^+X as a ratio of local L L -values for a large class of spherical varieties. 

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5. Persistent cup product structures and related invariants

   https://doi.org/10.1007/s41468-023-00138-5  

   Mémoli, Facundo; Stefanou, Anastasios; Zhou, Ling (October 2023, Journal of Applied and Computational Topology)

   Abstract One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and by utilizing cohomological ideas, e.g. the cohomological cup product. In this work, given a single parameter filtration, we investigate a certain 2-dimensional persistence module structure associated with persistent cohomology, where one parameter is the cup-length$$\ell \ge 0$$ $\ell \ge 0$and the other is the filtration parameter. This new persistence structure, called thepersistent cup module, is induced by the cohomological cup product and adapted to the persistence setting. Furthermore, we show that this persistence structure is stable. By fixing the cup-length parameter$$\ell $$ $\ell$, we obtain a 1-dimensional persistence module, called the persistent$$\ell $$ $\ell$-cup module, and again show it is stable in the interleaving distance sense, and study their associated generalized persistence diagrams. In addition, we consider a generalized notion of apersistent invariant, which extends both therank invariant(also referred to aspersistent Betti number), Puuska’s rank invariant induced by epi-mono-preserving invariants of abelian categories, and the recently-definedpersistent cup-length invariant, and we establish their stability. This generalized notion of persistent invariant also enables us to lift the Lyusternik-Schnirelmann (LS) category of topological spaces to a novel stable persistent invariant of filtrations, called thepersistent LS-category invariant. 

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