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1. Weighted homological regularities

   https://doi.org/10.1090/tran/8997  

   Kirkman, E.; Won, R.; Zhang, J. (August 2023, Transactions of the American Mathematical Society)

   Dan Abramovich (Ed.)

   Let $A$be a noetherian connected graded algebra. We introduce and study homological invariants that are weighted sums of the homological and internal degrees of cochain complexes of graded $A$-modules, providing weighted versions of Castelnuovo–Mumford regularity, Tor-regularity, Artin–Schelter regularity, and concavity. In some cases an invariant (such as Tor-regularity) that is infinite can be replaced with a weighted invariant that is finite, and several homological invariants of complexes can be expressed as weighted homological regularities. We prove a few weighted homological identities some of which unify different classical homological identities and produce interesting new ones. 

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2. Congruences with Eisenstein series and -invariants

   https://doi.org/10.1112/S0010437X19007127  

   Bellaïche, Joël; Pollack, Robert (May 2019, Compositio Mathematica)

   We study the variation of $$\unicode[STIX]{x1D707}$$ -invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the $$p$$ -adic zeta function. This lower bound forces these $$\unicode[STIX]{x1D707}$$ -invariants to be unbounded along the family, and we conjecture that this lower bound is an equality. When $$U_{p}-1$$ generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the $$p$$ -adic $$L$$ -function is simply a power of $$p$$ up to a unit (i.e.  $$\unicode[STIX]{x1D706}=0$$ ). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms. 

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3. Chord measures in integral geometry and their Minkowski problems

   https://doi.org/10.1002/cpa.22190  

   Lutwak, Erwin; Xi, Dongmeng; Yang, Deane; Zhang, Gaoyong (December 2023, Communications on Pure and Applied Mathematics)

   Abstract To the families of geometric measures of convex bodies (the area measures of Aleksandrov‐Fenchel‐Jessen, the curvature measures of Federer, and the recently discovered dual curvature measures) a new family is added. The new family of geometric measures, called chord measures, arises from the study of integral geometric invariants of convex bodies. The Minkowski problems for the new measures and their logarithmic variants are proposed and attacked. When the given ‘data’ is sufficiently regular, these problems are a new type of fully nonlinear partial differential equations involving dual quermassintegrals of functions. Major cases of these Minkowski problems are solved without regularity assumptions. 

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4. K-theoretic Gromov–Witten invariants of line degrees on flag varieties

   https://doi.org/10.1142/S0217751X24460138  

   Buch, Anders S; Chen, Linda; Xu, Weihong (November 2024, International Journal of Modern Physics A)

   A homology class [Formula: see text] of a complex flag variety [Formula: see text] is called a line degree if the moduli space [Formula: see text] of 0-pointed stable maps to X of degree d is also a flag variety [Formula: see text]. We prove a quantum equals classical formula stating that any n-pointed (equivariant, [Formula: see text]-theoretic, genus zero) Gromov–Witten invariant of line degree on X is equal to a classical intersection number computed on the flag variety [Formula: see text]. We also prove an n-pointed analogue of the Peterson comparison formula stating that these invariants coincide with Gromov–Witten invariants of the variety of complete flags [Formula: see text]. Our formulas make it straightforward to compute the big quantum [Formula: see text]-theory ring [Formula: see text] modulo the ideal [Formula: see text] generated by degrees d larger than line degrees. 

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5. Safety assessemt based on physically-viable data-driven models

   https://doi.org/10.1109/CDC.2017.8264626  

   Ahmadi, Mohamadreza; Israel, Arie; Topcu, Ufuk (December 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC))

   We consider the problem of safety assessment of a dynamical system for which no model and just limited data on the states is available. That is, given samples of the state {x(t i )} N i=1 at time instances t 1 ≤ t 2 ≤ ··· ≤ t N and some other side information in terms of the regularity of the state evolutions, we are interested in checking whether x(T) ∉ Xu, where T > t N and Xu ⊂ R n (the unsafe set) are pre-specified. To this end, we use piecewise-polynomial approximations of the trajectories based on the data along with the regularity side information to formulate a data-driven differential inclusion model. For these classes of data-driven differential inclusions, we propose a safety assessment theorem based on barrier certificates. The barrier certificates are then found using polynomial optimization. The method is illustrated by two examples. 

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