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1. Consistency of M-Theory on Non-Orientable Manifolds

   https://doi.org/10.1093/qmath/haab007  

   Freed, Daniel S; Hopkins, Michael J (April 2021, The Quarterly Journal of Mathematics)

   Abstract We prove that there is no parity anomaly in M-theory in the low-energy field theory approximation. Our approach is computational. We determine the generators for the 12-dimensional bordism group of pin manifolds with a w1-twisted integer lift of w4; these are the manifolds on which Wick-rotated M-theory exists. The anomaly cancellation comes down to computing a specific η-invariant and cubic form on these manifolds. Of interest beyond this specific problem are our expositions of computational techniques for η-invariants, the algebraic theory of cubic forms, Adams spectral sequence techniques and anomalies for spinor fields and Rarita–Schwinger fields. 

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2. Cohomological Invariants in Positive Characteristic

   https://doi.org/10.1093/imrn/rnaa321  

   Totaro, Burt (January 2021, International Mathematics Research Notices)

   Abstract We determine the mod $$p$$ cohomological invariants for several affine group schemes $$G$$ in characteristic $$p$$. These are invariants of $$G$$-torsors with values in étale motivic cohomology, or equivalently in Kato’s version of Galois cohomology based on differential forms. In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2. We also determine all operations on the mod $$p$$ étale motivic cohomology of fields, extending Vial’s computation of the operations on the mod $$p$$ Milnor K-theory of fields. 

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3. 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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4. Counting integral points on symmetric varieties with applications to arithmetic statistics

   https://doi.org/10.1112/plms.70039  

   Shankar, Arul; Siad, Artane; Swaminathan, Ashvin A (April 2025, Proceedings of the London Mathematical Society)

   Abstract In this article, we combine Bhargava's geometry‐of‐numbers methods with the dynamical point‐counting methods of Eskin–McMullen and Benoist–Oh to develop a new technique for counting integral points on symmetric varieties lying within fundamental domains for coregular representations. As applications, we study the distribution of the 2‐torsion subgroup of the class group in thin families of cubic number fields, as well as the distribution of the 2‐Selmer groups in thin families of elliptic curves over . For example, our results suggest that the existence of a generator of the ring of integers with small norm has an increasing effect on the average size of the 2‐torsion subgroup of the class group, relative to the Cohen–Lenstra predictions. 

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5. Stability conditions in families

   https://doi.org/10.1007/s10240-021-00124-6  

   Bayer, Arend; Lahoz, Martí; Macrì, Emanuele; Nuer, Howard; Perry, Alexander; Stellari, Paolo (June 2021, Publications mathématiques de l'IHÉS)

   Abstract We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers. Our main application is the generalization of Mukai’s theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type. Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration. 

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