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1. Structure constants in equivariant oriented cohomology of flag varieties

   https://doi.org/10.1007/s00209-024-03485-w  

   Goldin, Rebecca; Zhong, Changlong (July 2024, Mathematische Zeitschrift)

   We introduce generalized Demazure operators for the equivariant oriented cohomology of the flag variety, which have specializations to various Demazure operators and Demazure–Lusztig operators in both equivariant cohomology and equivariant K-theory. In the context of the geometric basis of the equivariant oriented cohomology given by certain Bott–Samelson classes, we use these operators to obtain formulas for the structure constants arising in different bases. Specializing to divided difference operators and Demazure operators in singular cohomology and K-theory, we recover the formulas for structure constants of Schubert classes obtained in Goldin and Knutson (Pure Appl Math Q 17(4):1345–1385, 2021). Two specific specializations result in formulas for the the structure constants for cohomological and K-theoretic stable bases as well; as a corollary we reproduce a formula for the structure constants of the Segre–Schwartz–MacPherson basis previously obtained by Su (Math Zeitschrift 298:193–213, 2021). Our methods involve the study of the formal affine Demazure algebra, providing a purely algebraic proof of these results. 

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2. The stable Galois correspondence for real closed fields

   https://doi.org/10.1090/conm/707/14250  

   Heller, J.; Ormsby, K. (January 2018, Contemporary mathematics - American Mathematical Society)

   In previous work [7], the authors constructed and studied a lift of the Galois correspondence to stable homotopy categories. In particular, if L/k is a finite Galois extension of fields with Galois group G, there is a functor c∗L/k : SHG → SHk from the G-equivariant stable homotopy category to the stable motivic homotopy category over k such that c∗L/k(G/H+) = Spec(LH)+. The main theorem of [7] says that when k is a real closed field and L = k[i], the restriction of c∗L/k to the η-complete subcategory is full and faithful. Here we “uncomplete” this theorem so that it applies to c∗L/k itself. Our main tools are Bachmann’s theorem on the (2,η)- periodic stable motivic homotopy category and an isomorphism range for the map πRS → πC2 S induced by C2-equivariant Betti realization. 

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3. Co-design of CSS Codes and Diagonal Gates

   https://doi.org/10.1109/ISIT50566.2022.9834511  

   Hu, Jingzhen; Liang, Qingzhong; Calderbank, Robert (June 2022, 2022 IEEE International Symposium on Information Theory)

   A diagonal physical gate U has 2^n diagonal entries, each indexed by a binary vector v of length n. A CSS codespace C on n qubits is specified by two classical code C1 and C2, where C2 provides the X-stabilizers and the dual of C1 provides the Z-stabilizers. We proved U preserves C if and only if entries indexed by the same coset of C2 in C1 (same X-logical) are identical. 

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4. Stable homotopy refinement of quantum annular homology

   https://doi.org/10.1112/S0010437X20007721  

   Akhmechet, Rostislav; Krushkal, Vyacheslav; Willis, Michael (April 2021, Compositio Mathematica)

   null (Ed.)

   We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $$r\geq ~2$$ we associate to an annular link $$L$$ a naive $$\mathbb {Z}/r\mathbb {Z}$$ -equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $$L$$ as modules over $$\mathbb {Z}[\mathbb {Z}/r\mathbb {Z}]$$ . The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology. 

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5. Characterizing the origin band spectrum of isoquinoline with resonance enhanced multiphoton ionization and electronic structure calculations

   https://doi.org/10.1063/5.0168421  

   Krogmeier, Timothy J.; Pappas, Emerson S.; Reardon, Kylie A.; Rivera, Marcos R.; Head-Marsden, Kade; Parsons, Bradley F.; Schlimgen, Anthony W. (October 2023, The Journal of Chemical Physics)

   We report the experimental resonance enhanced multiphoton ionization spectrum of isoquinoline between 315 and 310 nm, along with correlated electronic structure calculations on the ground and excited states of this species. This spectral region spans the origin transitions to a π–π* excited state, which previous work has suggested to be vibronically coupled with a lower lying singlet n–π* state. Our computational results corroborate previous density functional theory calculations that predict the vertical excitation energy for the n–π* state to be higher than the π–π* state; however, we find an increase in the C–N–C angle brings the n–π* state below the energy of the π–π* state. The calculations find two out-of-plane vibrational modes of the n–π* state, which may be brought into near resonance with the π–π* state as the C–N–C bond angle increases. Therefore, the C–N–C bond angle may be important in activating vibronic coupling between the states. We fit the experimental rotational contour with a genetic algorithm to determine the excited state rotational constants and orientation of the transition dipole moment. The fits show a mostly in-plane polarized transition, and the projection of the transition dipole moment in the a-b plane is about 84° away from the a axis. These results are consistent with the prediction of our electronic structure calculations for the transition dipole moment of the π–π* excited state. 

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