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1. A uniform identification of stable sheaf cohomology

   https://doi.org/10.1090/proc/17326  

   Fiorindo, Luca; Reed, Ethan; Roshan_Zamir, Shahriyar; Yu, Hongmiao (July 2025, Proceedings of the American Mathematical Society)

   This paper considers generalizations of certain arithmetic complexes appearing in the work of Raicu and VandeBogert in connection with the study of stable sheaf cohomology on flag varieties. Defined over the ring of integer valued polynomials, we prove an isomorphism of these complexes as conjectured by Gao, Raicu, and VandeBogert. In particular, this shows that a previously made identification between the stable sheaf cohomology of hook and two column partition Schur functors applied to the cotangent sheaf of projective space can be made to be uniform with respect to these complexes. These results are extended to the projective space defined over the integers. 

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2. Vanishing of quartic and sextic twists of L-functions

   https://doi.org/10.1007/s40993-023-00499-x  

   Berg, Jennifer; Ryan, Nathan C.; Young, Matthew P. (February 2024, Research in Number Theory)

   Abstract LetEbe an elliptic curve over$${{\mathbb {Q}}}$$ $Q$. We conjecture asymptotic estimates for the number of vanishings of$$L(E,1,\chi )$$ $L(E,1,\chi )$as$$\chi $$ $\chi$varies over all primitive Dirichlet characters of orders 4 and 6, subject to a mild hypothesis onE. Our conjectures about these families come from conjectures about random unitary matrices as predicted by the philosophy of Katz-Sarnak. We support our conjectures with numerical evidence. Compared to earlier work by David, Fearnley and Kisilevsky that formulated analogous conjectures for characters of any odd prime order, in the composite order case, we need to justify our use of random matrix theory heuristics by analyzing the equidistribution of the squares of normalized Gauss sums. To do this, we introduce the notion of totally order$$\ell $$ $\ell$characters to quantify how quickly the quartic and sextic Gauss sums become equidistributed. Surprisingly, the rate of equidistribution in the full family of quartic (resp., sextic) characters is much slower than in the sub-family of totally quartic (resp., sextic) characters. We provide a conceptual explanation for this phenomenon by observing that the full family of order$$\ell $$ $\ell$twisted elliptic curveL-functions, with$$\ell $$ $\ell$even and composite, is a mixed family with both unitary and orthogonal aspects. 

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3. Modeled Surface‐Atmosphere Fluxes From Paired Sites in the Upper Great Lakes Region Using Neural Networks

   https://doi.org/10.1029/2021JG006363  

   Reed, David E.; Poe, Jeralyn; Abraha, Michael; Dahlin, Kyla M.; Chen, Jiquan (August 2021, Journal of Geophysical Research: Biogeosciences)

   Abstract The eddy covariance (EC) method is one of the most widely used approaches to quantify surface‐atmosphere fluxes. However, scaling up from a single EC tower to the landscape remains an open challenge. To address this, we used 63 site years of data to examine simulated annual and growing season sums of carbon fluxes from three paired land‐cover type sites of corn, restored‐prairie, and switchgrass ecosystems. This was also done across the landscape by modeling fluxes using different land‐cover type input data. An artificial neural network (ANN) approach was used to model net ecosystem exchange (NEE), ecosystem respiration (R_eco), and gross primary production (GPP) at one paired site using environmental observations from the second site only. With a mean spatial separation of 11 km between paired sites, we were able to model annual sums of NEE,R_eco, and GPP with uncertainties of 20%, 22%, and 8%, respectively, relative to observation sums. When considering the growing season only, model uncertainties were 17%, 22%, and 9%, respectively for the three flux terms. We also show that ANN models can estimate sums ofR_ecoand GPP fluxes without needing the constraint of similar land‐cover‐type, with annual uncertainties of 12% and 10%. These results provide new insights to scaling up observations from one EC site beyond the footprint of the EC tower to multiple land‐cover types across the landscape. 

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4. Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment

   https://doi.org/10.1093/imrn/rnac291  

   Blekherman, Grigoriy; Kummer, Mario; Sanyal, Raman; Shu, Kevin; Sun, Shengding (November 2022, International Mathematics Research Notices)

   Abstract A linear principal minor polynomial or lpm polynomial is a linear combination of principal minors of a symmetric matrix. By restricting to the diagonal, lpm polynomials are in bijection with multiaffine polynomials. We show that this establishes a one-to-one correspondence between homogeneous multiaffine stable polynomials and PSD-stable lpm polynomials. This yields new construction techniques for hyperbolic polynomials and allows us to find an explicit degree 3 hyperbolic polynomial in six variables some of whose Rayleigh differences are not sums of squares. We further generalize the well-known Fisher–Hadamard and Koteljanskii inequalities from determinants to PSD-stable lpm polynomials. We investigate the relationship between the associated hyperbolicity cones and conjecture a relationship between the eigenvalues of a symmetric matrix and the values of certain lpm polynomials evaluated at that matrix. We refer to this relationship as spectral containment. 

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5. Coherent Springer theory and the categorical Deligne-Langlands correspondence

   https://doi.org/10.1007/s00222-023-01224-2  

   Ben-Zvi, David; Chen, Harrison; Helm, David; Nadler, David (February 2024, Inventiones mathematicae)

   Abstract Kazhdan and Lusztig identified the affine Hecke algebra ℋ with an equivariant$$K$$ $K$-group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations of reductive groups over nonarchimedean local fields$$F$$ $F$with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from$$K$$ $K$-theory to Hochschild homology and thereby identify ℋ with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, thecoherent Springer sheaf. As a result the derived category of ℋ-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Fargues-Scholze, Hellmann and Zhu). In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of$$\mathrm{GL}_{n}(F)$$ ${\mathrm{GL}}_n\left(F\right)$into coherent sheaves on the stack of Langlands parameters. 

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