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1. Equivariant quantum differential equation and qKZ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and B-Theorem

   Giordano Cotti, Alexander Varchenko (January 2021, Proceedings of symposia in pure mathematics)

   Novikov, Krichever (Ed.)

   In [TV19a] the equivariant quantum differential equation (qDE) for a projective space was considered and a compatible system of difference qKZ equations was introduced; the space of solutions to the joint system of the qDE and qKZ equations was identified with the space of the equivariant K-theory algebra of the projective space; Stokes bases in the space of solutions were identified with exceptional bases in the equivariant K-theory algebra. This paper is a continuation of [TV19a]. We describe the relation between solutions to the joint system of the qDE and qKZ equations and the topological-enumerative solution to the qDE only, defined as a generating function of equivariant descendant Gromov-Witten invariants. The relation is in terms of the equivariant graded Chern character on the equivariant K-theory algebra, the equivariant Gamma class of the projective space, and the equivariant first Chern class of the tangent bundle of the projective space. We consider a Stokes basis, the associated exceptional basis in the equivariant K-theory algebra, and the associated Stokes matrix. We show that the Stokes matrix equals the Gram matrix of the equivariant Grothendieck-Euler-Poincaré pairing wrt to the basis, which is the left dual to the associated exceptional basis. We identify the Stokes bases in the space of solutions with explicit full exceptional collections in the equivariant derived category of coherent sheaves on the projective space, where the elements of those exceptional collections are just line bundles on the projective space and exterior powers of the tangent bundle of the projective space. These statements are equivariant analogs of results of G. Cotti, B. Dubrovin, D. Guzzetti, and S. Galkin, V. Golyshev, H. Iritani. 

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2. Equivariant Vector Bundles on Complexity-One T -Varieties and Bruhat–Tits Buildings

   https://doi.org/10.1093/imrn/rnaf124  

   Dasgupta, Jyoti; Gangopadhyay, Chandranandan; Kaveh, Kiumars; Manon, Christopher (May 2025, International Mathematics Research Notices)

   Abstract We give a combinatorial classification of torus equivariant vector bundles on a (normal) projective $$T$$-variety of complexity-one. This extends the classification of equivariant line bundles on complexity-one $$T$$-varieties by Petersen–Süß on one hand, and Klyachko’s classification of equivariant vector bundles on toric varieties on the other hand. A main ingredient in our classification is the classification of torus equivariant vector bundles on toric schemes over a DVR in terms of piecewise affine maps to the (extended) Bruhat–Tits building of the general linear group. 

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3. On the existence of Parseval frames for vector bundles

   https://doi.org/10.1090/btran/223  

   Ballas, Samuel; Needham, Tom; Shonkwiler, Clayton (March 2025, Transactions of the American Mathematical Society, Series B)

   Frames in finite-dimensional vector spaces are spanning sets of vectors which provide redundant representations of signals. The Parseval frames are particularly useful and important, since they provide a simple reconstruction scheme and are maximally robust against certain types of noise. In this paper we describe a theory of frames on arbitrary vector bundles—this is the natural setting for signals which are realized as parameterized families of vectors rather than as single vectors—and discuss the existence of Parseval frames in this setting. Our approach is phrased in the language of G-bundles, which allows us to use many tools from classical algebraic topology. In particular, we show that orientable vector bundles always admit Parseval frames of sufficiently large size and provide an upper bound on the necessary size. We also give sufficient conditions for the existence of Parseval frames of smaller size for tangent bundles of several families of manifolds, and provide some numerical evidence that Parseval frames on vector bundles share the desirable reconstruction properties of classical Parseval frames. 

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4. Maximal Brill–Noether Loci via K3 Surfaces

   https://doi.org/10.1093/imrn/rnaf313  

   Auel, Asher; Haburcak, Richard (October 2025, International Mathematics Research Notices)

   Abstract The Brill–Noether loci $$\mathcal{M}^{r}_{g,d}$$ parameterize curves of genus $$g$$ admitting a linear system of rank $$r$$ and degree $$d$$. When the Brill–Noether number is negative, they are proper subvarieties of the moduli space of genus $$g$$ curves. We explain a strategy for distinguishing Brill–Noether loci by studying the lifting of linear systems on curves in polarized K3 surfaces, which motivates a conjecture identifying the maximal Brill–Noether loci. Via an analysis of the stability of Lazarsfeld–Mukai bundles, we obtain new lifting results for line bundles of type $$g^{3}_{d}$$ that suffice to prove the maximal Brill–Noether loci conjecture in genus $$3$$–$19$, $22$, $23$, and infinitely many cases. 

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5. Approximate and discrete Euclidean vector bundles

   https://doi.org/10.1017/fms.2023.16  

   Scoccola, Luis; Perea, Jose A. (January 2023, Forum of Mathematics, Sigma)

   Abstract We introduce $$\varepsilon $$ -approximate versions of the notion of a Euclidean vector bundle for $$\varepsilon \geq 0$$ , which recover the classical notion of a Euclidean vector bundle when $$\varepsilon = 0$$ . In particular, we study Čech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that $$\varepsilon $$ -approximate vector bundles can be used to represent classical vector bundles when $$\varepsilon> 0$$ is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data and also allows for some tolerance to noise when working with vector bundles in an applied setting. As an example, we prove a reconstruction theorem for vector bundles from finite samples. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples. 

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