@article{14930, abstract = {In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra R by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over R equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over R[t]/(t2). Using these results together with results of Geiss, Leclerc and Schröer we give, when k is algebraically closed, a classification of pairs (Q, R) such that the set of isomorphism classes of indecomposable locally free representations of Q over R is finite. Finally when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra Fq[t]/(tr). We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation.}, author = {Hausel, Tamás and Letellier, Emmanuel and Rodriguez-Villegas, Fernando}, issn = {1420-9020}, journal = {Selecta Mathematica}, number = {2}, publisher = {Springer Nature}, title = {{Locally free representations of quivers over commutative Frobenius algebras}}, doi = {10.1007/s00029-023-00914-2}, volume = {30}, year = {2024}, } @article{14986, abstract = {We prove a version of the tamely ramified geometric Langlands correspondence in positive characteristic for GLn(k). Let k be an algebraically closed field of characteristic p>n. Let X be a smooth projective curve over k with marked points, and fix a parabolic subgroup of GLn(k) at each marked point. We denote by Bunn,P the moduli stack of (quasi-)parabolic vector bundles on X, and by Locn,P the moduli stack of parabolic flat connections such that the residue is nilpotent with respect to the parabolic reduction at each marked point. We construct an equivalence between the bounded derived category Db(Qcoh(Loc0n,P)) of quasi-coherent sheaves on an open substack Loc0n,P⊂Locn,P, and the bounded derived category Db(D0Bunn,P-mod) of D0Bunn,P-modules, where D0Bunn,P is a localization of DBunn,P the sheaf of crystalline differential operators on Bunn,P. Thus we extend the work of Bezrukavnikov-Braverman to the tamely ramified case. We also prove a correspondence between flat connections on X with regular singularities and meromorphic Higgs bundles on the Frobenius twist X(1) of X with first order poles .}, author = {Shen, Shiyu}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, keywords = {General Mathematics}, publisher = {Oxford University Press}, title = {{Tamely ramified geometric Langlands correspondence in positive characteristic}}, doi = {10.1093/imrn/rnae005}, year = {2024}, } @article{12329, abstract = {In this article, we develop two independent and new approaches to model epidemic spread in a network. Contrary to the most studied models, those developed here allow for contacts with different probabilities of transmitting the disease (transmissibilities). We then examine each of these models using some mean field type approximations. The first model looks at the late-stage effects of an epidemic outbreak and allows for the computation of the probability that a given vertex was infected. This computation is based on a mean field approximation and only depends on the number of contacts and their transmissibilities. This approach shares many similarities with percolation models in networks. The second model we develop is a dynamic model which we analyze using a mean field approximation which highly reduces the dimensionality of the system. In particular, the original system which individually analyses each vertex of the network is reduced to one with as many equations as different transmissibilities. Perhaps the greatest contribution of this article is the observation that, in both these models, the existence and size of an epidemic outbreak are linked to the properties of a matrix which we call the R-matrix. This is a generalization of the basic reproduction number which more precisely characterizes the main routes of infection.}, author = {Gómez, Arturo and Oliveira, Goncalo}, issn = {2045-2322}, journal = {Scientific Reports}, publisher = {Springer Nature}, title = {{New approaches to epidemic modeling on networks}}, doi = {10.1038/s41598-022-19827-9}, volume = {13}, year = {2023}, } @article{13966, abstract = {We present a low-scaling diagrammatic Monte Carlo approach to molecular correlation energies. Using combinatorial graph theory to encode many-body Hugenholtz diagrams, we sample the Møller-Plesset (MPn) perturbation series, obtaining accurate correlation energies up to n=5, with quadratic scaling in the number of basis functions. Our technique reduces the computational complexity of the molecular many-fermion correlation problem, opening up the possibility of low-scaling, accurate stochastic computations for a wide class of many-body systems described by Hugenholtz diagrams.}, author = {Bighin, Giacomo and Ho, Quoc P and Lemeshko, Mikhail and Tscherbul, T. V.}, issn = {2469-9969}, journal = {Physical Review B}, number = {4}, publisher = {American Physical Society}, title = {{Diagrammatic Monte Carlo for electronic correlation in molecules: High-order many-body perturbation theory with low scaling}}, doi = {10.1103/PhysRevB.108.045115}, volume = {108}, year = {2023}, } @article{14239, abstract = {Given a resolution of rational singularities π:X~→X over a field of characteristic zero, we use a Hodge-theoretic argument to prove that the image of the functor Rπ∗:Db(X~)→Db(X) between bounded derived categories of coherent sheaves generates Db(X) as a triangulated category. This gives a weak version of the Bondal–Orlov localization conjecture [BO02], answering a question from [PS21]. The same result is established more generally for proper (not necessarily birational) morphisms π:X~→X , with X~ smooth, satisfying Rπ∗(OX~)=OX .}, author = {Mauri, Mirko and Shinder, Evgeny}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{Homological Bondal-Orlov localization conjecture for rational singularities}}, doi = {10.1017/fms.2023.65}, volume = {11}, year = {2023}, } @article{13268, abstract = {We give a simple argument to prove Nagai’s conjecture for type II degenerations of compact hyperkähler manifolds and cohomology classes of middle degree. Under an additional assumption, the techniques yield the conjecture in arbitrary degree. This would complete the proof of Nagai’s conjecture in general, as it was proved already for type I degenerations by Kollár, Laza, Saccà, and Voisin [10] and independently by Soldatenkov [18], while it is immediate for type III degenerations. Our arguments are close in spirit to a recent paper by Harder [8] proving similar results for the restrictive class of good degenerations.}, author = {Huybrechts, D. and Mauri, Mirko}, issn = {1945-001X}, journal = {Mathematical Research Letters}, number = {1}, pages = {125--141}, publisher = {International Press}, title = {{On type II degenerations of hyperkähler manifolds}}, doi = {10.4310/mrl.2023.v30.n1.a6}, volume = {30}, year = {2023}, } @article{14244, abstract = {In this paper, we determine the motivic class — in particular, the weight polynomial and conjecturally the Poincaré polynomial — of the open de Rham space, defined and studied by Boalch, of certain moduli spaces of irregular meromorphic connections on the trivial rank bundle on P1. The computation is by motivic Fourier transform. We show that the result satisfies the purity conjecture, that is, it agrees with the pure part of the conjectured mixed Hodge polynomial of the corresponding wild character variety. We also identify the open de Rham spaces with quiver varieties with multiplicities of Yamakawa and Geiss–Leclerc–Schröer. We finish with constructing natural complete hyperkähler metrics on them, which in the four-dimensional cases are expected to be of type ALF.}, author = {Hausel, Tamás and Wong, Michael Lennox and Wyss, Dimitri}, issn = {1460-244X}, journal = {Proceedings of the London Mathematical Society}, number = {4}, pages = {958--1027}, publisher = {Wiley}, title = {{Arithmetic and metric aspects of open de Rham spaces}}, doi = {10.1112/plms.12555}, volume = {127}, year = {2023}, } @inbook{12303, abstract = {We construct for each choice of a quiver Q, a cohomology theory A, and a poset P a “loop Grassmannian” GP(Q,A). This generalizes loop Grassmannians of semisimple groups and the loop Grassmannians of based quadratic forms. The addition of a “dilation” torus D⊆G2m gives a quantization GPD(Q,A). This construction is motivated by the program of introducing an inner cohomology theory in algebraic geometry adequate for the Geometric Langlands program (Mirković, Some extensions of the notion of loop Grassmannians. Rad Hrvat. Akad. Znan. Umjet. Mat. Znan., the Mardešić issue. No. 532, 53–74, 2017) and on the construction of affine quantum groups from generalized cohomology theories (Yang and Zhao, Quiver varieties and elliptic quantum groups, preprint. arxiv1708.01418).}, author = {Mirković, Ivan and Yang, Yaping and Zhao, Gufang}, booktitle = {Representation Theory and Algebraic Geometry}, editor = {Baranovskky, Vladimir and Guay, Nicolas and Schedler, Travis}, isbn = {9783030820060}, issn = {2297-024X}, pages = {347--392}, publisher = {Springer Nature; Birkhäuser}, title = {{Loop Grassmannians of Quivers and Affine Quantum Groups}}, doi = {10.1007/978-3-030-82007-7_8}, year = {2022}, } @article{9977, abstract = {For a Seifert fibered homology sphere X we show that the q-series invariant Zˆ0(X; q) introduced by Gukov-Pei-Putrov-Vafa, is a resummation of the Ohtsuki series Z0(X). We show that for every even k ∈ N there exists a full asymptotic expansion of Zˆ0(X; q) for q tending to e 2πi/k, and in particular that the limit Zˆ0(X; e 2πi/k) exists and is equal to the WRT quantum invariant τk(X). We show that the poles of the Borel transform of Z0(X) coincide with the classical complex Chern-Simons values, which we further show classifies the corresponding components of the moduli space of flat SL(2, C)-connections.}, author = {Mistegaard, William and Andersen, Jørgen Ellegaard}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {2}, pages = {709--764}, publisher = {Wiley}, title = {{Resurgence analysis of quantum invariants of Seifert fibered homology spheres}}, doi = {10.1112/jlms.12506}, volume = {105}, year = {2022}, } @article{10704, abstract = {We define and study the existence of very stable Higgs bundles on Riemann surfaces, how it implies a precise formula for the multiplicity of the very stable components of the global nilpotent cone and its relationship to mirror symmetry. The main ingredients are the Bialynicki-Birula theory of C∗-actions on semiprojective varieties, C∗ characters of indices of C∗-equivariant coherent sheaves, Hecke transformation for Higgs bundles, relative Fourier–Mukai transform along the Hitchin fibration, hyperholomorphic structures on universal bundles and cominuscule Higgs bundles.}, author = {Hausel, Tamás and Hitchin, Nigel}, issn = {1432-1297}, journal = {Inventiones Mathematicae}, pages = {893--989}, publisher = {Springer Nature}, title = {{Very stable Higgs bundles, equivariant multiplicity and mirror symmetry}}, doi = {10.1007/s00222-021-01093-7}, volume = {228}, year = {2022}, } @article{10772, abstract = {We introduce tropical corals, balanced trees in a half-space, and show that they correspond to holomorphic polygons capturing the product rule in Lagrangian Floer theory for the elliptic curve. We then prove a correspondence theorem equating counts of tropical corals to punctured log Gromov–Witten invariants of the Tate curve. This implies that the homogeneous coordinate ring of the mirror to the Tate curve is isomorphic to the degree-zero part of symplectic cohomology, confirming a prediction of homological mirror symmetry.}, author = {Arguez, Nuroemuer Huelya}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {1}, pages = {343--411}, publisher = {London Mathematical Society}, title = {{Mirror symmetry for the Tate curve via tropical and log corals}}, doi = {10.1112/jlms.12515}, volume = {105}, year = {2022}, } @article{12793, abstract = {Let F be a global function field with constant field Fq. Let G be a reductive group over Fq. We establish a variant of Arthur's truncated kernel for G and for its Lie algebra which generalizes Arthur's original construction. We establish a coarse geometric expansion for our variant truncation. As applications, we consider some existence and uniqueness problems of some cuspidal automorphic representations for the functions field of the projective line P1Fq with two points of ramifications.}, author = {Yu, Hongjie}, issn = {1945-5844}, journal = {Pacific Journal of Mathematics}, keywords = {Arthur–Selberg trace formula, cuspidal automorphic representations, global function fields}, number = {1}, pages = {193--237}, publisher = {Mathematical Sciences Publishers}, title = {{ A coarse geometric expansion of a variant of Arthur's truncated traces and some applications}}, doi = {10.2140/pjm.2022.321.193}, volume = {321}, year = {2022}, } @article{6965, abstract = {The central object of investigation of this paper is the Hirzebruch class, a deformation of the Todd class, given by Hirzebruch (for smooth varieties). The generalization for singular varieties is due to Brasselet–Schürmann–Yokura. Following the work of Weber, we investigate its equivariant version for (possibly singular) toric varieties. The local decomposition of the Hirzebruch class to the fixed points of the torus action and a formula for the local class in terms of the defining fan are recalled. After this review part, we prove the positivity of local Hirzebruch classes for all toric varieties, thus proving false the alleged counterexample given by Weber.}, author = {Rychlewicz, Kamil P}, issn = {1469-2120}, journal = {Bulletin of the London Mathematical Society}, number = {2}, pages = {560--574}, publisher = {Wiley}, title = {{The positivity of local equivariant Hirzebruch class for toric varieties}}, doi = {10.1112/blms.12442}, volume = {53}, year = {2021}, } @article{9099, abstract = {We show that on an Abelian variety over an algebraically closed field of positive characteristic, the obstruction to lifting an automorphism to a field of characteristic zero as a morphism vanishes if and only if it vanishes for lifting it as a derived autoequivalence. We also compare the deformation space of these two types of deformations.}, author = {Srivastava, Tanya K}, issn = {14208938}, journal = {Archiv der Mathematik}, number = {5}, pages = {515--527}, publisher = {Springer Nature}, title = {{Lifting automorphisms on Abelian varieties as derived autoequivalences}}, doi = {10.1007/s00013-020-01564-y}, volume = {116}, year = {2021}, } @article{9173, abstract = {We show that Hilbert schemes of points on supersingular Enriques surface in characteristic 2, Hilbn(X), for n ≥ 2 are simply connected, symplectic varieties but are not irreducible symplectic as the hodge number h2,0 > 1, even though a supersingular Enriques surface is an irreducible symplectic variety. These are the classes of varieties which appear only in characteristic 2 and they show that the hodge number formula for G¨ottsche-Soergel does not hold over haracteristic 2. It also gives examples of varieties with trivial canonical class which are neither irreducible symplectic nor Calabi-Yau, thereby showing that there are strictly more classes of simply connected varieties with trivial canonical class in characteristic 2 than over C as given by Beauville-Bogolomov decomposition theorem.}, author = {Srivastava, Tanya K}, issn = {0007-4497}, journal = {Bulletin des Sciences Mathematiques}, number = {03}, publisher = {Elsevier}, title = {{Pathologies of the Hilbert scheme of points of a supersingular Enriques surface}}, doi = {10.1016/j.bulsci.2021.102957}, volume = {167}, year = {2021}, } @article{9359, abstract = {We prove that the factorization homologies of a scheme with coefficients in truncated polynomial algebras compute the cohomologies of its generalized configuration spaces. Using Koszul duality between commutative algebras and Lie algebras, we obtain new expressions for the cohomologies of the latter. As a consequence, we obtain a uniform and conceptual approach for treating homological stability, homological densities, and arithmetic densities of generalized configuration spaces. Our results categorify, generalize, and in fact provide a conceptual understanding of the coincidences appearing in the work of Farb--Wolfson--Wood. Our computation of the stable homological densities also yields rational homotopy types, answering a question posed by Vakil--Wood. Our approach hinges on the study of homological stability of cohomological Chevalley complexes, which is of independent interest. }, author = {Ho, Quoc P}, issn = {1364-0380}, journal = {Geometry & Topology}, keywords = {Generalized configuration spaces, homological stability, homological densities, chiral algebras, chiral homology, factorization algebras, Koszul duality, Ran space}, number = {2}, pages = {813--912}, publisher = {Mathematical Sciences Publishers}, title = {{Homological stability and densities of generalized configuration spaces}}, doi = {10.2140/gt.2021.25.813}, volume = {25}, year = {2021}, } @article{9998, abstract = {We define quantum equivariant K-theory of Nakajima quiver varieties. We discuss type A in detail as well as its connections with quantum XXZ spin chains and trigonometric Ruijsenaars-Schneider models. Finally we study a limit which produces a K-theoretic version of results of Givental and Kim, connecting quantum geometry of flag varieties and Toda lattice.}, author = {Koroteev, Peter and Pushkar, Petr and Smirnov, Andrey V. and Zeitlin, Anton M.}, issn = {1420-9020}, journal = {Selecta Mathematica}, number = {5}, publisher = {Springer Nature}, title = {{Quantum K-theory of quiver varieties and many-body systems}}, doi = {10.1007/s00029-021-00698-3}, volume = {27}, year = {2021}, } @article{10033, abstract = {The ⊗*-monoidal structure on the category of sheaves on the Ran space is not pro-nilpotent in the sense of [3]. However, under some connectivity assumptions, we prove that Koszul duality induces an equivalence of categories and that this equivalence behaves nicely with respect to Verdier duality on the Ran space and integrating along the Ran space, i.e. taking factorization homology. Based on ideas sketched in [4], we show that these results also offer a simpler alternative to one of the two main steps in the proof of the Atiyah-Bott formula given in [7] and [5].}, author = {Ho, Quoc P}, issn = {1090-2082}, journal = {Advances in Mathematics}, keywords = {Chiral algebras, Chiral homology, Factorization algebras, Koszul duality, Ran space}, publisher = {Elsevier}, title = {{The Atiyah-Bott formula and connectivity in chiral Koszul duality}}, doi = {10.1016/j.aim.2021.107992}, volume = {392}, year = {2021}, } @article{7004, abstract = {We define an action of the (double of) Cohomological Hall algebra of Kontsevich and Soibelman on the cohomology of the moduli space of spiked instantons of Nekrasov. We identify this action with the one of the affine Yangian of gl(1). Based on that we derive the vertex algebra at the corner Wr1,r2,r3 of Gaiotto and Rapčák. We conjecture that our approach works for a big class of Calabi–Yau categories, including those associated with toric Calabi–Yau 3-folds.}, author = {Rapcak, Miroslav and Soibelman, Yan and Yang, Yaping and Zhao, Gufang}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, pages = {1803--1873}, publisher = {Springer Nature}, title = {{Cohomological Hall algebras, vertex algebras and instantons}}, doi = {10.1007/s00220-019-03575-5}, volume = {376}, year = {2020}, } @article{7683, abstract = {For any free oriented Borel–Moore homology theory A, we construct an associative product on the A-theory of the stack of Higgs torsion sheaves over a projective curve C. We show that the resulting algebra AHa0C admits a natural shuffle presentation, and prove it is faithful when A is replaced with usual Borel–Moore homology groups. We also introduce moduli spaces of stable triples, heavily inspired by Nakajima quiver varieties, whose A-theory admits an AHa0C-action. These triples can be interpreted as certain sheaves on PC(ωC⊕OC). In particular, we obtain an action of AHa0C on the cohomology of Hilbert schemes of points on T∗C.}, author = {Minets, Sasha}, issn = {14209020}, journal = {Selecta Mathematica, New Series}, number = {2}, publisher = {Springer Nature}, title = {{Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces}}, doi = {10.1007/s00029-020-00553-x}, volume = {26}, year = {2020}, }