TY - JOUR AB - 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. AU - Hausel, Tamás AU - Letellier, Emmanuel AU - Rodriguez-Villegas, Fernando ID - 14930 IS - 2 JF - Selecta Mathematica SN - 1022-1824 TI - Locally free representations of quivers over commutative Frobenius algebras VL - 30 ER - TY - JOUR AB - 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 . AU - Shen, Shiyu ID - 14986 JF - International Mathematics Research Notices KW - General Mathematics SN - 1073-7928 TI - Tamely ramified geometric Langlands correspondence in positive characteristic ER - TY - JOUR AB - 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. AU - Gómez, Arturo AU - Oliveira, Goncalo ID - 12329 JF - Scientific Reports TI - New approaches to epidemic modeling on networks VL - 13 ER - TY - JOUR AB - 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. AU - Bighin, Giacomo AU - Ho, Quoc P AU - Lemeshko, Mikhail AU - Tscherbul, T. V. ID - 13966 IS - 4 JF - Physical Review B SN - 2469-9950 TI - Diagrammatic Monte Carlo for electronic correlation in molecules: High-order many-body perturbation theory with low scaling VL - 108 ER - TY - JOUR AB - 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 . AU - Mauri, Mirko AU - Shinder, Evgeny ID - 14239 JF - Forum of Mathematics, Sigma TI - Homological Bondal-Orlov localization conjecture for rational singularities VL - 11 ER - TY - JOUR AB - 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. AU - Huybrechts, D. AU - Mauri, Mirko ID - 13268 IS - 1 JF - Mathematical Research Letters SN - 1073-2780 TI - On type II degenerations of hyperkähler manifolds VL - 30 ER - TY - JOUR AB - 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. AU - Hausel, Tamás AU - Wong, Michael Lennox AU - Wyss, Dimitri ID - 14244 IS - 4 JF - Proceedings of the London Mathematical Society SN - 0024-6115 TI - Arithmetic and metric aspects of open de Rham spaces VL - 127 ER - TY - CHAP AB - 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). AU - Mirković, Ivan AU - Yang, Yaping AU - Zhao, Gufang ED - Baranovskky, Vladimir ED - Guay, Nicolas ED - Schedler, Travis ID - 12303 SN - 2297-0215 T2 - Representation Theory and Algebraic Geometry TI - Loop Grassmannians of Quivers and Affine Quantum Groups ER - TY - JOUR AB - 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. AU - Mistegaard, William AU - Andersen, Jørgen Ellegaard ID - 9977 IS - 2 JF - Journal of the London Mathematical Society TI - Resurgence analysis of quantum invariants of Seifert fibered homology spheres VL - 105 ER - TY - JOUR AB - 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. AU - Hausel, Tamás AU - Hitchin, Nigel ID - 10704 JF - Inventiones Mathematicae SN - 0020-9910 TI - Very stable Higgs bundles, equivariant multiplicity and mirror symmetry VL - 228 ER - TY - JOUR AB - 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. AU - Arguez, Nuroemuer Huelya ID - 10772 IS - 1 JF - Journal of the London Mathematical Society SN - 0024-6107 TI - Mirror symmetry for the Tate curve via tropical and log corals VL - 105 ER - TY - JOUR AB - 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. AU - Yu, Hongjie ID - 12793 IS - 1 JF - Pacific Journal of Mathematics KW - Arthur–Selberg trace formula KW - cuspidal automorphic representations KW - global function fields SN - 0030-8730 TI - A coarse geometric expansion of a variant of Arthur's truncated traces and some applications VL - 321 ER - TY - JOUR AB - 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. AU - Rychlewicz, Kamil P ID - 6965 IS - 2 JF - Bulletin of the London Mathematical Society SN - 0024-6093 TI - The positivity of local equivariant Hirzebruch class for toric varieties VL - 53 ER - TY - JOUR AB - 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. AU - Srivastava, Tanya K ID - 9099 IS - 5 JF - Archiv der Mathematik SN - 0003889X TI - Lifting automorphisms on Abelian varieties as derived autoequivalences VL - 116 ER - TY - JOUR AB - 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. AU - Srivastava, Tanya K ID - 9173 IS - 03 JF - Bulletin des Sciences Mathematiques SN - 0007-4497 TI - Pathologies of the Hilbert scheme of points of a supersingular Enriques surface VL - 167 ER - TY - JOUR AB - 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. AU - Ho, Quoc P ID - 9359 IS - 2 JF - Geometry & Topology KW - Generalized configuration spaces KW - homological stability KW - homological densities KW - chiral algebras KW - chiral homology KW - factorization algebras KW - Koszul duality KW - Ran space SN - 1364-0380 TI - Homological stability and densities of generalized configuration spaces VL - 25 ER - TY - JOUR AB - 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. AU - Koroteev, Peter AU - Pushkar, Petr AU - Smirnov, Andrey V. AU - Zeitlin, Anton M. ID - 9998 IS - 5 JF - Selecta Mathematica SN - 1022-1824 TI - Quantum K-theory of quiver varieties and many-body systems VL - 27 ER - TY - JOUR AB - 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]. AU - Ho, Quoc P ID - 10033 JF - Advances in Mathematics KW - Chiral algebras KW - Chiral homology KW - Factorization algebras KW - Koszul duality KW - Ran space SN - 0001-8708 TI - The Atiyah-Bott formula and connectivity in chiral Koszul duality VL - 392 ER - TY - JOUR AB - 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. AU - Rapcak, Miroslav AU - Soibelman, Yan AU - Yang, Yaping AU - Zhao, Gufang ID - 7004 JF - Communications in Mathematical Physics SN - 0010-3616 TI - Cohomological Hall algebras, vertex algebras and instantons VL - 376 ER - TY - JOUR AB - 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. AU - Minets, Sasha ID - 7683 IS - 2 JF - Selecta Mathematica, New Series SN - 10221824 TI - Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces VL - 26 ER - TY - JOUR AB - We prove that the Yangian associated to an untwisted symmetric affine Kac–Moody Lie algebra is isomorphic to the Drinfeld double of a shuffle algebra. The latter is constructed in [YZ14] as an algebraic formalism of cohomological Hall algebras. As a consequence, we obtain the Poincare–Birkhoff–Witt (PBW) theorem for this class of affine Yangians. Another independent proof of the PBW theorem is given recently by Guay, Regelskis, and Wendlandt [GRW18]. AU - Yang, Yaping AU - Zhao, Gufang ID - 7940 JF - Transformation Groups SN - 10834362 TI - The PBW theorem for affine Yangians VL - 25 ER - TY - JOUR AB - Let 𝐹:ℤ2→ℤ be the pointwise minimum of several linear functions. The theory of smoothing allows us to prove that under certain conditions there exists the pointwise minimal function among all integer-valued superharmonic functions coinciding with F “at infinity”. We develop such a theory to prove existence of so-called solitons (or strings) in a sandpile model, studied by S. Caracciolo, G. Paoletti, and A. Sportiello. Thus we made a step towards understanding the phenomena of the identity in the sandpile group for planar domains where solitons appear according to experiments. We prove that sandpile states, defined using our smoothing procedure, move changeless when we apply the wave operator (that is why we call them solitons), and can interact, forming triads and nodes. AU - Kalinin, Nikita AU - Shkolnikov, Mikhail ID - 8325 IS - 9 JF - Communications in Mathematical Physics SN - 00103616 TI - Sandpile solitons via smoothing of superharmonic functions VL - 378 ER - TY - JOUR AB - Cohomological and K-theoretic stable bases originated from the study of quantum cohomology and quantum K-theory. Restriction formula for cohomological stable bases played an important role in computing the quantum connection of cotangent bundle of partial flag varieties. In this paper we study the K-theoretic stable bases of cotangent bundles of flag varieties. We describe these bases in terms of the action of the affine Hecke algebra and the twisted group algebra of KostantKumar. Using this algebraic description and the method of root polynomials, we give a restriction formula of the stable bases. We apply it to obtain the restriction formula for partial flag varieties. We also build a relation between the stable basis and the Casselman basis in the principal series representations of the Langlands dual group. As an application, we give a closed formula for the transition matrix between Casselman basis and the characteristic functions. AU - Su, C. AU - Zhao, Gufang AU - Zhong, C. ID - 8539 IS - 3 JF - Annales Scientifiques de l'Ecole Normale Superieure SN - 0012-9593 TI - On the K-theory stable bases of the springer resolution VL - 53 ER - TY - JOUR AB - This workshop focused on interactions between the various perspectives on the moduli space of Higgs bundles over a Riemann surface. This subject draws on algebraic geometry, geometric topology, geometric analysis and mathematical physics, and the goal was to promote interactions between these various branches of the subject. The main current directions of research were well represented by the participants, and the talks included many from both senior and junior participants. AU - Anderson, Lara AU - Hausel, Tamás AU - Mazzeo, Rafe AU - Schaposnik, Laura ID - 15070 IS - 2 JF - Oberwolfach Reports KW - Organic Chemistry KW - Biochemistry SN - 1660-8933 TI - Geometry and physics of Higgs bundles VL - 16 ER - TY - JOUR AU - Kalinin, Nikita AU - Shkolnikov, Mikhail ID - 441 IS - 3 JF - European Journal of Mathematics SN - 2199-675X TI - Tropical formulae for summation over a part of SL(2,Z) VL - 5 ER - TY - JOUR AB - We count points over a finite field on wild character varieties,of Riemann surfaces for singularities with regular semisimple leading term. The new feature in our counting formulas is the appearance of characters of Yokonuma–Hecke algebras. Our result leads to the conjecture that the mixed Hodge polynomials of these character varieties agree with previously conjectured perverse Hodge polynomials of certain twisted parabolic Higgs moduli spaces, indicating the possibility of a P = W conjecture for a suitable wild Hitchin system. AU - Hausel, Tamas AU - Mereb, Martin AU - Wong, Michael ID - 439 IS - 10 JF - Journal of the European Mathematical Society TI - Arithmetic and representation theory of wild character varieties VL - 21 ER - TY - JOUR AB - Li-Nadler proposed a conjecture about traces of Hecke categories, which implies the semistable part of the Betti geometric Langlands conjecture of Ben-Zvi-Nadler in genus 1. We prove a Weyl group analogue of this conjecture. Our theorem holds in the natural generality of reflection groups in Euclidean or hyperbolic space. As a corollary, we give an expression of the centralizer of a finite order element in a reflection group using homotopy theory. AU - Li, Penghui ID - 6986 IS - 11 JF - Proceedings of the American Mathematical Society SN - 0002-9939 TI - A colimit of traces of reflection groups VL - 147 ER - TY - JOUR AB - The abelian sandpile serves as a model to study self-organized criticality, a phenomenon occurring in biological, physical and social processes. The identity of the abelian group is a fractal composed of self-similar patches, and its limit is subject of extensive collaborative research. Here, we analyze the evolution of the sandpile identity under harmonic fields of different orders. We show that this evolution corresponds to periodic cycles through the abelian group characterized by the smooth transformation and apparent conservation of the patches constituting the identity. The dynamics induced by second and third order harmonics resemble smooth stretchings, respectively translations, of the identity, while the ones induced by fourth order harmonics resemble magnifications and rotations. Starting with order three, the dynamics pass through extended regions of seemingly random configurations which spontaneously reassemble into accentuated patterns. We show that the space of harmonic functions projects to the extended analogue of the sandpile group, thus providing a set of universal coordinates identifying configurations between different domains. Since the original sandpile group is a subgroup of the extended one, this directly implies that it admits a natural renormalization. Furthermore, we show that the harmonic fields can be induced by simple Markov processes, and that the corresponding stochastic dynamics show remarkable robustness over hundreds of periods. Finally, we encode information into seemingly random configurations, and decode this information with an algorithm requiring minimal prior knowledge. Our results suggest that harmonic fields might split the sandpile group into sub-sets showing different critical coefficients, and that it might be possible to extend the fractal structure of the identity beyond the boundaries of its domain. AU - Lang, Moritz AU - Shkolnikov, Mikhail ID - 196 IS - 8 JF - Proceedings of the National Academy of Sciences TI - Harmonic dynamics of the Abelian sandpile VL - 116 ER - TY - JOUR AB - In this paper, we introduce a quantum version of the wonderful compactification of a group as a certain noncommutative projective scheme. Our approach stems from the fact that the wonderful compactification encodes the asymptotics of matrix coefficients, and from its realization as a GIT quotient of the Vinberg semigroup. In order to define the wonderful compactification for a quantum group, we adopt a generalized formalism of Proj categories in the spirit of Artin and Zhang. Key to our construction is a quantum version of the Vinberg semigroup, which we define as a q-deformation of a certain Rees algebra, compatible with a standard Poisson structure. Furthermore, we discuss quantum analogues of the stratification of the wonderful compactification by orbits for a certain group action, and provide explicit computations in the case of SL2. AU - Ganev, Iordan V ID - 5 IS - 3 JF - Journal of the London Mathematical Society TI - The wonderful compactification for quantum groups VL - 99 ER - TY - JOUR AB - For an ordinary K3 surface over an algebraically closed field of positive characteristic we show that every automorphism lifts to characteristic zero. Moreover, we show that the Fourier-Mukai partners of an ordinary K3 surface are in one-to-one correspondence with the Fourier-Mukai partners of the geometric generic fiber of its canonical lift. We also prove that the explicit counting formula for Fourier-Mukai partners of the K3 surfaces with Picard rank two and with discriminant equal to minus of a prime number, in terms of the class number of the prime, holds over a field of positive characteristic as well. We show that the image of the derived autoequivalence group of a K3 surface of finite height in the group of isometries of its crystalline cohomology has index at least two. Moreover, we provide a conditional upper bound on the kernel of this natural cohomological descent map. Further, we give an extended remark in the appendix on the possibility of an F-crystal structure on the crystalline cohomology of a K3 surface over an algebraically closed field of positive characteristic and show that the naive F-crystal structure fails in being compatible with inner product. AU - Srivastava, Tanya K ID - 7436 JF - Documenta Mathematica SN - 1431-0635 TI - On derived equivalences of k3 surfaces in positive characteristic VL - 24 ER - TY - CHAP AB - We prove that there is no strongly regular graph (SRG) with parameters (460; 153; 32; 60). The proof is based on a recent lower bound on the number of 4-cliques in a SRG and some applications of Euclidean representation of SRGs. AU - Bondarenko, Andriy AU - Mellit, Anton AU - Prymak, Andriy AU - Radchenko, Danylo AU - Viazovska, Maryna ID - 61 T2 - Contemporary Computational Mathematics TI - There is no strongly regular graph with parameters (460; 153; 32; 60) ER - TY - CHAP AB - This chapter finds an agreement of equivariant indices of semi-classical homomorphisms between pairwise mirror branes in the GL2 Higgs moduli space on a Riemann surface. On one side of the agreement, components of the Lagrangian brane of U(1,1) Higgs bundles, whose mirror was proposed by Hitchin to be certain even exterior powers of the hyperholomorphic Dirac bundle on the SL2 Higgs moduli space, are present. The agreement arises from a mysterious functional equation. This gives strong computational evidence for Hitchin’s proposal. AU - Hausel, Tamás AU - Mellit, Anton AU - Pei, Du ID - 6525 SN - 9780198802013 T2 - Geometry and Physics: Volume I TI - Mirror symmetry with branes by equivariant verlinde formulas ER - TY - JOUR AB - The theory of tropical series, that we develop here, firstly appeared in the study of the growth of pluriharmonic functions. Motivated by waves in sandpile models we introduce a dynamic on the set of tropical series, and it is experimentally observed that this dynamic obeys a power law. So, this paper serves as a compilation of results we need for other articles and also introduces several objects interesting by themselves. AU - Kalinin, Nikita AU - Shkolnikov, Mikhail ID - 303 IS - 6 JF - Discrete and Continuous Dynamical Systems- Series A TI - Introduction to tropical series and wave dynamic on them VL - 38 ER - TY - JOUR AB - We construct quantizations of multiplicative hypertoric varieties using an algebra of q-difference operators on affine space, where q is a root of unity in C. The quantization defines a matrix bundle (i.e. Azumaya algebra) over the multiplicative hypertoric variety and admits an explicit finite étale splitting. The global sections of this Azumaya algebra is a hypertoric quantum group, and we prove a localization theorem. We introduce a general framework of Frobenius quantum moment maps and their Hamiltonian reductions; our results shed light on an instance of this framework. AU - Ganev, Iordan V ID - 322 JF - Journal of Algebra TI - Quantizations of multiplicative hypertoric varieties at a root of unity VL - 506 ER - TY - JOUR AB - Tropical geometry, an established field in pure mathematics, is a place where string theory, mirror symmetry, computational algebra, auction theory, and so forth meet and influence one another. In this paper, we report on our discovery of a tropical model with self-organized criticality (SOC) behavior. Our model is continuous, in contrast to all known models of SOC, and is a certain scaling limit of the sandpile model, the first and archetypical model of SOC. We describe how our model is related to pattern formation and proportional growth phenomena and discuss the dichotomy between continuous and discrete models in several contexts. Our aim in this context is to present an idealized tropical toy model (cf. Turing reaction-diffusion model), requiring further investigation. AU - Kalinin, Nikita AU - Guzmán Sáenz, Aldo AU - Prieto, Y AU - Shkolnikov, Mikhail AU - Kalinina, V AU - Lupercio, Ernesto ID - 64 IS - 35 JF - PNAS: Proceedings of the National Academy of Sciences of the United States of America SN - 00278424 TI - Self-organized criticality and pattern emergence through the lens of tropical geometry VL - 115 ER - TY - JOUR AB - We introduce for each quiver Q and each algebraic oriented cohomology theory A, the cohomological Hall algebra (CoHA) of Q, as the A-homology of the moduli of representations of the preprojective algebra of Q. This generalizes the K-theoretic Hall algebra of commuting varieties defined by Schiffmann-Vasserot. When A is the Morava K-theory, we show evidence that this algebra is a candidate for Lusztig's reformulated conjecture on modular representations of algebraic groups. We construct an action of the preprojective CoHA on the A-homology of Nakajima quiver varieties. We compare this with the action of the Borel subalgebra of Yangian when A is the intersection theory. We also give a shuffle algebra description of this CoHA in terms of the underlying formal group law of A. As applications, we obtain a shuffle description of the Yangian. AU - Yang, Yaping AU - Zhao, Gufang ID - 5999 IS - 5 JF - Proceedings of the London Mathematical Society SN - 0024-6115 TI - The cohomological Hall algebra of a preprojective algebra VL - 116 ER - TY - JOUR AB - Pursuing the similarity between the Kontsevich-Soibelman construction of the cohomological Hall algebra (CoHA) of BPS states and Lusztig's construction of canonical bases for quantum enveloping algebras, and the similarity between the integrality conjecture for motivic Donaldson-Thomas invariants and the PBW theorem for quantum enveloping algebras, we build a coproduct on the CoHA associated to a quiver with potential. We also prove a cohomological dimensional reduction theorem, further linking a special class of CoHAs with Yangians, and explaining how to connect the study of character varieties with the study of CoHAs. AU - Davison, Ben ID - 687 IS - 2 JF - Quarterly Journal of Mathematics SN - 00335606 TI - The critical CoHA of a quiver with potential VL - 68 ER -