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 - 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 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 - 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 - 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 - 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 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
JF - Journal of the London Mathematical Society
TI - Resurgence analysis of quantum invariants of Seifert fibered homology spheres
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 - 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 - 14310635
TI - On derived equivalences of k3 surfaces in positive characteristic
VL - 24
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 - 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 - 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
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 - 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 - 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 - 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 -